TPTP Problem File: ITP178^1.p
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%------------------------------------------------------------------------------
% File : ITP178^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_429__5391446_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : StandardRules/prob_429__5391446_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 541 ( 143 unt; 185 typ; 0 def)
% Number of atoms : 993 ( 367 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 3581 ( 95 ~; 1 |; 131 &;2941 @)
% ( 0 <=>; 413 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 272 ( 272 >; 0 *; 0 +; 0 <<)
% Number of symbols : 145 ( 144 usr; 12 con; 0-6 aty)
% Number of variables : 1063 ( 62 ^; 951 !; 50 ?;1063 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:28.807
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
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thf(sy_c_RulesAndChains_Oset__of__graph__rules_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Nat__Onat,type,
set_of1384085797_a_nat: set_Pr1987088711_a_nat > $o ).
thf(sy_c_RulesAndChains_Oset__of__graph__rules_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
set_of195930477at_nat: set_Pr665622551at_nat > $o ).
thf(sy_c_RulesAndChains_Oset__of__graph__rules_001tf__b_001t__Nat__Onat,type,
set_of41538795_b_nat: set_Pr551076371_b_nat > $o ).
thf(sy_c_RulesAndChains_Oweak__universal_001t__Nat__Onat_001tf__b_001t__Nat__Onat_001t__Nat__Onat,type,
weak_u2026406106at_nat: itself_nat > produc1235635379_b_nat > labeled_graph_b_nat > labeled_graph_b_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_Mt__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_J,type,
collec135640594term_b: ( produc1478835367term_b > $o ) > set_Pr1163220871term_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
collec357096914_a_nat: ( produc1871334759_a_nat > $o ) > set_Pr1987088711_a_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_J,type,
collec1615000990_b_nat: ( produc1235635379_b_nat > $o ) > set_Pr551076371_b_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
collec1701899602_a_nat: ( produc398057191_a_nat > $o ) > set_Pr924198087_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oconstant__rules_001tf__a,type,
standa1897115818ules_a: set_a > set_Pr1987088711_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oidentity__rules_001tf__a,type,
standa1568205540ules_a: set_St761939237tant_a > set_Pr1987088711_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ononempty__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa1410829644tant_a: produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ononempty__rule_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
standa214789732at_nat: produc1391440311at_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ononempty__rule_001tf__b,type,
standa879863266rule_b: produc1235635379_b_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oreflexivity__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa63370785tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oreflexivity__rule_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
standa2131591247at_nat: product_prod_nat_nat > produc1391440311at_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oreflexivity__rule_001tf__b,type,
standa1329480013rule_b: b > produc1235635379_b_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Otop__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa305748545tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Otop__rule_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
standa153097263at_nat: product_prod_nat_nat > produc1391440311at_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Otop__rule_001tf__b,type,
standa1360217389rule_b: b > produc1235635379_b_nat ).
thf(sy_c_member_001t__LabeledGraphSemantics__Oallegorical____term_Itf__b_J,type,
member93680451term_b: allegorical_term_b > set_al1193902458term_b > $o ).
thf(sy_c_member_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
member964390942_a_nat: labele935650037_a_nat > set_la1083530965_a_nat > $o ).
thf(sy_c_member_001t__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J,type,
member1483953152_b_nat: labeled_graph_b_nat > set_la1976028319_b_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_Mt__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_J,type,
member516522448term_b: produc1478835367term_b > set_Pr1163220871term_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
member832397200_a_nat: produc1871334759_a_nat > set_Pr1987088711_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member1129678944at_nat: produc1391440311at_nat > set_Pr665622551at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_J,type,
member963855452_b_nat: produc1235635379_b_nat > set_Pr551076371_b_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_Mt__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_Mt__LabeledGraphSemantics__Oallegorical____term_Itf__b_J_J_J,type,
member1449757456term_b: produc1116408039term_b > set_Pr1839611079term_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
member584645392_a_nat: produc398057191_a_nat > set_Pr924198087_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__b_Mt__Nat__Onat_J_J_J,type,
member889223696_b_nat: produc446386919_b_nat > set_Pr1173424071_b_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2027625872at_nat: produc842455143at_nat > set_Pr1490359111at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J,type,
member829905680_a_nat: produc116665063_a_nat > set_Pr2123625671_a_nat > $o ).
thf(sy_v_C,type,
c: set_a ).
thf(sy_v_u,type,
u: allegorical_term_b ).
thf(sy_v_v,type,
v: allegorical_term_b ).
thf(sy_v_x,type,
x: produc1871334759_a_nat ).
% Relevant facts (355)
thf(fact_0_assms,axiom,
member832397200_a_nat @ x @ ( standa1897115818ules_a @ c ) ).
% assms
thf(fact_1_graph__rule__translation,axiom,
! [X: allego1565409692at_nat,Y: allego1565409692at_nat] :
( ( graph_2111906684at_nat @ ( produc1995789403at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) ) @ ( produc1564126365at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) ) @ ( id_on_nat @ ( labele560327297at_nat @ ( produc1995789403at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) ) ) ) )
& ( ( produc1564126365at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) )
= ( restri321299017at_nat @ ( produc1564126365at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) ) ) )
& ( finite_finite_nat @ ( labele560327297at_nat @ ( produc1564126365at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) ) ) )
& ( finite48957584at_nat @ ( labele2032268018at_nat @ ( produc1564126365at_nat @ ( produc590202991at_nat @ ( transl490985778at_nat @ X ) @ ( transl490985778at_nat @ ( allego266765201at_nat @ X @ Y ) ) ) ) ) ) ) ).
% graph_rule_translation
thf(fact_2_graph__rule__translation,axiom,
! [X: allego510293162tant_a,Y: allego510293162tant_a] :
( ( graph_2130075512at_nat @ ( produc719117507_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) ) @ ( produc880161797_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) ) ) ) )
& ( ( produc880161797_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ ( produc1676969687_a_nat @ ( transl1275713022tant_a @ X ) @ ( transl1275713022tant_a @ ( allego745587551tant_a @ X @ Y ) ) ) ) ) ) ) ).
% graph_rule_translation
thf(fact_3_graph__rule__translation,axiom,
! [X: allegorical_term_b,Y: allegorical_term_b] :
( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) ) @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) ) ) ) )
& ( ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) )
= ( restrict_b_nat @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ X ) @ ( translation_b @ ( allegorical_A_Int_b @ X @ Y ) ) ) ) ) ) ) ).
% graph_rule_translation
thf(fact_4_subgraph__refl,axiom,
! [G: labele935650037_a_nat] :
( ( graph_2130075512at_nat @ G @ G @ ( id_on_nat @ ( labele1810595089_a_nat @ G ) ) )
= ( G
= ( restri572569417_a_nat @ G ) ) ) ).
% subgraph_refl
thf(fact_5_subgraph__refl,axiom,
! [G: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ G @ G @ ( id_on_nat @ ( labele460410879_b_nat @ G ) ) )
= ( G
= ( restrict_b_nat @ G ) ) ) ).
% subgraph_refl
thf(fact_6_subgraph__restrict,axiom,
! [G: labele935650037_a_nat] :
( ( graph_2130075512at_nat @ G @ ( restri572569417_a_nat @ G ) @ ( id_on_nat @ ( labele1810595089_a_nat @ G ) ) )
= ( G
= ( restri572569417_a_nat @ G ) ) ) ).
% subgraph_restrict
thf(fact_7_subgraph__restrict,axiom,
! [G: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ G @ ( restrict_b_nat @ G ) @ ( id_on_nat @ ( labele460410879_b_nat @ G ) ) )
= ( G
= ( restrict_b_nat @ G ) ) ) ).
% subgraph_restrict
thf(fact_8_graph__homomorphism__Id,axiom,
! [A: labele935650037_a_nat] : ( graph_2130075512at_nat @ ( restri572569417_a_nat @ A ) @ ( restri572569417_a_nat @ A ) @ ( id_on_nat @ ( labele1810595089_a_nat @ A ) ) ) ).
% graph_homomorphism_Id
thf(fact_9_graph__homomorphism__Id,axiom,
! [A: labeled_graph_b_nat] : ( graph_529870330at_nat @ ( restrict_b_nat @ A ) @ ( restrict_b_nat @ A ) @ ( id_on_nat @ ( labele460410879_b_nat @ A ) ) ) ).
% graph_homomorphism_Id
thf(fact_10_verts__in__translation__finite_I2_J,axiom,
! [X: allego1565409692at_nat] : ( finite48957584at_nat @ ( labele2032268018at_nat @ ( transl490985778at_nat @ X ) ) ) ).
% verts_in_translation_finite(2)
thf(fact_11_verts__in__translation__finite_I2_J,axiom,
! [X: allego510293162tant_a] : ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( transl1275713022tant_a @ X ) ) ) ).
% verts_in_translation_finite(2)
thf(fact_12_verts__in__translation__finite_I2_J,axiom,
! [X: allegorical_term_b] : ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( translation_b @ X ) ) ) ).
% verts_in_translation_finite(2)
thf(fact_13_are__rules_I1_J,axiom,
( ( graph_2111906684at_nat @ ( produc1995789403at_nat @ standa214789732at_nat ) @ ( produc1564126365at_nat @ standa214789732at_nat ) @ ( id_on_nat @ ( labele560327297at_nat @ ( produc1995789403at_nat @ standa214789732at_nat ) ) ) )
& ( ( produc1564126365at_nat @ standa214789732at_nat )
= ( restri321299017at_nat @ ( produc1564126365at_nat @ standa214789732at_nat ) ) )
& ( finite_finite_nat @ ( labele560327297at_nat @ ( produc1564126365at_nat @ standa214789732at_nat ) ) )
& ( finite48957584at_nat @ ( labele2032268018at_nat @ ( produc1564126365at_nat @ standa214789732at_nat ) ) ) ) ).
% are_rules(1)
thf(fact_14_are__rules_I1_J,axiom,
( ( graph_2130075512at_nat @ ( produc719117507_a_nat @ standa1410829644tant_a ) @ ( produc880161797_a_nat @ standa1410829644tant_a ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ standa1410829644tant_a ) ) ) )
& ( ( produc880161797_a_nat @ standa1410829644tant_a )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ standa1410829644tant_a ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ standa1410829644tant_a ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ standa1410829644tant_a ) ) ) ) ).
% are_rules(1)
thf(fact_15_are__rules_I1_J,axiom,
( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ standa879863266rule_b ) @ ( produc194497945_b_nat @ standa879863266rule_b ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ standa879863266rule_b ) ) ) )
& ( ( produc194497945_b_nat @ standa879863266rule_b )
= ( restrict_b_nat @ ( produc194497945_b_nat @ standa879863266rule_b ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ standa879863266rule_b ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ standa879863266rule_b ) ) ) ) ).
% are_rules(1)
thf(fact_16_are__rules_I2_J,axiom,
! [T: product_prod_nat_nat] :
( ( graph_2111906684at_nat @ ( produc1995789403at_nat @ ( standa153097263at_nat @ T ) ) @ ( produc1564126365at_nat @ ( standa153097263at_nat @ T ) ) @ ( id_on_nat @ ( labele560327297at_nat @ ( produc1995789403at_nat @ ( standa153097263at_nat @ T ) ) ) ) )
& ( ( produc1564126365at_nat @ ( standa153097263at_nat @ T ) )
= ( restri321299017at_nat @ ( produc1564126365at_nat @ ( standa153097263at_nat @ T ) ) ) )
& ( finite_finite_nat @ ( labele560327297at_nat @ ( produc1564126365at_nat @ ( standa153097263at_nat @ T ) ) ) )
& ( finite48957584at_nat @ ( labele2032268018at_nat @ ( produc1564126365at_nat @ ( standa153097263at_nat @ T ) ) ) ) ) ).
% are_rules(2)
thf(fact_17_are__rules_I2_J,axiom,
! [T: standard_Constant_a] :
( ( graph_2130075512at_nat @ ( produc719117507_a_nat @ ( standa305748545tant_a @ T ) ) @ ( produc880161797_a_nat @ ( standa305748545tant_a @ T ) ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ ( standa305748545tant_a @ T ) ) ) ) )
& ( ( produc880161797_a_nat @ ( standa305748545tant_a @ T ) )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ ( standa305748545tant_a @ T ) ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ ( standa305748545tant_a @ T ) ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ ( standa305748545tant_a @ T ) ) ) ) ) ).
% are_rules(2)
thf(fact_18_are__rules_I2_J,axiom,
! [T: b] :
( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ ( standa1360217389rule_b @ T ) ) @ ( produc194497945_b_nat @ ( standa1360217389rule_b @ T ) ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ ( standa1360217389rule_b @ T ) ) ) ) )
& ( ( produc194497945_b_nat @ ( standa1360217389rule_b @ T ) )
= ( restrict_b_nat @ ( produc194497945_b_nat @ ( standa1360217389rule_b @ T ) ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ ( standa1360217389rule_b @ T ) ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ ( standa1360217389rule_b @ T ) ) ) ) ) ).
% are_rules(2)
thf(fact_19_verts__in__translation__finite_I1_J,axiom,
! [X: allego510293162tant_a] : ( finite_finite_nat @ ( labele1810595089_a_nat @ ( transl1275713022tant_a @ X ) ) ) ).
% verts_in_translation_finite(1)
thf(fact_20_verts__in__translation__finite_I1_J,axiom,
! [X: allegorical_term_b] : ( finite_finite_nat @ ( labele460410879_b_nat @ ( translation_b @ X ) ) ) ).
% verts_in_translation_finite(1)
thf(fact_21_are__rules_I3_J,axiom,
! [I: product_prod_nat_nat] :
( ( graph_2111906684at_nat @ ( produc1995789403at_nat @ ( standa2131591247at_nat @ I ) ) @ ( produc1564126365at_nat @ ( standa2131591247at_nat @ I ) ) @ ( id_on_nat @ ( labele560327297at_nat @ ( produc1995789403at_nat @ ( standa2131591247at_nat @ I ) ) ) ) )
& ( ( produc1564126365at_nat @ ( standa2131591247at_nat @ I ) )
= ( restri321299017at_nat @ ( produc1564126365at_nat @ ( standa2131591247at_nat @ I ) ) ) )
& ( finite_finite_nat @ ( labele560327297at_nat @ ( produc1564126365at_nat @ ( standa2131591247at_nat @ I ) ) ) )
& ( finite48957584at_nat @ ( labele2032268018at_nat @ ( produc1564126365at_nat @ ( standa2131591247at_nat @ I ) ) ) ) ) ).
% are_rules(3)
thf(fact_22_are__rules_I3_J,axiom,
! [I: standard_Constant_a] :
( ( graph_2130075512at_nat @ ( produc719117507_a_nat @ ( standa63370785tant_a @ I ) ) @ ( produc880161797_a_nat @ ( standa63370785tant_a @ I ) ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ ( standa63370785tant_a @ I ) ) ) ) )
& ( ( produc880161797_a_nat @ ( standa63370785tant_a @ I ) )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ ( standa63370785tant_a @ I ) ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ ( standa63370785tant_a @ I ) ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ ( standa63370785tant_a @ I ) ) ) ) ) ).
% are_rules(3)
thf(fact_23_are__rules_I3_J,axiom,
! [I: b] :
( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ ( standa1329480013rule_b @ I ) ) @ ( produc194497945_b_nat @ ( standa1329480013rule_b @ I ) ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ ( standa1329480013rule_b @ I ) ) ) ) )
& ( ( produc194497945_b_nat @ ( standa1329480013rule_b @ I ) )
= ( restrict_b_nat @ ( produc194497945_b_nat @ ( standa1329480013rule_b @ I ) ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ ( standa1329480013rule_b @ I ) ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ ( standa1329480013rule_b @ I ) ) ) ) ) ).
% are_rules(3)
thf(fact_24_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_25_prod_Ocollapse,axiom,
! [Prod: produc398057191_a_nat] :
( ( produc1677124439_a_nat @ ( produc1049080131_a_nat @ Prod ) @ ( produc1022852229_a_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_26_prod_Ocollapse,axiom,
! [Prod: produc1871334759_a_nat] :
( ( produc1676969687_a_nat @ ( produc719117507_a_nat @ Prod ) @ ( produc880161797_a_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_27_prod_Ocollapse,axiom,
! [Prod: produc1478835367term_b] :
( ( produc1990145943term_b @ ( produc854192515term_b @ Prod ) @ ( produc1223098053term_b @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_28_prod_Ocollapse,axiom,
! [Prod: produc1235635379_b_nat] :
( ( produc951298923_b_nat @ ( produc1542243159_b_nat @ Prod ) @ ( produc194497945_b_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_29_translation__graph,axiom,
( transl1275713022tant_a
= ( ^ [X2: allego510293162tant_a] : ( restri572569417_a_nat @ ( transl1275713022tant_a @ X2 ) ) ) ) ).
% translation_graph
thf(fact_30_translation__graph,axiom,
( translation_b
= ( ^ [X2: allegorical_term_b] : ( restrict_b_nat @ ( translation_b @ X2 ) ) ) ) ).
% translation_graph
thf(fact_31_vertices__restrict,axiom,
! [G: labele935650037_a_nat] :
( ( labele1810595089_a_nat @ ( restri572569417_a_nat @ G ) )
= ( labele1810595089_a_nat @ G ) ) ).
% vertices_restrict
thf(fact_32_vertices__restrict,axiom,
! [G: labeled_graph_b_nat] :
( ( labele460410879_b_nat @ ( restrict_b_nat @ G ) )
= ( labele460410879_b_nat @ G ) ) ).
% vertices_restrict
thf(fact_33_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
! [A: produc1871334759_a_nat,B: produc1871334759_a_nat,A2: produc1871334759_a_nat,B2: produc1871334759_a_nat] :
( ( ( produc1677124439_a_nat @ A @ B )
= ( produc1677124439_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_35_old_Oprod_Oinject,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,A2: labele935650037_a_nat,B2: labele935650037_a_nat] :
( ( ( produc1676969687_a_nat @ A @ B )
= ( produc1676969687_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_36_old_Oprod_Oinject,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,A2: labeled_graph_b_nat,B2: labeled_graph_b_nat] :
( ( ( produc951298923_b_nat @ A @ B )
= ( produc951298923_b_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_37_old_Oprod_Oinject,axiom,
! [A: allegorical_term_b,B: allegorical_term_b,A2: allegorical_term_b,B2: allegorical_term_b] :
( ( ( produc1990145943term_b @ A @ B )
= ( produc1990145943term_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_38_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_39_prod_Oinject,axiom,
! [X1: produc1871334759_a_nat,X22: produc1871334759_a_nat,Y1: produc1871334759_a_nat,Y2: produc1871334759_a_nat] :
( ( ( produc1677124439_a_nat @ X1 @ X22 )
= ( produc1677124439_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_40_prod_Oinject,axiom,
! [X1: labele935650037_a_nat,X22: labele935650037_a_nat,Y1: labele935650037_a_nat,Y2: labele935650037_a_nat] :
( ( ( produc1676969687_a_nat @ X1 @ X22 )
= ( produc1676969687_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_41_prod_Oinject,axiom,
! [X1: labeled_graph_b_nat,X22: labeled_graph_b_nat,Y1: labeled_graph_b_nat,Y2: labeled_graph_b_nat] :
( ( ( produc951298923_b_nat @ X1 @ X22 )
= ( produc951298923_b_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_42_prod_Oinject,axiom,
! [X1: allegorical_term_b,X22: allegorical_term_b,Y1: allegorical_term_b,Y2: allegorical_term_b] :
( ( ( produc1990145943term_b @ X1 @ X22 )
= ( produc1990145943term_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_43_restrict__idemp,axiom,
! [X3: labele935650037_a_nat] :
( ( restri572569417_a_nat @ ( restri572569417_a_nat @ X3 ) )
= ( restri572569417_a_nat @ X3 ) ) ).
% restrict_idemp
thf(fact_44_restrict__idemp,axiom,
! [X3: labeled_graph_b_nat] :
( ( restrict_b_nat @ ( restrict_b_nat @ X3 ) )
= ( restrict_b_nat @ X3 ) ) ).
% restrict_idemp
thf(fact_45_old_Oprod_Oinducts,axiom,
! [P: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_46_old_Oprod_Oinducts,axiom,
! [P: produc398057191_a_nat > $o,Prod: produc398057191_a_nat] :
( ! [A3: produc1871334759_a_nat,B3: produc1871334759_a_nat] : ( P @ ( produc1677124439_a_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_47_old_Oprod_Oinducts,axiom,
! [P: produc1871334759_a_nat > $o,Prod: produc1871334759_a_nat] :
( ! [A3: labele935650037_a_nat,B3: labele935650037_a_nat] : ( P @ ( produc1676969687_a_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_48_old_Oprod_Oinducts,axiom,
! [P: produc1235635379_b_nat > $o,Prod: produc1235635379_b_nat] :
( ! [A3: labeled_graph_b_nat,B3: labeled_graph_b_nat] : ( P @ ( produc951298923_b_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_49_old_Oprod_Oinducts,axiom,
! [P: produc1478835367term_b > $o,Prod: produc1478835367term_b] :
( ! [A3: allegorical_term_b,B3: allegorical_term_b] : ( P @ ( produc1990145943term_b @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_50_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y3
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust,axiom,
! [Y3: produc398057191_a_nat] :
~ ! [A3: produc1871334759_a_nat,B3: produc1871334759_a_nat] :
( Y3
!= ( produc1677124439_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [Y3: produc1871334759_a_nat] :
~ ! [A3: labele935650037_a_nat,B3: labele935650037_a_nat] :
( Y3
!= ( produc1676969687_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
! [Y3: produc1235635379_b_nat] :
~ ! [A3: labeled_graph_b_nat,B3: labeled_graph_b_nat] :
( Y3
!= ( produc951298923_b_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_54_old_Oprod_Oexhaust,axiom,
! [Y3: produc1478835367term_b] :
~ ! [A3: allegorical_term_b,B3: allegorical_term_b] :
( Y3
!= ( produc1990145943term_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_55_prod__induct3,axiom,
! [P: produc398057191_a_nat > $o,X3: produc398057191_a_nat] :
( ! [A3: produc1871334759_a_nat,B3: labele935650037_a_nat,C: labele935650037_a_nat] : ( P @ ( produc1677124439_a_nat @ A3 @ ( produc1676969687_a_nat @ B3 @ C ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_56_prod__cases3,axiom,
! [Y3: produc398057191_a_nat] :
~ ! [A3: produc1871334759_a_nat,B3: labele935650037_a_nat,C: labele935650037_a_nat] :
( Y3
!= ( produc1677124439_a_nat @ A3 @ ( produc1676969687_a_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_57_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_58_Pair__inject,axiom,
! [A: produc1871334759_a_nat,B: produc1871334759_a_nat,A2: produc1871334759_a_nat,B2: produc1871334759_a_nat] :
( ( ( produc1677124439_a_nat @ A @ B )
= ( produc1677124439_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_59_Pair__inject,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,A2: labele935650037_a_nat,B2: labele935650037_a_nat] :
( ( ( produc1676969687_a_nat @ A @ B )
= ( produc1676969687_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_60_Pair__inject,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,A2: labeled_graph_b_nat,B2: labeled_graph_b_nat] :
( ( ( produc951298923_b_nat @ A @ B )
= ( produc951298923_b_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_61_Pair__inject,axiom,
! [A: allegorical_term_b,B: allegorical_term_b,A2: allegorical_term_b,B2: allegorical_term_b] :
( ( ( produc1990145943term_b @ A @ B )
= ( produc1990145943term_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_62_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_63_prod__cases,axiom,
! [P: produc398057191_a_nat > $o,P2: produc398057191_a_nat] :
( ! [A3: produc1871334759_a_nat,B3: produc1871334759_a_nat] : ( P @ ( produc1677124439_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_64_prod__cases,axiom,
! [P: produc1871334759_a_nat > $o,P2: produc1871334759_a_nat] :
( ! [A3: labele935650037_a_nat,B3: labele935650037_a_nat] : ( P @ ( produc1676969687_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_65_prod__cases,axiom,
! [P: produc1235635379_b_nat > $o,P2: produc1235635379_b_nat] :
( ! [A3: labeled_graph_b_nat,B3: labeled_graph_b_nat] : ( P @ ( produc951298923_b_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_66_prod__cases,axiom,
! [P: produc1478835367term_b > $o,P2: produc1478835367term_b] :
( ! [A3: allegorical_term_b,B3: allegorical_term_b] : ( P @ ( produc1990145943term_b @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_67_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X4: nat,Y4: nat] :
( P2
= ( product_Pair_nat_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_68_surj__pair,axiom,
! [P2: produc398057191_a_nat] :
? [X4: produc1871334759_a_nat,Y4: produc1871334759_a_nat] :
( P2
= ( produc1677124439_a_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_69_surj__pair,axiom,
! [P2: produc1871334759_a_nat] :
? [X4: labele935650037_a_nat,Y4: labele935650037_a_nat] :
( P2
= ( produc1676969687_a_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_70_surj__pair,axiom,
! [P2: produc1235635379_b_nat] :
? [X4: labeled_graph_b_nat,Y4: labeled_graph_b_nat] :
( P2
= ( produc951298923_b_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_71_surj__pair,axiom,
! [P2: produc1478835367term_b] :
? [X4: allegorical_term_b,Y4: allegorical_term_b] :
( P2
= ( produc1990145943term_b @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_72_fst__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_73_fst__conv,axiom,
! [X1: produc1871334759_a_nat,X22: produc1871334759_a_nat] :
( ( produc1049080131_a_nat @ ( produc1677124439_a_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_74_fst__conv,axiom,
! [X1: labele935650037_a_nat,X22: labele935650037_a_nat] :
( ( produc719117507_a_nat @ ( produc1676969687_a_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_75_fst__conv,axiom,
! [X1: labeled_graph_b_nat,X22: labeled_graph_b_nat] :
( ( produc1542243159_b_nat @ ( produc951298923_b_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_76_fst__conv,axiom,
! [X1: allegorical_term_b,X22: allegorical_term_b] :
( ( produc854192515term_b @ ( produc1990145943term_b @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_77_fst__eqD,axiom,
! [X3: nat,Y3: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_78_fst__eqD,axiom,
! [X3: produc1871334759_a_nat,Y3: produc1871334759_a_nat,A: produc1871334759_a_nat] :
( ( ( produc1049080131_a_nat @ ( produc1677124439_a_nat @ X3 @ Y3 ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_79_fst__eqD,axiom,
! [X3: labele935650037_a_nat,Y3: labele935650037_a_nat,A: labele935650037_a_nat] :
( ( ( produc719117507_a_nat @ ( produc1676969687_a_nat @ X3 @ Y3 ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_80_fst__eqD,axiom,
! [X3: labeled_graph_b_nat,Y3: labeled_graph_b_nat,A: labeled_graph_b_nat] :
( ( ( produc1542243159_b_nat @ ( produc951298923_b_nat @ X3 @ Y3 ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_81_fst__eqD,axiom,
! [X3: allegorical_term_b,Y3: allegorical_term_b,A: allegorical_term_b] :
( ( ( produc854192515term_b @ ( produc1990145943term_b @ X3 @ Y3 ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_82_snd__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_83_snd__conv,axiom,
! [X1: produc1871334759_a_nat,X22: produc1871334759_a_nat] :
( ( produc1022852229_a_nat @ ( produc1677124439_a_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_84_snd__conv,axiom,
! [X1: labele935650037_a_nat,X22: labele935650037_a_nat] :
( ( produc880161797_a_nat @ ( produc1676969687_a_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_85_snd__conv,axiom,
! [X1: allegorical_term_b,X22: allegorical_term_b] :
( ( produc1223098053term_b @ ( produc1990145943term_b @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_86_snd__conv,axiom,
! [X1: labeled_graph_b_nat,X22: labeled_graph_b_nat] :
( ( produc194497945_b_nat @ ( produc951298923_b_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_87_snd__eqD,axiom,
! [X3: nat,Y3: nat,A: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_88_snd__eqD,axiom,
! [X3: produc1871334759_a_nat,Y3: produc1871334759_a_nat,A: produc1871334759_a_nat] :
( ( ( produc1022852229_a_nat @ ( produc1677124439_a_nat @ X3 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_89_snd__eqD,axiom,
! [X3: labele935650037_a_nat,Y3: labele935650037_a_nat,A: labele935650037_a_nat] :
( ( ( produc880161797_a_nat @ ( produc1676969687_a_nat @ X3 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_90_snd__eqD,axiom,
! [X3: allegorical_term_b,Y3: allegorical_term_b,A: allegorical_term_b] :
( ( ( produc1223098053term_b @ ( produc1990145943term_b @ X3 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_91_snd__eqD,axiom,
! [X3: labeled_graph_b_nat,Y3: labeled_graph_b_nat,A: labeled_graph_b_nat] :
( ( ( produc194497945_b_nat @ ( produc951298923_b_nat @ X3 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_92_prod__eq__iff,axiom,
( ( ^ [Y5: produc1871334759_a_nat,Z: produc1871334759_a_nat] : Y5 = Z )
= ( ^ [S: produc1871334759_a_nat,T2: produc1871334759_a_nat] :
( ( ( produc719117507_a_nat @ S )
= ( produc719117507_a_nat @ T2 ) )
& ( ( produc880161797_a_nat @ S )
= ( produc880161797_a_nat @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_93_prod__eq__iff,axiom,
( ( ^ [Y5: produc1478835367term_b,Z: produc1478835367term_b] : Y5 = Z )
= ( ^ [S: produc1478835367term_b,T2: produc1478835367term_b] :
( ( ( produc854192515term_b @ S )
= ( produc854192515term_b @ T2 ) )
& ( ( produc1223098053term_b @ S )
= ( produc1223098053term_b @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_94_prod__eq__iff,axiom,
( ( ^ [Y5: produc1235635379_b_nat,Z: produc1235635379_b_nat] : Y5 = Z )
= ( ^ [S: produc1235635379_b_nat,T2: produc1235635379_b_nat] :
( ( ( produc1542243159_b_nat @ S )
= ( produc1542243159_b_nat @ T2 ) )
& ( ( produc194497945_b_nat @ S )
= ( produc194497945_b_nat @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_95_prod_Oexpand,axiom,
! [Prod: produc1871334759_a_nat,Prod2: produc1871334759_a_nat] :
( ( ( ( produc719117507_a_nat @ Prod )
= ( produc719117507_a_nat @ Prod2 ) )
& ( ( produc880161797_a_nat @ Prod )
= ( produc880161797_a_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_96_prod_Oexpand,axiom,
! [Prod: produc1478835367term_b,Prod2: produc1478835367term_b] :
( ( ( ( produc854192515term_b @ Prod )
= ( produc854192515term_b @ Prod2 ) )
& ( ( produc1223098053term_b @ Prod )
= ( produc1223098053term_b @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_97_prod_Oexpand,axiom,
! [Prod: produc1235635379_b_nat,Prod2: produc1235635379_b_nat] :
( ( ( ( produc1542243159_b_nat @ Prod )
= ( produc1542243159_b_nat @ Prod2 ) )
& ( ( produc194497945_b_nat @ Prod )
= ( produc194497945_b_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_98_prod__eqI,axiom,
! [P2: produc1871334759_a_nat,Q: produc1871334759_a_nat] :
( ( ( produc719117507_a_nat @ P2 )
= ( produc719117507_a_nat @ Q ) )
=> ( ( ( produc880161797_a_nat @ P2 )
= ( produc880161797_a_nat @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_99_prod__eqI,axiom,
! [P2: produc1478835367term_b,Q: produc1478835367term_b] :
( ( ( produc854192515term_b @ P2 )
= ( produc854192515term_b @ Q ) )
=> ( ( ( produc1223098053term_b @ P2 )
= ( produc1223098053term_b @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_100_prod__eqI,axiom,
! [P2: produc1235635379_b_nat,Q: produc1235635379_b_nat] :
( ( ( produc1542243159_b_nat @ P2 )
= ( produc1542243159_b_nat @ Q ) )
=> ( ( ( produc194497945_b_nat @ P2 )
= ( produc194497945_b_nat @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_101_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labele935650037_a_nat,Labeled_graph2: labele935650037_a_nat] :
( ( ( ( labele195203296_a_nat @ Labeled_graph )
= ( labele195203296_a_nat @ Labeled_graph2 ) )
& ( ( labele1810595089_a_nat @ Labeled_graph )
= ( labele1810595089_a_nat @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_102_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labeled_graph_b_nat,Labeled_graph2: labeled_graph_b_nat] :
( ( ( ( labeled_edges_b_nat @ Labeled_graph )
= ( labeled_edges_b_nat @ Labeled_graph2 ) )
& ( ( labele460410879_b_nat @ Labeled_graph )
= ( labele460410879_b_nat @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_103_surjective__pairing,axiom,
! [T: product_prod_nat_nat] :
( T
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_104_surjective__pairing,axiom,
! [T: produc398057191_a_nat] :
( T
= ( produc1677124439_a_nat @ ( produc1049080131_a_nat @ T ) @ ( produc1022852229_a_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_105_surjective__pairing,axiom,
! [T: produc1871334759_a_nat] :
( T
= ( produc1676969687_a_nat @ ( produc719117507_a_nat @ T ) @ ( produc880161797_a_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_106_surjective__pairing,axiom,
! [T: produc1478835367term_b] :
( T
= ( produc1990145943term_b @ ( produc854192515term_b @ T ) @ ( produc1223098053term_b @ T ) ) ) ).
% surjective_pairing
thf(fact_107_surjective__pairing,axiom,
! [T: produc1235635379_b_nat] :
( T
= ( produc951298923_b_nat @ ( produc1542243159_b_nat @ T ) @ ( produc194497945_b_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_108_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_109_prod_Oexhaust__sel,axiom,
! [Prod: produc398057191_a_nat] :
( Prod
= ( produc1677124439_a_nat @ ( produc1049080131_a_nat @ Prod ) @ ( produc1022852229_a_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_110_prod_Oexhaust__sel,axiom,
! [Prod: produc1871334759_a_nat] :
( Prod
= ( produc1676969687_a_nat @ ( produc719117507_a_nat @ Prod ) @ ( produc880161797_a_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_111_prod_Oexhaust__sel,axiom,
! [Prod: produc1478835367term_b] :
( Prod
= ( produc1990145943term_b @ ( produc854192515term_b @ Prod ) @ ( produc1223098053term_b @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_112_prod_Oexhaust__sel,axiom,
! [Prod: produc1235635379_b_nat] :
( Prod
= ( produc951298923_b_nat @ ( produc1542243159_b_nat @ Prod ) @ ( produc194497945_b_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_113_subgraph__preserves__hom,axiom,
! [A4: labele935650037_a_nat,B4: labele935650037_a_nat,X: labele935650037_a_nat,H: set_Pr1986765409at_nat] :
( ( graph_2130075512at_nat @ A4 @ B4 @ ( id_on_nat @ ( labele1810595089_a_nat @ A4 ) ) )
=> ( ( graph_2130075512at_nat @ X @ A4 @ H )
=> ( graph_2130075512at_nat @ X @ B4 @ H ) ) ) ).
% subgraph_preserves_hom
thf(fact_114_subgraph__preserves__hom,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,X: labeled_graph_b_nat,H: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ A4 @ B4 @ ( id_on_nat @ ( labele460410879_b_nat @ A4 ) ) )
=> ( ( graph_529870330at_nat @ X @ A4 @ H )
=> ( graph_529870330at_nat @ X @ B4 @ H ) ) ) ).
% subgraph_preserves_hom
thf(fact_115_subgraph__trans,axiom,
! [G_1: labele935650037_a_nat,G_2: labele935650037_a_nat,G_3: labele935650037_a_nat] :
( ( graph_2130075512at_nat @ G_1 @ G_2 @ ( id_on_nat @ ( labele1810595089_a_nat @ G_1 ) ) )
=> ( ( graph_2130075512at_nat @ G_2 @ G_3 @ ( id_on_nat @ ( labele1810595089_a_nat @ G_2 ) ) )
=> ( graph_2130075512at_nat @ G_1 @ G_3 @ ( id_on_nat @ ( labele1810595089_a_nat @ G_1 ) ) ) ) ) ).
% subgraph_trans
thf(fact_116_subgraph__trans,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat,G_3: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ G_1 @ G_2 @ ( id_on_nat @ ( labele460410879_b_nat @ G_1 ) ) )
=> ( ( graph_529870330at_nat @ G_2 @ G_3 @ ( id_on_nat @ ( labele460410879_b_nat @ G_2 ) ) )
=> ( graph_529870330at_nat @ G_1 @ G_3 @ ( id_on_nat @ ( labele460410879_b_nat @ G_1 ) ) ) ) ) ).
% subgraph_trans
thf(fact_117_Id__onI,axiom,
! [A: produc1478835367term_b,A4: set_Pr1163220871term_b] :
( ( member516522448term_b @ A @ A4 )
=> ( member1449757456term_b @ ( produc859843415term_b @ A @ A ) @ ( id_on_1664915780term_b @ A4 ) ) ) ).
% Id_onI
thf(fact_118_Id__onI,axiom,
! [A: produc1235635379_b_nat,A4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ A @ A4 )
=> ( member889223696_b_nat @ ( produc1754969175_b_nat @ A @ A ) @ ( id_on_138931664_b_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_119_Id__onI,axiom,
! [A: produc398057191_a_nat,A4: set_Pr924198087_a_nat] :
( ( member584645392_a_nat @ A @ A4 )
=> ( member829905680_a_nat @ ( produc170611543_a_nat @ A @ A ) @ ( id_on_1395957380_a_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_120_Id__onI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ A @ A4 )
=> ( member2027625872at_nat @ ( produc1168807639at_nat @ A @ A ) @ ( id_on_2144791838at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_121_Id__onI,axiom,
! [A: labele935650037_a_nat,A4: set_la1083530965_a_nat] :
( ( member964390942_a_nat @ A @ A4 )
=> ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ A ) @ ( id_on_689842066_a_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_122_Id__onI,axiom,
! [A: produc1871334759_a_nat,A4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ A @ A4 )
=> ( member584645392_a_nat @ ( produc1677124439_a_nat @ A @ A ) @ ( id_on_1651096324_a_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_123_Id__onI,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_124_Id__onI,axiom,
! [A: labeled_graph_b_nat,A4: set_la1976028319_b_nat] :
( ( member1483953152_b_nat @ A @ A4 )
=> ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ A ) @ ( id_on_583275916_b_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_125_Id__onI,axiom,
! [A: allegorical_term_b,A4: set_al1193902458term_b] :
( ( member93680451term_b @ A @ A4 )
=> ( member516522448term_b @ ( produc1990145943term_b @ A @ A ) @ ( id_on_1536886967term_b @ A4 ) ) ) ).
% Id_onI
thf(fact_126_set__of__graph__rules__def,axiom,
( set_of195930477at_nat
= ( ^ [Rs: set_Pr665622551at_nat] :
! [X5: produc1391440311at_nat] :
( ( member1129678944at_nat @ X5 @ Rs )
=> ( ( graph_2111906684at_nat @ ( produc1995789403at_nat @ X5 ) @ ( produc1564126365at_nat @ X5 ) @ ( id_on_nat @ ( labele560327297at_nat @ ( produc1995789403at_nat @ X5 ) ) ) )
& ( ( produc1564126365at_nat @ X5 )
= ( restri321299017at_nat @ ( produc1564126365at_nat @ X5 ) ) )
& ( finite_finite_nat @ ( labele560327297at_nat @ ( produc1564126365at_nat @ X5 ) ) )
& ( finite48957584at_nat @ ( labele2032268018at_nat @ ( produc1564126365at_nat @ X5 ) ) ) ) ) ) ) ).
% set_of_graph_rules_def
thf(fact_127_set__of__graph__rules__def,axiom,
( set_of1384085797_a_nat
= ( ^ [Rs: set_Pr1987088711_a_nat] :
! [X5: produc1871334759_a_nat] :
( ( member832397200_a_nat @ X5 @ Rs )
=> ( ( graph_2130075512at_nat @ ( produc719117507_a_nat @ X5 ) @ ( produc880161797_a_nat @ X5 ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ X5 ) ) ) )
& ( ( produc880161797_a_nat @ X5 )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ X5 ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ X5 ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ X5 ) ) ) ) ) ) ) ).
% set_of_graph_rules_def
thf(fact_128_set__of__graph__rules__def,axiom,
( set_of41538795_b_nat
= ( ^ [Rs: set_Pr551076371_b_nat] :
! [X5: produc1235635379_b_nat] :
( ( member963855452_b_nat @ X5 @ Rs )
=> ( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ X5 ) @ ( produc194497945_b_nat @ X5 ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ X5 ) ) ) )
& ( ( produc194497945_b_nat @ X5 )
= ( restrict_b_nat @ ( produc194497945_b_nat @ X5 ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ X5 ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ X5 ) ) ) ) ) ) ) ).
% set_of_graph_rules_def
thf(fact_129_mem__Collect__eq,axiom,
! [A: produc1478835367term_b,P: produc1478835367term_b > $o] :
( ( member516522448term_b @ A @ ( collec135640594term_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
! [A: produc1235635379_b_nat,P: produc1235635379_b_nat > $o] :
( ( member963855452_b_nat @ A @ ( collec1615000990_b_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
! [A: produc398057191_a_nat,P: produc398057191_a_nat > $o] :
( ( member584645392_a_nat @ A @ ( collec1701899602_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A: produc1871334759_a_nat,P: produc1871334759_a_nat > $o] :
( ( member832397200_a_nat @ A @ ( collec357096914_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A4: set_Pr1163220871term_b] :
( ( collec135640594term_b
@ ^ [X5: produc1478835367term_b] : ( member516522448term_b @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A4: set_Pr551076371_b_nat] :
( ( collec1615000990_b_nat
@ ^ [X5: produc1235635379_b_nat] : ( member963855452_b_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A4: set_Pr924198087_a_nat] :
( ( collec1701899602_a_nat
@ ^ [X5: produc398057191_a_nat] : ( member584645392_a_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A4: set_Pr1986765409at_nat] :
( ( collec7649004at_nat
@ ^ [X5: product_prod_nat_nat] : ( member701585322at_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A4: set_Pr1987088711_a_nat] :
( ( collec357096914_a_nat
@ ^ [X5: produc1871334759_a_nat] : ( member832397200_a_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_141_Collect__cong,axiom,
! [P: produc1871334759_a_nat > $o,Q2: produc1871334759_a_nat > $o] :
( ! [X4: produc1871334759_a_nat] :
( ( P @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collec357096914_a_nat @ P )
= ( collec357096914_a_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_142_verts__in__translation,axiom,
! [X: allego510293162tant_a] : ( inv_translation @ ( labele1810595089_a_nat @ ( transl1275713022tant_a @ X ) ) ) ).
% verts_in_translation
thf(fact_143_verts__in__translation,axiom,
! [X: allegorical_term_b] : ( inv_translation @ ( labele460410879_b_nat @ ( translation_b @ X ) ) ) ).
% verts_in_translation
thf(fact_144_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: allegorical_term_b > allegorical_term_b > $o,X3: allegorical_term_b,Y3: allegorical_term_b,A: produc1478835367term_b] :
( ( P @ X3 @ Y3 )
=> ( ( A
= ( produc1990145943term_b @ X3 @ Y3 ) )
=> ( P @ ( produc854192515term_b @ A ) @ ( produc1223098053term_b @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_145_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: labeled_graph_b_nat > labeled_graph_b_nat > $o,X3: labeled_graph_b_nat,Y3: labeled_graph_b_nat,A: produc1235635379_b_nat] :
( ( P @ X3 @ Y3 )
=> ( ( A
= ( produc951298923_b_nat @ X3 @ Y3 ) )
=> ( P @ ( produc1542243159_b_nat @ A ) @ ( produc194497945_b_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_146_conjI__realizer,axiom,
! [P: allegorical_term_b > $o,P2: allegorical_term_b,Q2: allegorical_term_b > $o,Q: allegorical_term_b] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc854192515term_b @ ( produc1990145943term_b @ P2 @ Q ) ) )
& ( Q2 @ ( produc1223098053term_b @ ( produc1990145943term_b @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_147_conjI__realizer,axiom,
! [P: labeled_graph_b_nat > $o,P2: labeled_graph_b_nat,Q2: labeled_graph_b_nat > $o,Q: labeled_graph_b_nat] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc1542243159_b_nat @ ( produc951298923_b_nat @ P2 @ Q ) ) )
& ( Q2 @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_148_exI__realizer,axiom,
! [P: allegorical_term_b > allegorical_term_b > $o,Y3: allegorical_term_b,X3: allegorical_term_b] :
( ( P @ Y3 @ X3 )
=> ( P @ ( produc1223098053term_b @ ( produc1990145943term_b @ X3 @ Y3 ) ) @ ( produc854192515term_b @ ( produc1990145943term_b @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_149_exI__realizer,axiom,
! [P: labeled_graph_b_nat > labeled_graph_b_nat > $o,Y3: labeled_graph_b_nat,X3: labeled_graph_b_nat] :
( ( P @ Y3 @ X3 )
=> ( P @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ X3 @ Y3 ) ) @ ( produc1542243159_b_nat @ ( produc951298923_b_nat @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_150_allegorical__term_Oinject_I1_J,axiom,
! [X11: allegorical_term_b,X12: allegorical_term_b,Y11: allegorical_term_b,Y12: allegorical_term_b] :
( ( ( allegorical_A_Int_b @ X11 @ X12 )
= ( allegorical_A_Int_b @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% allegorical_term.inject(1)
thf(fact_151_fin__maintainedI,axiom,
! [R: produc1235635379_b_nat,G: labeled_graph_b_nat] :
( ! [F: labeled_graph_b_nat,F2: set_Pr1986765409at_nat] :
( ( ( F
= ( restrict_b_nat @ F ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ F ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ F ) ) )
=> ( ( graph_529870330at_nat @ F @ ( produc1542243159_b_nat @ R ) @ ( id_on_nat @ ( labele460410879_b_nat @ F ) ) )
=> ( ( extensible_b_nat_nat @ ( produc951298923_b_nat @ F @ ( produc1542243159_b_nat @ R ) ) @ G @ F2 )
=> ( ( graph_529870330at_nat @ F @ G @ F2 )
=> ( extensible_b_nat_nat @ ( produc951298923_b_nat @ F @ ( produc194497945_b_nat @ R ) ) @ G @ F2 ) ) ) ) )
=> ( fin_ma971967913at_nat @ R @ G ) ) ).
% fin_maintainedI
thf(fact_152_set__of__graph__rulesD_I1_J,axiom,
! [Rs2: set_Pr1987088711_a_nat,R: produc1871334759_a_nat] :
( ( set_of1384085797_a_nat @ Rs2 )
=> ( ( member832397200_a_nat @ R @ Rs2 )
=> ( ( ( produc719117507_a_nat @ R )
= ( restri572569417_a_nat @ ( produc719117507_a_nat @ R ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ R ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc719117507_a_nat @ R ) ) ) ) ) ) ).
% set_of_graph_rulesD(1)
thf(fact_153_set__of__graph__rulesD_I1_J,axiom,
! [Rs2: set_Pr551076371_b_nat,R: produc1235635379_b_nat] :
( ( set_of41538795_b_nat @ Rs2 )
=> ( ( member963855452_b_nat @ R @ Rs2 )
=> ( ( ( produc1542243159_b_nat @ R )
= ( restrict_b_nat @ ( produc1542243159_b_nat @ R ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ R ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc1542243159_b_nat @ R ) ) ) ) ) ) ).
% set_of_graph_rulesD(1)
thf(fact_154_set__of__graph__rulesD_I2_J,axiom,
! [Rs2: set_Pr1987088711_a_nat,R: produc1871334759_a_nat] :
( ( set_of1384085797_a_nat @ Rs2 )
=> ( ( member832397200_a_nat @ R @ Rs2 )
=> ( ( ( produc880161797_a_nat @ R )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ R ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ R ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ R ) ) ) ) ) ) ).
% set_of_graph_rulesD(2)
thf(fact_155_set__of__graph__rulesD_I2_J,axiom,
! [Rs2: set_Pr551076371_b_nat,R: produc1235635379_b_nat] :
( ( set_of41538795_b_nat @ Rs2 )
=> ( ( member963855452_b_nat @ R @ Rs2 )
=> ( ( ( produc194497945_b_nat @ R )
= ( restrict_b_nat @ ( produc194497945_b_nat @ R ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ R ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ R ) ) ) ) ) ) ).
% set_of_graph_rulesD(2)
thf(fact_156_extensible__refl,axiom,
! [R: labeled_graph_b_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ R @ G @ F3 )
=> ( extensible_b_nat_nat @ ( produc951298923_b_nat @ R @ R ) @ G @ F3 ) ) ).
% extensible_refl
thf(fact_157_set__of__graph__rulesD_I3_J,axiom,
! [Rs2: set_Pr1987088711_a_nat,R: produc1871334759_a_nat] :
( ( set_of1384085797_a_nat @ Rs2 )
=> ( ( member832397200_a_nat @ R @ Rs2 )
=> ( graph_2130075512at_nat @ ( produc719117507_a_nat @ R ) @ ( produc880161797_a_nat @ R ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ R ) ) ) ) ) ) ).
% set_of_graph_rulesD(3)
thf(fact_158_set__of__graph__rulesD_I3_J,axiom,
! [Rs2: set_Pr551076371_b_nat,R: produc1235635379_b_nat] :
( ( set_of41538795_b_nat @ Rs2 )
=> ( ( member963855452_b_nat @ R @ Rs2 )
=> ( graph_529870330at_nat @ ( produc1542243159_b_nat @ R ) @ ( produc194497945_b_nat @ R ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ R ) ) ) ) ) ) ).
% set_of_graph_rulesD(3)
thf(fact_159_exE__realizer_H,axiom,
! [P: allegorical_term_b > allegorical_term_b > $o,P2: produc1478835367term_b] :
( ( P @ ( produc1223098053term_b @ P2 ) @ ( produc854192515term_b @ P2 ) )
=> ~ ! [X4: allegorical_term_b,Y4: allegorical_term_b] :
~ ( P @ Y4 @ X4 ) ) ).
% exE_realizer'
thf(fact_160_exE__realizer_H,axiom,
! [P: labeled_graph_b_nat > labeled_graph_b_nat > $o,P2: produc1235635379_b_nat] :
( ( P @ ( produc194497945_b_nat @ P2 ) @ ( produc1542243159_b_nat @ P2 ) )
=> ~ ! [X4: labeled_graph_b_nat,Y4: labeled_graph_b_nat] :
~ ( P @ Y4 @ X4 ) ) ).
% exE_realizer'
thf(fact_161_Id__onE,axiom,
! [C2: produc398057191_a_nat,A4: set_Pr1987088711_a_nat] :
( ( member584645392_a_nat @ C2 @ ( id_on_1651096324_a_nat @ A4 ) )
=> ~ ! [X4: produc1871334759_a_nat] :
( ( member832397200_a_nat @ X4 @ A4 )
=> ( C2
!= ( produc1677124439_a_nat @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_162_Id__onE,axiom,
! [C2: produc1871334759_a_nat,A4: set_la1083530965_a_nat] :
( ( member832397200_a_nat @ C2 @ ( id_on_689842066_a_nat @ A4 ) )
=> ~ ! [X4: labele935650037_a_nat] :
( ( member964390942_a_nat @ X4 @ A4 )
=> ( C2
!= ( produc1676969687_a_nat @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_163_Id__onE,axiom,
! [C2: product_prod_nat_nat,A4: set_nat] :
( ( member701585322at_nat @ C2 @ ( id_on_nat @ A4 ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( C2
!= ( product_Pair_nat_nat @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_164_Id__onE,axiom,
! [C2: produc1235635379_b_nat,A4: set_la1976028319_b_nat] :
( ( member963855452_b_nat @ C2 @ ( id_on_583275916_b_nat @ A4 ) )
=> ~ ! [X4: labeled_graph_b_nat] :
( ( member1483953152_b_nat @ X4 @ A4 )
=> ( C2
!= ( produc951298923_b_nat @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_165_Id__onE,axiom,
! [C2: produc1478835367term_b,A4: set_al1193902458term_b] :
( ( member516522448term_b @ C2 @ ( id_on_1536886967term_b @ A4 ) )
=> ~ ! [X4: allegorical_term_b] :
( ( member93680451term_b @ X4 @ A4 )
=> ( C2
!= ( produc1990145943term_b @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_166_Id__on__eqI,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,A4: set_la1083530965_a_nat] :
( ( A = B )
=> ( ( member964390942_a_nat @ A @ A4 )
=> ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B ) @ ( id_on_689842066_a_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_167_Id__on__eqI,axiom,
! [A: produc1871334759_a_nat,B: produc1871334759_a_nat,A4: set_Pr1987088711_a_nat] :
( ( A = B )
=> ( ( member832397200_a_nat @ A @ A4 )
=> ( member584645392_a_nat @ ( produc1677124439_a_nat @ A @ B ) @ ( id_on_1651096324_a_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_168_Id__on__eqI,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( A = B )
=> ( ( member_nat @ A @ A4 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_169_Id__on__eqI,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,A4: set_la1976028319_b_nat] :
( ( A = B )
=> ( ( member1483953152_b_nat @ A @ A4 )
=> ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B ) @ ( id_on_583275916_b_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_170_Id__on__eqI,axiom,
! [A: allegorical_term_b,B: allegorical_term_b,A4: set_al1193902458term_b] :
( ( A = B )
=> ( ( member93680451term_b @ A @ A4 )
=> ( member516522448term_b @ ( produc1990145943term_b @ A @ B ) @ ( id_on_1536886967term_b @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_171_Id__on__iff,axiom,
! [X3: produc1871334759_a_nat,Y3: produc1871334759_a_nat,A4: set_Pr1987088711_a_nat] :
( ( member584645392_a_nat @ ( produc1677124439_a_nat @ X3 @ Y3 ) @ ( id_on_1651096324_a_nat @ A4 ) )
= ( ( X3 = Y3 )
& ( member832397200_a_nat @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_172_Id__on__iff,axiom,
! [X3: labele935650037_a_nat,Y3: labele935650037_a_nat,A4: set_la1083530965_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X3 @ Y3 ) @ ( id_on_689842066_a_nat @ A4 ) )
= ( ( X3 = Y3 )
& ( member964390942_a_nat @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_173_Id__on__iff,axiom,
! [X3: nat,Y3: nat,A4: set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( id_on_nat @ A4 ) )
= ( ( X3 = Y3 )
& ( member_nat @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_174_Id__on__iff,axiom,
! [X3: labeled_graph_b_nat,Y3: labeled_graph_b_nat,A4: set_la1976028319_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ X3 @ Y3 ) @ ( id_on_583275916_b_nat @ A4 ) )
= ( ( X3 = Y3 )
& ( member1483953152_b_nat @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_175_Id__on__iff,axiom,
! [X3: allegorical_term_b,Y3: allegorical_term_b,A4: set_al1193902458term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ X3 @ Y3 ) @ ( id_on_1536886967term_b @ A4 ) )
= ( ( X3 = Y3 )
& ( member93680451term_b @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_176_fin__maintained__def,axiom,
( fin_ma971967913at_nat
= ( ^ [R2: produc1235635379_b_nat,G2: labeled_graph_b_nat] :
! [F4: labeled_graph_b_nat,F5: set_Pr1986765409at_nat] :
( ( ( F4
= ( restrict_b_nat @ F4 ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ F4 ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ F4 ) ) )
=> ( ( graph_529870330at_nat @ F4 @ ( produc1542243159_b_nat @ R2 ) @ ( id_on_nat @ ( labele460410879_b_nat @ F4 ) ) )
=> ( ( extensible_b_nat_nat @ ( produc951298923_b_nat @ F4 @ ( produc1542243159_b_nat @ R2 ) ) @ G2 @ F5 )
=> ( ( graph_529870330at_nat @ F4 @ G2 @ F5 )
=> ( extensible_b_nat_nat @ ( produc951298923_b_nat @ F4 @ ( produc194497945_b_nat @ R2 ) ) @ G2 @ F5 ) ) ) ) ) ) ) ).
% fin_maintained_def
thf(fact_177_fair__chainD_I2_J,axiom,
! [Rs2: set_Pr551076371_b_nat,S2: nat > labeled_graph_b_nat,R: produc1235635379_b_nat,I: nat,F3: set_Pr1986765409at_nat] :
( ( fair_chain_b_nat_nat @ Rs2 @ S2 )
=> ( ( member963855452_b_nat @ R @ Rs2 )
=> ( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ R ) @ ( S2 @ I ) @ F3 )
=> ? [J: nat] : ( extensible_b_nat_nat @ R @ ( S2 @ J ) @ F3 ) ) ) ) ).
% fair_chainD(2)
thf(fact_178_identity__rules__graph__rule,axiom,
! [X3: produc1871334759_a_nat,L: set_St761939237tant_a] :
( ( member832397200_a_nat @ X3 @ ( standa1568205540ules_a @ L ) )
=> ( ( graph_2130075512at_nat @ ( produc719117507_a_nat @ X3 ) @ ( produc880161797_a_nat @ X3 ) @ ( id_on_nat @ ( labele1810595089_a_nat @ ( produc719117507_a_nat @ X3 ) ) ) )
& ( ( produc880161797_a_nat @ X3 )
= ( restri572569417_a_nat @ ( produc880161797_a_nat @ X3 ) ) )
& ( finite_finite_nat @ ( labele1810595089_a_nat @ ( produc880161797_a_nat @ X3 ) ) )
& ( finite1242387294at_nat @ ( labele195203296_a_nat @ ( produc880161797_a_nat @ X3 ) ) ) ) ) ).
% identity_rules_graph_rule
thf(fact_179_maintainedI,axiom,
! [A4: labeled_graph_b_nat,G: labeled_graph_b_nat,B4: labeled_graph_b_nat] :
( ! [F2: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ A4 @ G @ F2 )
=> ( extensible_b_nat_nat @ ( produc951298923_b_nat @ A4 @ B4 ) @ G @ F2 ) )
=> ( maintained_b_nat_nat @ ( produc951298923_b_nat @ A4 @ B4 ) @ G ) ) ).
% maintainedI
thf(fact_180_sndI,axiom,
! [X3: produc1478835367term_b,Y3: allegorical_term_b,Z2: allegorical_term_b] :
( ( X3
= ( produc1990145943term_b @ Y3 @ Z2 ) )
=> ( ( produc1223098053term_b @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_181_sndI,axiom,
! [X3: produc1235635379_b_nat,Y3: labeled_graph_b_nat,Z2: labeled_graph_b_nat] :
( ( X3
= ( produc951298923_b_nat @ Y3 @ Z2 ) )
=> ( ( produc194497945_b_nat @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_182_consequence__graphI,axiom,
! [Rs2: set_Pr551076371_b_nat,G: labeled_graph_b_nat] :
( ! [R3: produc1235635379_b_nat] :
( ( member963855452_b_nat @ R3 @ Rs2 )
=> ( maintained_b_nat_nat @ R3 @ G ) )
=> ( ! [R3: produc1235635379_b_nat] :
( ( member963855452_b_nat @ R3 @ Rs2 )
=> ( graph_529870330at_nat @ ( produc1542243159_b_nat @ R3 ) @ ( produc194497945_b_nat @ R3 ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ R3 ) ) ) ) )
=> ( ( G
= ( restrict_b_nat @ G ) )
=> ( conseq1730780375at_nat @ Rs2 @ G ) ) ) ) ).
% consequence_graphI
thf(fact_183_consequence__graph__def,axiom,
( conseq1730780375at_nat
= ( ^ [Rs: set_Pr551076371_b_nat,G2: labeled_graph_b_nat] :
( ( G2
= ( restrict_b_nat @ G2 ) )
& ! [X5: produc1235635379_b_nat] :
( ( member963855452_b_nat @ X5 @ Rs )
=> ( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ X5 ) @ ( produc194497945_b_nat @ X5 ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ X5 ) ) ) )
& ( maintained_b_nat_nat @ X5 @ G2 ) ) ) ) ) ) ).
% consequence_graph_def
thf(fact_184_maintained__def,axiom,
( maintained_b_nat_nat
= ( ^ [R2: produc1235635379_b_nat,G2: labeled_graph_b_nat] :
! [F5: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ R2 ) @ G2 @ F5 )
=> ( extensible_b_nat_nat @ R2 @ G2 @ F5 ) ) ) ) ).
% maintained_def
thf(fact_185_maintainedD,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( maintained_b_nat_nat @ ( produc951298923_b_nat @ A4 @ B4 ) @ G )
=> ( ( graph_529870330at_nat @ A4 @ G @ F3 )
=> ( extensible_b_nat_nat @ ( produc951298923_b_nat @ A4 @ B4 ) @ G @ F3 ) ) ) ).
% maintainedD
thf(fact_186_fstI,axiom,
! [X3: produc1235635379_b_nat,Y3: labeled_graph_b_nat,Z2: labeled_graph_b_nat] :
( ( X3
= ( produc951298923_b_nat @ Y3 @ Z2 ) )
=> ( ( produc1542243159_b_nat @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_187_fstI,axiom,
! [X3: produc1478835367term_b,Y3: allegorical_term_b,Z2: allegorical_term_b] :
( ( X3
= ( produc1990145943term_b @ Y3 @ Z2 ) )
=> ( ( produc854192515term_b @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_188_fair__chainI,axiom,
! [S2: nat > labeled_graph_b_nat,Rs2: set_Pr551076371_b_nat] :
( ( chain_b_nat @ S2 )
=> ( ! [R3: produc1235635379_b_nat,F2: set_Pr1986765409at_nat,I2: nat] :
( ( member963855452_b_nat @ R3 @ Rs2 )
=> ( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ R3 ) @ ( S2 @ I2 ) @ F2 )
=> ? [J2: nat] : ( extensible_b_nat_nat @ R3 @ ( S2 @ J2 ) @ F2 ) ) )
=> ( fair_chain_b_nat_nat @ Rs2 @ S2 ) ) ) ).
% fair_chainI
thf(fact_189_maintained__holds__iff,axiom,
! [G: labeled_graph_b_nat,E_L: allegorical_term_b,E_R: allegorical_term_b] :
( ( G
= ( restrict_b_nat @ G ) )
=> ( ( maintained_b_nat_nat @ ( produc951298923_b_nat @ ( translation_b @ E_L ) @ ( translation_b @ ( allegorical_A_Int_b @ E_L @ E_R ) ) ) @ G )
= ( ( semantics_b_nat @ G @ ( produc854192515term_b @ ( produc1990145943term_b @ E_L @ ( allegorical_A_Int_b @ E_L @ E_R ) ) ) )
= ( semantics_b_nat @ G @ ( produc1223098053term_b @ ( produc1990145943term_b @ E_L @ ( allegorical_A_Int_b @ E_L @ E_R ) ) ) ) ) ) ) ).
% maintained_holds_iff
thf(fact_190_fair__chain__def,axiom,
( fair_chain_b_nat_nat
= ( ^ [Rs: set_Pr551076371_b_nat,S3: nat > labeled_graph_b_nat] :
( ( chain_b_nat @ S3 )
& ! [R2: produc1235635379_b_nat,F5: set_Pr1986765409at_nat,I3: nat] :
( ( ( member963855452_b_nat @ R2 @ Rs )
& ( graph_529870330at_nat @ ( produc1542243159_b_nat @ R2 ) @ ( S3 @ I3 ) @ F5 ) )
=> ? [J3: nat] : ( extensible_b_nat_nat @ R2 @ ( S3 @ J3 ) @ F5 ) ) ) ) ) ).
% fair_chain_def
thf(fact_191_chain__sup__graph,axiom,
! [S2: nat > labeled_graph_b_nat] :
( ( chain_b_nat @ S2 )
=> ( ( chain_sup_b_nat @ S2 )
= ( restrict_b_nat @ ( chain_sup_b_nat @ S2 ) ) ) ) ).
% chain_sup_graph
thf(fact_192_chain__sup__subgraph,axiom,
! [S2: nat > labeled_graph_b_nat,J4: nat] :
( ( chain_b_nat @ S2 )
=> ( graph_529870330at_nat @ ( S2 @ J4 ) @ ( chain_sup_b_nat @ S2 ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( S2 @ J4 ) ) ) ) ) ).
% chain_sup_subgraph
thf(fact_193_chain__then__restrict,axiom,
! [S2: nat > labeled_graph_b_nat,I: nat] :
( ( chain_b_nat @ S2 )
=> ( ( S2 @ I )
= ( restrict_b_nat @ ( S2 @ I ) ) ) ) ).
% chain_then_restrict
thf(fact_194_graph__homomorphism__semantics,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,F3: set_Pr1986765409at_nat,A: nat,B: nat,E: allegorical_term_b,A2: nat,B2: nat] :
( ( graph_529870330at_nat @ A4 @ B4 @ F3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( semantics_b_nat @ A4 @ E ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ F3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ F3 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( semantics_b_nat @ B4 @ E ) ) ) ) ) ) ).
% graph_homomorphism_semantics
thf(fact_195_semantics__in__vertices_I2_J,axiom,
! [A4: labeled_graph_b_nat,A: nat,B: nat,E: allegorical_term_b] :
( ( A4
= ( restrict_b_nat @ A4 ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( semantics_b_nat @ A4 @ E ) )
=> ( member_nat @ B @ ( labele460410879_b_nat @ A4 ) ) ) ) ).
% semantics_in_vertices(2)
thf(fact_196_semantics__in__vertices_I1_J,axiom,
! [A4: labeled_graph_b_nat,A: nat,B: nat,E: allegorical_term_b] :
( ( A4
= ( restrict_b_nat @ A4 ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( semantics_b_nat @ A4 @ E ) )
=> ( member_nat @ A @ ( labele460410879_b_nat @ A4 ) ) ) ) ).
% semantics_in_vertices(1)
thf(fact_197_subgraph__semantics,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,A: nat,B: nat,E: allegorical_term_b] :
( ( graph_529870330at_nat @ A4 @ B4 @ ( id_on_nat @ ( labele460410879_b_nat @ A4 ) ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( semantics_b_nat @ A4 @ E ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( semantics_b_nat @ B4 @ E ) ) ) ) ).
% subgraph_semantics
thf(fact_198_maintained__holds__subset__iff,axiom,
! [G: labeled_graph_b_nat,E_L: allegorical_term_b,E_R: allegorical_term_b] :
( ( G
= ( restrict_b_nat @ G ) )
=> ( ( maintained_b_nat_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ E_L @ ( allegorical_A_Int_b @ E_L @ E_R ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ E_L @ ( allegorical_A_Int_b @ E_L @ E_R ) ) ) ) ) @ G )
= ( ord_le841296385at_nat @ ( semantics_b_nat @ G @ E_L ) @ ( semantics_b_nat @ G @ E_R ) ) ) ) ).
% maintained_holds_subset_iff
thf(fact_199_find__graph__occurence,axiom,
! [S2: nat > labeled_graph_b_nat,E2: set_Pr9961929at_nat,V: set_nat,F3: set_Pr1986765409at_nat] :
( ( chain_b_nat @ S2 )
=> ( ( finite1987068434at_nat @ E2 )
=> ( ( finite_finite_nat @ V )
=> ( ( graph_529870330at_nat @ ( labeled_LG_b_nat @ E2 @ V ) @ ( chain_sup_b_nat @ S2 ) @ F3 )
=> ? [I2: nat] : ( graph_529870330at_nat @ ( labeled_LG_b_nat @ E2 @ V ) @ ( S2 @ I2 ) @ F3 ) ) ) ) ) ).
% find_graph_occurence
thf(fact_200_eq__snd__iff,axiom,
! [B: allegorical_term_b,P2: produc1478835367term_b] :
( ( B
= ( produc1223098053term_b @ P2 ) )
= ( ? [A5: allegorical_term_b] :
( P2
= ( produc1990145943term_b @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_201_eq__snd__iff,axiom,
! [B: labeled_graph_b_nat,P2: produc1235635379_b_nat] :
( ( B
= ( produc194497945_b_nat @ P2 ) )
= ( ? [A5: labeled_graph_b_nat] :
( P2
= ( produc951298923_b_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_202_eq__fst__iff,axiom,
! [A: labeled_graph_b_nat,P2: produc1235635379_b_nat] :
( ( A
= ( produc1542243159_b_nat @ P2 ) )
= ( ? [B5: labeled_graph_b_nat] :
( P2
= ( produc951298923_b_nat @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_203_eq__fst__iff,axiom,
! [A: allegorical_term_b,P2: produc1478835367term_b] :
( ( A
= ( produc854192515term_b @ P2 ) )
= ( ? [B5: allegorical_term_b] :
( P2
= ( produc1990145943term_b @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_204_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labeled_graph_b_nat] :
( ( labeled_LG_b_nat @ ( labeled_edges_b_nat @ Labeled_graph ) @ ( labele460410879_b_nat @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_205_extensible__refl__concr,axiom,
! [E_1: set_Pr9961929at_nat,V2: set_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat,E_2: set_Pr9961929at_nat] :
( ( graph_529870330at_nat @ ( labeled_LG_b_nat @ E_1 @ V2 ) @ G @ F3 )
=> ( ( extensible_b_nat_nat @ ( produc951298923_b_nat @ ( labeled_LG_b_nat @ E_1 @ V2 ) @ ( labeled_LG_b_nat @ E_2 @ V2 ) ) @ G @ F3 )
= ( graph_529870330at_nat @ ( labeled_LG_b_nat @ E_2 @ V2 ) @ G @ F3 ) ) ) ).
% extensible_refl_concr
thf(fact_206_subrelI,axiom,
! [R4: set_Pr1987088711_a_nat,S4: set_Pr1987088711_a_nat] :
( ! [X4: labele935650037_a_nat,Y4: labele935650037_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X4 @ Y4 ) @ R4 )
=> ( member832397200_a_nat @ ( produc1676969687_a_nat @ X4 @ Y4 ) @ S4 ) )
=> ( ord_le1718765799_a_nat @ R4 @ S4 ) ) ).
% subrelI
thf(fact_207_subrelI,axiom,
! [R4: set_Pr551076371_b_nat,S4: set_Pr551076371_b_nat] :
( ! [X4: labeled_graph_b_nat,Y4: labeled_graph_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ X4 @ Y4 ) @ R4 )
=> ( member963855452_b_nat @ ( produc951298923_b_nat @ X4 @ Y4 ) @ S4 ) )
=> ( ord_le13035955_b_nat @ R4 @ S4 ) ) ).
% subrelI
thf(fact_208_subrelI,axiom,
! [R4: set_Pr1163220871term_b,S4: set_Pr1163220871term_b] :
( ! [X4: allegorical_term_b,Y4: allegorical_term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ X4 @ Y4 ) @ R4 )
=> ( member516522448term_b @ ( produc1990145943term_b @ X4 @ Y4 ) @ S4 ) )
=> ( ord_le138473255term_b @ R4 @ S4 ) ) ).
% subrelI
thf(fact_209_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_210_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_211_finite__subset,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_212_finite__subset,axiom,
! [A4: set_Pr9961929at_nat,B4: set_Pr9961929at_nat] :
( ( ord_le910748009at_nat @ A4 @ B4 )
=> ( ( finite1987068434at_nat @ B4 )
=> ( finite1987068434at_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_213_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_214_infinite__super,axiom,
! [S2: set_Pr9961929at_nat,T3: set_Pr9961929at_nat] :
( ( ord_le910748009at_nat @ S2 @ T3 )
=> ( ~ ( finite1987068434at_nat @ S2 )
=> ~ ( finite1987068434at_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_215_rev__finite__subset,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_216_rev__finite__subset,axiom,
! [B4: set_Pr9961929at_nat,A4: set_Pr9961929at_nat] :
( ( finite1987068434at_nat @ B4 )
=> ( ( ord_le910748009at_nat @ A4 @ B4 )
=> ( finite1987068434at_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_217_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr9961929at_nat,X22: set_nat] :
( ( labele460410879_b_nat @ ( labeled_LG_b_nat @ X1 @ X22 ) )
= X22 ) ).
% labeled_graph.sel(2)
thf(fact_218_labeled__graph_Osel_I1_J,axiom,
! [X1: set_Pr9961929at_nat,X22: set_nat] :
( ( labeled_edges_b_nat @ ( labeled_LG_b_nat @ X1 @ X22 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_219_restrict__subsD,axiom,
! [G: labeled_graph_b_nat] :
( ( ord_le910748009at_nat @ ( labeled_edges_b_nat @ G ) @ ( labeled_edges_b_nat @ ( restrict_b_nat @ G ) ) )
=> ( G
= ( restrict_b_nat @ G ) ) ) ).
% restrict_subsD
thf(fact_220_subgraph__def2,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( G_1
= ( restrict_b_nat @ G_1 ) )
=> ( ( G_2
= ( restrict_b_nat @ G_2 ) )
=> ( ( graph_529870330at_nat @ G_1 @ G_2 @ ( id_on_nat @ ( labele460410879_b_nat @ G_1 ) ) )
= ( ( ord_less_eq_set_nat @ ( labele460410879_b_nat @ G_1 ) @ ( labele460410879_b_nat @ G_2 ) )
& ( ord_le910748009at_nat @ ( labeled_edges_b_nat @ G_1 ) @ ( labeled_edges_b_nat @ G_2 ) ) ) ) ) ) ).
% subgraph_def2
thf(fact_221_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labeled_graph_b_nat] :
( Labeled_graph
= ( labeled_LG_b_nat @ ( labeled_edges_b_nat @ Labeled_graph ) @ ( labele460410879_b_nat @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_222_subgraph__subset_I1_J,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ G_1 @ G_2 @ ( id_on_nat @ ( labele460410879_b_nat @ G_1 ) ) )
=> ( ord_less_eq_set_nat @ ( labele460410879_b_nat @ G_1 ) @ ( labele460410879_b_nat @ G_2 ) ) ) ).
% subgraph_subset(1)
thf(fact_223_maintainedD2,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( maintained_b_nat_nat @ ( produc951298923_b_nat @ A4 @ B4 ) @ G )
=> ( ( graph_529870330at_nat @ A4 @ G @ F3 )
=> ~ ! [G3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ B4 @ G @ G3 )
=> ~ ( ord_le841296385at_nat @ F3 @ G3 ) ) ) ) ).
% maintainedD2
thf(fact_224_subgraph__subset_I2_J,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ G_1 @ G_2 @ ( id_on_nat @ ( labele460410879_b_nat @ G_1 ) ) )
=> ( ord_le910748009at_nat @ ( labeled_edges_b_nat @ ( restrict_b_nat @ G_1 ) ) @ ( labeled_edges_b_nat @ G_2 ) ) ) ).
% subgraph_subset(2)
thf(fact_225_subsetI,axiom,
! [A4: set_Pr1987088711_a_nat,B4: set_Pr1987088711_a_nat] :
( ! [X4: produc1871334759_a_nat] :
( ( member832397200_a_nat @ X4 @ A4 )
=> ( member832397200_a_nat @ X4 @ B4 ) )
=> ( ord_le1718765799_a_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_226_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_227_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_228_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_229_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_230_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_231_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_232_order__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_233_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_234_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_235_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_236_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_237_antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_238_le__cases3,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_239_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_240_le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% le_cases
thf(fact_241_eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_242_linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linear
thf(fact_243_antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_244_eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_245_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_246_ord__eq__le__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_247_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_248_order__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_249_subset__iff,axiom,
( ord_le1718765799_a_nat
= ( ^ [A6: set_Pr1987088711_a_nat,B6: set_Pr1987088711_a_nat] :
! [T2: produc1871334759_a_nat] :
( ( member832397200_a_nat @ T2 @ A6 )
=> ( member832397200_a_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_250_subset__eq,axiom,
( ord_le1718765799_a_nat
= ( ^ [A6: set_Pr1987088711_a_nat,B6: set_Pr1987088711_a_nat] :
! [X5: produc1871334759_a_nat] :
( ( member832397200_a_nat @ X5 @ A6 )
=> ( member832397200_a_nat @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_251_subsetD,axiom,
! [A4: set_Pr1987088711_a_nat,B4: set_Pr1987088711_a_nat,C2: produc1871334759_a_nat] :
( ( ord_le1718765799_a_nat @ A4 @ B4 )
=> ( ( member832397200_a_nat @ C2 @ A4 )
=> ( member832397200_a_nat @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_252_in__mono,axiom,
! [A4: set_Pr1987088711_a_nat,B4: set_Pr1987088711_a_nat,X3: produc1871334759_a_nat] :
( ( ord_le1718765799_a_nat @ A4 @ B4 )
=> ( ( member832397200_a_nat @ X3 @ A4 )
=> ( member832397200_a_nat @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_253_chain,axiom,
! [S2: nat > labeled_graph_b_nat,I: nat,J4: nat] :
( ( chain_b_nat @ S2 )
=> ( ( ord_less_eq_nat @ I @ J4 )
=> ( graph_529870330at_nat @ ( S2 @ I ) @ ( S2 @ J4 ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( S2 @ I ) ) ) ) ) ) ).
% chain
thf(fact_254_chain__def2,axiom,
( chain_b_nat
= ( ^ [S3: nat > labeled_graph_b_nat] :
! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( graph_529870330at_nat @ ( S3 @ I3 ) @ ( S3 @ J3 ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( S3 @ I3 ) ) ) ) ) ) ) ).
% chain_def2
thf(fact_255_graph__unionI,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( ord_le910748009at_nat @ ( labeled_edges_b_nat @ G_1 ) @ ( labeled_edges_b_nat @ G_2 ) )
=> ( ( ord_less_eq_set_nat @ ( labele460410879_b_nat @ G_1 ) @ ( labele460410879_b_nat @ G_2 ) )
=> ( ( graph_union_b_nat @ G_1 @ G_2 )
= G_2 ) ) ) ).
% graph_unionI
thf(fact_256_extensibleD,axiom,
! [R: produc1235635379_b_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( extensible_b_nat_nat @ R @ G @ F3 )
=> ~ ! [G3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( produc194497945_b_nat @ R ) @ G @ G3 )
=> ~ ( agree_on_b_nat_nat @ ( produc1542243159_b_nat @ R ) @ F3 @ G3 ) ) ) ).
% extensibleD
thf(fact_257_extensible__def,axiom,
( extensible_b_nat_nat
= ( ^ [R2: produc1235635379_b_nat,G2: labeled_graph_b_nat,F5: set_Pr1986765409at_nat] :
? [G4: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( produc194497945_b_nat @ R2 ) @ G2 @ G4 )
& ( agree_on_b_nat_nat @ ( produc1542243159_b_nat @ R2 ) @ F5 @ G4 ) ) ) ) ).
% extensible_def
thf(fact_258_extensibleI,axiom,
! [R22: labeled_graph_b_nat,G: labeled_graph_b_nat,G5: set_Pr1986765409at_nat,R1: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ R22 @ G @ G5 )
=> ( ( agree_on_b_nat_nat @ R1 @ F3 @ G5 )
=> ( extensible_b_nat_nat @ ( produc951298923_b_nat @ R1 @ R22 ) @ G @ F3 ) ) ) ).
% extensibleI
thf(fact_259_graph__union__preserves__restrict,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( G_1
= ( restrict_b_nat @ G_1 ) )
=> ( ( G_2
= ( restrict_b_nat @ G_2 ) )
=> ( ( graph_union_b_nat @ G_1 @ G_2 )
= ( restrict_b_nat @ ( graph_union_b_nat @ G_1 @ G_2 ) ) ) ) ) ).
% graph_union_preserves_restrict
thf(fact_260_subgraph__def,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ G_1 @ G_2 @ ( id_on_nat @ ( labele460410879_b_nat @ G_1 ) ) )
= ( ( G_1
= ( restrict_b_nat @ G_1 ) )
& ( G_2
= ( restrict_b_nat @ G_2 ) )
& ( ( graph_union_b_nat @ G_1 @ G_2 )
= G_2 ) ) ) ).
% subgraph_def
thf(fact_261_graph__union__iff,axiom,
! [G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat] :
( ( ( graph_union_b_nat @ G_1 @ G_2 )
= G_2 )
= ( ( ord_le910748009at_nat @ ( labeled_edges_b_nat @ G_1 ) @ ( labeled_edges_b_nat @ G_2 ) )
& ( ord_less_eq_set_nat @ ( labele460410879_b_nat @ G_1 ) @ ( labele460410879_b_nat @ G_2 ) ) ) ) ).
% graph_union_iff
thf(fact_262_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N: set_nat] :
? [M: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N )
=> ( ord_less_eq_nat @ X5 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_263_nonempty__rule,axiom,
! [G: labeled_graph_b_nat] :
( ( G
= ( restrict_b_nat @ G ) )
=> ( ( maintained_b_nat_nat @ standa879863266rule_b @ G )
= ( ( labele460410879_b_nat @ G )
!= bot_bot_set_nat ) ) ) ).
% nonempty_rule
thf(fact_264_weak__universalI,axiom,
! [R: produc1235635379_b_nat,G_1: labeled_graph_b_nat,F_1: set_Pr1986765409at_nat,G_2: labeled_graph_b_nat,T: itself_nat,F_2: set_Pr1986765409at_nat] :
( ! [H_1: set_Pr1986765409at_nat,H_2: set_Pr1986765409at_nat,G6: labeled_graph_b_nat] :
( ( graph_529870330at_nat @ ( produc194497945_b_nat @ R ) @ G6 @ H_1 )
=> ( ( graph_529870330at_nat @ G_1 @ G6 @ H_2 )
=> ( ( ord_le841296385at_nat @ ( relcomp_nat_nat_nat @ F_1 @ H_2 ) @ H_1 )
=> ? [H2: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ G_2 @ G6 @ H2 )
& ( ord_le841296385at_nat @ H_2 @ H2 ) ) ) ) )
=> ( weak_u2026406106at_nat @ T @ R @ G_1 @ G_2 @ F_1 @ F_2 ) ) ).
% weak_universalI
thf(fact_265_Id__on__empty,axiom,
( ( id_on_nat @ bot_bot_set_nat )
= bot_bo2130386637at_nat ) ).
% Id_on_empty
thf(fact_266_graph__homomorphism__composes,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,X3: set_Pr1986765409at_nat,C2: labeled_graph_b_nat,Y3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ A @ B @ X3 )
=> ( ( graph_529870330at_nat @ B @ C2 @ Y3 )
=> ( graph_529870330at_nat @ A @ C2 @ ( relcomp_nat_nat_nat @ X3 @ Y3 ) ) ) ) ).
% graph_homomorphism_composes
thf(fact_267_graph__empty__e,axiom,
! [V2: set_nat] :
( ( labeled_LG_b_nat @ bot_bo1626616373at_nat @ V2 )
= ( restrict_b_nat @ ( labeled_LG_b_nat @ bot_bo1626616373at_nat @ V2 ) ) ) ).
% graph_empty_e
thf(fact_268_graph__homomorphism__empty,axiom,
! [G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( labeled_LG_b_nat @ bot_bo1626616373at_nat @ bot_bot_set_nat ) @ G @ F3 )
= ( ( F3 = bot_bo2130386637at_nat )
& ( G
= ( restrict_b_nat @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_269_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_270_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_271_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_272_relcomp_OrelcompI,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,C2: labele935650037_a_nat,S4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B ) @ R4 )
=> ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ B @ C2 ) @ S4 )
=> ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ C2 ) @ ( relcom1338300020_a_nat @ R4 @ S4 ) ) ) ) ).
% relcomp.relcompI
thf(fact_273_relcomp_OrelcompI,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,C2: labeled_graph_b_nat,S4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B ) @ R4 )
=> ( ( member963855452_b_nat @ ( produc951298923_b_nat @ B @ C2 ) @ S4 )
=> ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ C2 ) @ ( relcom1426860350_b_nat @ R4 @ S4 ) ) ) ) ).
% relcomp.relcompI
thf(fact_274_relcomp_OrelcompI,axiom,
! [A: allegorical_term_b,B: allegorical_term_b,R4: set_Pr1163220871term_b,C2: allegorical_term_b,S4: set_Pr1163220871term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A @ B ) @ R4 )
=> ( ( member516522448term_b @ ( produc1990145943term_b @ B @ C2 ) @ S4 )
=> ( member516522448term_b @ ( produc1990145943term_b @ A @ C2 ) @ ( relcom1955155673term_b @ R4 @ S4 ) ) ) ) ).
% relcomp.relcompI
thf(fact_275_relcomp_Oinducts,axiom,
! [X1: labele935650037_a_nat,X22: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,S4: set_Pr1987088711_a_nat,P: labele935650037_a_nat > labele935650037_a_nat > $o] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X1 @ X22 ) @ ( relcom1338300020_a_nat @ R4 @ S4 ) )
=> ( ! [A3: labele935650037_a_nat,B3: labele935650037_a_nat,C: labele935650037_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A3 @ B3 ) @ R4 )
=> ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ B3 @ C ) @ S4 )
=> ( P @ A3 @ C ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_276_relcomp_Oinducts,axiom,
! [X1: labeled_graph_b_nat,X22: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,S4: set_Pr551076371_b_nat,P: labeled_graph_b_nat > labeled_graph_b_nat > $o] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ X1 @ X22 ) @ ( relcom1426860350_b_nat @ R4 @ S4 ) )
=> ( ! [A3: labeled_graph_b_nat,B3: labeled_graph_b_nat,C: labeled_graph_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A3 @ B3 ) @ R4 )
=> ( ( member963855452_b_nat @ ( produc951298923_b_nat @ B3 @ C ) @ S4 )
=> ( P @ A3 @ C ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_277_relcomp_Oinducts,axiom,
! [X1: allegorical_term_b,X22: allegorical_term_b,R4: set_Pr1163220871term_b,S4: set_Pr1163220871term_b,P: allegorical_term_b > allegorical_term_b > $o] :
( ( member516522448term_b @ ( produc1990145943term_b @ X1 @ X22 ) @ ( relcom1955155673term_b @ R4 @ S4 ) )
=> ( ! [A3: allegorical_term_b,B3: allegorical_term_b,C: allegorical_term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A3 @ B3 ) @ R4 )
=> ( ( member516522448term_b @ ( produc1990145943term_b @ B3 @ C ) @ S4 )
=> ( P @ A3 @ C ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_278_relcomp_Osimps,axiom,
! [A1: labele935650037_a_nat,A22: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,S4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A1 @ A22 ) @ ( relcom1338300020_a_nat @ R4 @ S4 ) )
= ( ? [A5: labele935650037_a_nat,B5: labele935650037_a_nat,C3: labele935650037_a_nat] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member832397200_a_nat @ ( produc1676969687_a_nat @ A5 @ B5 ) @ R4 )
& ( member832397200_a_nat @ ( produc1676969687_a_nat @ B5 @ C3 ) @ S4 ) ) ) ) ).
% relcomp.simps
thf(fact_279_relcomp_Osimps,axiom,
! [A1: labeled_graph_b_nat,A22: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,S4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A1 @ A22 ) @ ( relcom1426860350_b_nat @ R4 @ S4 ) )
= ( ? [A5: labeled_graph_b_nat,B5: labeled_graph_b_nat,C3: labeled_graph_b_nat] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member963855452_b_nat @ ( produc951298923_b_nat @ A5 @ B5 ) @ R4 )
& ( member963855452_b_nat @ ( produc951298923_b_nat @ B5 @ C3 ) @ S4 ) ) ) ) ).
% relcomp.simps
thf(fact_280_relcomp_Osimps,axiom,
! [A1: allegorical_term_b,A22: allegorical_term_b,R4: set_Pr1163220871term_b,S4: set_Pr1163220871term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A1 @ A22 ) @ ( relcom1955155673term_b @ R4 @ S4 ) )
= ( ? [A5: allegorical_term_b,B5: allegorical_term_b,C3: allegorical_term_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member516522448term_b @ ( produc1990145943term_b @ A5 @ B5 ) @ R4 )
& ( member516522448term_b @ ( produc1990145943term_b @ B5 @ C3 ) @ S4 ) ) ) ) ).
% relcomp.simps
thf(fact_281_relcomp_Ocases,axiom,
! [A1: labele935650037_a_nat,A22: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,S4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A1 @ A22 ) @ ( relcom1338300020_a_nat @ R4 @ S4 ) )
=> ~ ! [B3: labele935650037_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A1 @ B3 ) @ R4 )
=> ~ ( member832397200_a_nat @ ( produc1676969687_a_nat @ B3 @ A22 ) @ S4 ) ) ) ).
% relcomp.cases
thf(fact_282_relcomp_Ocases,axiom,
! [A1: labeled_graph_b_nat,A22: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,S4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A1 @ A22 ) @ ( relcom1426860350_b_nat @ R4 @ S4 ) )
=> ~ ! [B3: labeled_graph_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A1 @ B3 ) @ R4 )
=> ~ ( member963855452_b_nat @ ( produc951298923_b_nat @ B3 @ A22 ) @ S4 ) ) ) ).
% relcomp.cases
thf(fact_283_relcomp_Ocases,axiom,
! [A1: allegorical_term_b,A22: allegorical_term_b,R4: set_Pr1163220871term_b,S4: set_Pr1163220871term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A1 @ A22 ) @ ( relcom1955155673term_b @ R4 @ S4 ) )
=> ~ ! [B3: allegorical_term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A1 @ B3 ) @ R4 )
=> ~ ( member516522448term_b @ ( produc1990145943term_b @ B3 @ A22 ) @ S4 ) ) ) ).
% relcomp.cases
thf(fact_284_relcompEpair,axiom,
! [A: labele935650037_a_nat,C2: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,S4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ C2 ) @ ( relcom1338300020_a_nat @ R4 @ S4 ) )
=> ~ ! [B3: labele935650037_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B3 ) @ R4 )
=> ~ ( member832397200_a_nat @ ( produc1676969687_a_nat @ B3 @ C2 ) @ S4 ) ) ) ).
% relcompEpair
thf(fact_285_relcompEpair,axiom,
! [A: labeled_graph_b_nat,C2: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,S4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ C2 ) @ ( relcom1426860350_b_nat @ R4 @ S4 ) )
=> ~ ! [B3: labeled_graph_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B3 ) @ R4 )
=> ~ ( member963855452_b_nat @ ( produc951298923_b_nat @ B3 @ C2 ) @ S4 ) ) ) ).
% relcompEpair
thf(fact_286_relcompEpair,axiom,
! [A: allegorical_term_b,C2: allegorical_term_b,R4: set_Pr1163220871term_b,S4: set_Pr1163220871term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A @ C2 ) @ ( relcom1955155673term_b @ R4 @ S4 ) )
=> ~ ! [B3: allegorical_term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A @ B3 ) @ R4 )
=> ~ ( member516522448term_b @ ( produc1990145943term_b @ B3 @ C2 ) @ S4 ) ) ) ).
% relcompEpair
thf(fact_287_relcompE,axiom,
! [Xz: produc1871334759_a_nat,R4: set_Pr1987088711_a_nat,S4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ Xz @ ( relcom1338300020_a_nat @ R4 @ S4 ) )
=> ~ ! [X4: labele935650037_a_nat,Y4: labele935650037_a_nat,Z3: labele935650037_a_nat] :
( ( Xz
= ( produc1676969687_a_nat @ X4 @ Z3 ) )
=> ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X4 @ Y4 ) @ R4 )
=> ~ ( member832397200_a_nat @ ( produc1676969687_a_nat @ Y4 @ Z3 ) @ S4 ) ) ) ) ).
% relcompE
thf(fact_288_relcompE,axiom,
! [Xz: produc1235635379_b_nat,R4: set_Pr551076371_b_nat,S4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ Xz @ ( relcom1426860350_b_nat @ R4 @ S4 ) )
=> ~ ! [X4: labeled_graph_b_nat,Y4: labeled_graph_b_nat,Z3: labeled_graph_b_nat] :
( ( Xz
= ( produc951298923_b_nat @ X4 @ Z3 ) )
=> ( ( member963855452_b_nat @ ( produc951298923_b_nat @ X4 @ Y4 ) @ R4 )
=> ~ ( member963855452_b_nat @ ( produc951298923_b_nat @ Y4 @ Z3 ) @ S4 ) ) ) ) ).
% relcompE
thf(fact_289_relcompE,axiom,
! [Xz: produc1478835367term_b,R4: set_Pr1163220871term_b,S4: set_Pr1163220871term_b] :
( ( member516522448term_b @ Xz @ ( relcom1955155673term_b @ R4 @ S4 ) )
=> ~ ! [X4: allegorical_term_b,Y4: allegorical_term_b,Z3: allegorical_term_b] :
( ( Xz
= ( produc1990145943term_b @ X4 @ Z3 ) )
=> ( ( member516522448term_b @ ( produc1990145943term_b @ X4 @ Y4 ) @ R4 )
=> ~ ( member516522448term_b @ ( produc1990145943term_b @ Y4 @ Z3 ) @ S4 ) ) ) ) ).
% relcompE
thf(fact_290_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_291_finite_OemptyI,axiom,
finite1987068434at_nat @ bot_bo1626616373at_nat ).
% finite.emptyI
thf(fact_292_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_293_infinite__imp__nonempty,axiom,
! [S2: set_Pr9961929at_nat] :
( ~ ( finite1987068434at_nat @ S2 )
=> ( S2 != bot_bo1626616373at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_294_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_295_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_296_graph__homomorphism__nonempty,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,F3: set_Pr1986765409at_nat,E: allegorical_term_b] :
( ( graph_529870330at_nat @ A4 @ B4 @ F3 )
=> ( ( ( semantics_b_nat @ A4 @ E )
!= bot_bo2130386637at_nat )
=> ( ( semantics_b_nat @ B4 @ E )
!= bot_bo2130386637at_nat ) ) ) ).
% graph_homomorphism_nonempty
thf(fact_297_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_298_Id__on__vertices__identity_I2_J,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,F3: set_Pr1986765409at_nat,Aa: nat,Ba: nat] :
( ( graph_529870330at_nat @ A @ B @ F3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ F3 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( relcomp_nat_nat_nat @ F3 @ ( id_on_nat @ ( labele460410879_b_nat @ B ) ) ) ) ) ) ).
% Id_on_vertices_identity(2)
thf(fact_299_Id__on__vertices__identity_I1_J,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,F3: set_Pr1986765409at_nat,Aa: nat,Ba: nat] :
( ( graph_529870330at_nat @ A @ B @ F3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ F3 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( relcomp_nat_nat_nat @ ( id_on_nat @ ( labele460410879_b_nat @ A ) ) @ F3 ) ) ) ) ).
% Id_on_vertices_identity(1)
thf(fact_300_translation__homomorphism_I2_J,axiom,
! [E: allegorical_term_b,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( translation_b @ E ) @ G @ F3 )
=> ( ( semantics_b_nat @ G @ E )
!= bot_bo2130386637at_nat ) ) ).
% translation_homomorphism(2)
thf(fact_301_weak__universalD,axiom,
! [T: itself_nat,R: produc1235635379_b_nat,G_1: labeled_graph_b_nat,G_2: labeled_graph_b_nat,F_1: set_Pr1986765409at_nat,F_2: set_Pr1986765409at_nat,G: labeled_graph_b_nat,H_12: set_Pr1986765409at_nat,H_22: set_Pr1986765409at_nat] :
( ( weak_u2026406106at_nat @ T @ R @ G_1 @ G_2 @ F_1 @ F_2 )
=> ( ( graph_529870330at_nat @ ( produc194497945_b_nat @ R ) @ G @ H_12 )
=> ( ( graph_529870330at_nat @ G_1 @ G @ H_22 )
=> ( ( ord_le841296385at_nat @ ( relcomp_nat_nat_nat @ F_1 @ H_22 ) @ H_12 )
=> ? [H3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ G_2 @ G @ H3 )
& ( ord_le841296385at_nat @ H_22 @ H3 ) ) ) ) ) ) ).
% weak_universalD
thf(fact_302_weak__universal__def,axiom,
( weak_u2026406106at_nat
= ( ^ [Uu: itself_nat,R2: produc1235635379_b_nat,G_12: labeled_graph_b_nat,G_22: labeled_graph_b_nat,F_12: set_Pr1986765409at_nat,F_22: set_Pr1986765409at_nat] :
! [H_13: set_Pr1986765409at_nat,H_23: set_Pr1986765409at_nat,G2: labeled_graph_b_nat] :
( ( ( graph_529870330at_nat @ ( produc194497945_b_nat @ R2 ) @ G2 @ H_13 )
& ( graph_529870330at_nat @ G_12 @ G2 @ H_23 )
& ( ord_le841296385at_nat @ ( relcomp_nat_nat_nat @ F_12 @ H_23 ) @ H_13 ) )
=> ? [H4: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ G_22 @ G2 @ H4 )
& ( ord_le841296385at_nat @ H_23 @ H4 ) ) ) ) ) ).
% weak_universal_def
thf(fact_303_graph__homo__union__id_I2_J,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( graph_union_b_nat @ A4 @ B4 ) @ G @ F3 )
=> ( ( B4
= ( restrict_b_nat @ B4 ) )
=> ( graph_529870330at_nat @ B4 @ G @ ( relcomp_nat_nat_nat @ ( id_on_nat @ ( labele460410879_b_nat @ B4 ) ) @ F3 ) ) ) ) ).
% graph_homo_union_id(2)
thf(fact_304_graph__homo__union__id_I1_J,axiom,
! [A4: labeled_graph_b_nat,B4: labeled_graph_b_nat,G: labeled_graph_b_nat,F3: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ ( graph_union_b_nat @ A4 @ B4 ) @ G @ F3 )
=> ( ( A4
= ( restrict_b_nat @ A4 ) )
=> ( graph_529870330at_nat @ A4 @ G @ ( relcomp_nat_nat_nat @ ( id_on_nat @ ( labele460410879_b_nat @ A4 ) ) @ F3 ) ) ) ) ).
% graph_homo_union_id(1)
thf(fact_305_finite__relcomp,axiom,
! [R: set_Product_prod_b_b,S2: set_Pr9961929at_nat] :
( ( finite1015599120od_b_b @ R )
=> ( ( finite1987068434at_nat @ S2 )
=> ( finite1987068434at_nat @ ( relcom14055552at_nat @ R @ S2 ) ) ) ) ).
% finite_relcomp
thf(fact_306_finite__relcomp,axiom,
! [R: set_Pr9961929at_nat,S2: set_Pr1490359111at_nat] :
( ( finite1987068434at_nat @ R )
=> ( ( finite48957584at_nat @ S2 )
=> ( finite1987068434at_nat @ ( relcom195261566at_nat @ R @ S2 ) ) ) ) ).
% finite_relcomp
thf(fact_307_subset__emptyI,axiom,
! [A4: set_Pr1987088711_a_nat] :
( ! [X4: produc1871334759_a_nat] :
~ ( member832397200_a_nat @ X4 @ A4 )
=> ( ord_le1718765799_a_nat @ A4 @ bot_bo1836341171_a_nat ) ) ).
% subset_emptyI
thf(fact_308_bot__empty__eq,axiom,
( bot_bo1024461546_nat_o
= ( ^ [X5: produc1871334759_a_nat] : ( member832397200_a_nat @ X5 @ bot_bo1836341171_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_309_ssubst__Pair__rhs,axiom,
! [R4: labele935650037_a_nat,S4: labele935650037_a_nat,R: set_Pr1987088711_a_nat,S5: labele935650037_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ R4 @ S4 ) @ R )
=> ( ( S5 = S4 )
=> ( member832397200_a_nat @ ( produc1676969687_a_nat @ R4 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_310_ssubst__Pair__rhs,axiom,
! [R4: labeled_graph_b_nat,S4: labeled_graph_b_nat,R: set_Pr551076371_b_nat,S5: labeled_graph_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ R4 @ S4 ) @ R )
=> ( ( S5 = S4 )
=> ( member963855452_b_nat @ ( produc951298923_b_nat @ R4 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_311_ssubst__Pair__rhs,axiom,
! [R4: allegorical_term_b,S4: allegorical_term_b,R: set_Pr1163220871term_b,S5: allegorical_term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ R4 @ S4 ) @ R )
=> ( ( S5 = S4 )
=> ( member516522448term_b @ ( produc1990145943term_b @ R4 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_312_agree__iff__subset,axiom,
! [G: labeled_graph_b_nat,X: labeled_graph_b_nat,F3: set_Pr1986765409at_nat,G5: set_Pr1986765409at_nat] :
( ( graph_529870330at_nat @ G @ X @ F3 )
=> ( ( univalent_nat_nat @ G5 )
=> ( ( agree_on_b_nat_nat @ G @ F3 @ G5 )
= ( ord_le841296385at_nat @ F3 @ G5 ) ) ) ) ).
% agree_iff_subset
thf(fact_313_id__univalent,axiom,
! [X3: set_nat] : ( univalent_nat_nat @ ( id_on_nat @ X3 ) ) ).
% id_univalent
thf(fact_314_univalentD,axiom,
! [R: set_Pr1987088711_a_nat,X3: labele935650037_a_nat,Y3: labele935650037_a_nat,Z2: labele935650037_a_nat] :
( ( unival1637751524_a_nat @ R )
=> ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X3 @ Y3 ) @ R )
=> ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X3 @ Z2 ) @ R )
=> ( Z2 = Y3 ) ) ) ) ).
% univalentD
thf(fact_315_univalentD,axiom,
! [R: set_Pr551076371_b_nat,X3: labeled_graph_b_nat,Y3: labeled_graph_b_nat,Z2: labeled_graph_b_nat] :
( ( unival857119480_b_nat @ R )
=> ( ( member963855452_b_nat @ ( produc951298923_b_nat @ X3 @ Y3 ) @ R )
=> ( ( member963855452_b_nat @ ( produc951298923_b_nat @ X3 @ Z2 ) @ R )
=> ( Z2 = Y3 ) ) ) ) ).
% univalentD
thf(fact_316_univalentD,axiom,
! [R: set_Pr1163220871term_b,X3: allegorical_term_b,Y3: allegorical_term_b,Z2: allegorical_term_b] :
( ( unival1191217828term_b @ R )
=> ( ( member516522448term_b @ ( produc1990145943term_b @ X3 @ Y3 ) @ R )
=> ( ( member516522448term_b @ ( produc1990145943term_b @ X3 @ Z2 ) @ R )
=> ( Z2 = Y3 ) ) ) ) ).
% univalentD
thf(fact_317_univalent__def,axiom,
( unival1637751524_a_nat
= ( ^ [R2: set_Pr1987088711_a_nat] :
! [X5: labele935650037_a_nat,Y6: labele935650037_a_nat,Z4: labele935650037_a_nat] :
( ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ X5 @ Y6 ) @ R2 )
& ( member832397200_a_nat @ ( produc1676969687_a_nat @ X5 @ Z4 ) @ R2 ) )
=> ( Z4 = Y6 ) ) ) ) ).
% univalent_def
thf(fact_318_univalent__def,axiom,
( unival857119480_b_nat
= ( ^ [R2: set_Pr551076371_b_nat] :
! [X5: labeled_graph_b_nat,Y6: labeled_graph_b_nat,Z4: labeled_graph_b_nat] :
( ( ( member963855452_b_nat @ ( produc951298923_b_nat @ X5 @ Y6 ) @ R2 )
& ( member963855452_b_nat @ ( produc951298923_b_nat @ X5 @ Z4 ) @ R2 ) )
=> ( Z4 = Y6 ) ) ) ) ).
% univalent_def
thf(fact_319_univalent__def,axiom,
( unival1191217828term_b
= ( ^ [R2: set_Pr1163220871term_b] :
! [X5: allegorical_term_b,Y6: allegorical_term_b,Z4: allegorical_term_b] :
( ( ( member516522448term_b @ ( produc1990145943term_b @ X5 @ Y6 ) @ R2 )
& ( member516522448term_b @ ( produc1990145943term_b @ X5 @ Z4 ) @ R2 ) )
=> ( Z4 = Y6 ) ) ) ) ).
% univalent_def
thf(fact_320_find__graph__occurence__vertices,axiom,
! [S2: nat > labeled_graph_b_nat,V: set_nat,F3: set_Pr1986765409at_nat] :
( ( chain_b_nat @ S2 )
=> ( ( finite_finite_nat @ V )
=> ( ( univalent_nat_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ V ) @ ( labele460410879_b_nat @ ( chain_sup_b_nat @ S2 ) ) )
=> ? [I2: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ V ) @ ( labele460410879_b_nat @ ( S2 @ I2 ) ) ) ) ) ) ) ).
% find_graph_occurence_vertices
thf(fact_321_find__graph__occurence__vertices,axiom,
! [S2: nat > labeled_graph_b_nat,V: set_Pr9961929at_nat,F3: set_Pr2041158302at_nat] :
( ( chain_b_nat @ S2 )
=> ( ( finite1987068434at_nat @ V )
=> ( ( unival633212949at_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_1356842150at_nat @ F3 @ V ) @ ( labele460410879_b_nat @ ( chain_sup_b_nat @ S2 ) ) )
=> ? [I2: nat] : ( ord_less_eq_set_nat @ ( image_1356842150at_nat @ F3 @ V ) @ ( labele460410879_b_nat @ ( S2 @ I2 ) ) ) ) ) ) ) ).
% find_graph_occurence_vertices
thf(fact_322_ImageI,axiom,
! [A: produc1871334759_a_nat,B: produc1871334759_a_nat,R4: set_Pr924198087_a_nat,A4: set_Pr1987088711_a_nat] :
( ( member584645392_a_nat @ ( produc1677124439_a_nat @ A @ B ) @ R4 )
=> ( ( member832397200_a_nat @ A @ A4 )
=> ( member832397200_a_nat @ B @ ( image_1168831379_a_nat @ R4 @ A4 ) ) ) ) ).
% ImageI
thf(fact_323_ImageI,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,A4: set_la1083530965_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B ) @ R4 )
=> ( ( member964390942_a_nat @ A @ A4 )
=> ( member964390942_a_nat @ B @ ( image_1971191571_a_nat @ R4 @ A4 ) ) ) ) ).
% ImageI
thf(fact_324_ImageI,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,A4: set_la1976028319_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B ) @ R4 )
=> ( ( member1483953152_b_nat @ A @ A4 )
=> ( member1483953152_b_nat @ B @ ( image_1183964583_b_nat @ R4 @ A4 ) ) ) ) ).
% ImageI
thf(fact_325_ImageI,axiom,
! [A: allegorical_term_b,B: allegorical_term_b,R4: set_Pr1163220871term_b,A4: set_al1193902458term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A @ B ) @ R4 )
=> ( ( member93680451term_b @ A @ A4 )
=> ( member93680451term_b @ B @ ( image_329221075term_b @ R4 @ A4 ) ) ) ) ).
% ImageI
thf(fact_326_Domain__Id__on,axiom,
! [A4: set_nat] :
( ( domain_nat_nat @ ( id_on_nat @ A4 ) )
= A4 ) ).
% Domain_Id_on
thf(fact_327_univalent__finite_I2_J,axiom,
! [R: set_Pr9961929at_nat] :
( ( unival989235430at_nat @ R )
=> ( ( finite_finite_b @ ( domain1101989710at_nat @ R ) )
= ( finite1987068434at_nat @ R ) ) ) ).
% univalent_finite(2)
thf(fact_328_finite__Domain,axiom,
! [R4: set_Pr9961929at_nat] :
( ( finite1987068434at_nat @ R4 )
=> ( finite_finite_b @ ( domain1101989710at_nat @ R4 ) ) ) ).
% finite_Domain
thf(fact_329_finite__Image,axiom,
! [R: set_Pr9961929at_nat,A4: set_b] :
( ( finite1987068434at_nat @ R )
=> ( finite772653738at_nat @ ( image_2112855445at_nat @ R @ A4 ) ) ) ).
% finite_Image
thf(fact_330_Domain_Oinducts,axiom,
! [X3: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,P: labele935650037_a_nat > $o] :
( ( member964390942_a_nat @ X3 @ ( domain1068567884_a_nat @ R4 ) )
=> ( ! [A3: labele935650037_a_nat,B3: labele935650037_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A3 @ B3 ) @ R4 )
=> ( P @ A3 ) )
=> ( P @ X3 ) ) ) ).
% Domain.inducts
thf(fact_331_Domain_Oinducts,axiom,
! [X3: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,P: labeled_graph_b_nat > $o] :
( ( member1483953152_b_nat @ X3 @ ( domain767519072_b_nat @ R4 ) )
=> ( ! [A3: labeled_graph_b_nat,B3: labeled_graph_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A3 @ B3 ) @ R4 )
=> ( P @ A3 ) )
=> ( P @ X3 ) ) ) ).
% Domain.inducts
thf(fact_332_Domain_Oinducts,axiom,
! [X3: allegorical_term_b,R4: set_Pr1163220871term_b,P: allegorical_term_b > $o] :
( ( member93680451term_b @ X3 @ ( domain859272460term_b @ R4 ) )
=> ( ! [A3: allegorical_term_b,B3: allegorical_term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A3 @ B3 ) @ R4 )
=> ( P @ A3 ) )
=> ( P @ X3 ) ) ) ).
% Domain.inducts
thf(fact_333_Domain_ODomainI,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,R4: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B ) @ R4 )
=> ( member964390942_a_nat @ A @ ( domain1068567884_a_nat @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_334_Domain_ODomainI,axiom,
! [A: labeled_graph_b_nat,B: labeled_graph_b_nat,R4: set_Pr551076371_b_nat] :
( ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B ) @ R4 )
=> ( member1483953152_b_nat @ A @ ( domain767519072_b_nat @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_335_Domain_ODomainI,axiom,
! [A: allegorical_term_b,B: allegorical_term_b,R4: set_Pr1163220871term_b] :
( ( member516522448term_b @ ( produc1990145943term_b @ A @ B ) @ R4 )
=> ( member93680451term_b @ A @ ( domain859272460term_b @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_336_Domain_Osimps,axiom,
! [A: labele935650037_a_nat,R4: set_Pr1987088711_a_nat] :
( ( member964390942_a_nat @ A @ ( domain1068567884_a_nat @ R4 ) )
= ( ? [A5: labele935650037_a_nat,B5: labele935650037_a_nat] :
( ( A = A5 )
& ( member832397200_a_nat @ ( produc1676969687_a_nat @ A5 @ B5 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_337_Domain_Osimps,axiom,
! [A: labeled_graph_b_nat,R4: set_Pr551076371_b_nat] :
( ( member1483953152_b_nat @ A @ ( domain767519072_b_nat @ R4 ) )
= ( ? [A5: labeled_graph_b_nat,B5: labeled_graph_b_nat] :
( ( A = A5 )
& ( member963855452_b_nat @ ( produc951298923_b_nat @ A5 @ B5 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_338_Domain_Osimps,axiom,
! [A: allegorical_term_b,R4: set_Pr1163220871term_b] :
( ( member93680451term_b @ A @ ( domain859272460term_b @ R4 ) )
= ( ? [A5: allegorical_term_b,B5: allegorical_term_b] :
( ( A = A5 )
& ( member516522448term_b @ ( produc1990145943term_b @ A5 @ B5 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_339_Domain_Ocases,axiom,
! [A: labele935650037_a_nat,R4: set_Pr1987088711_a_nat] :
( ( member964390942_a_nat @ A @ ( domain1068567884_a_nat @ R4 ) )
=> ~ ! [B3: labele935650037_a_nat] :
~ ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_340_Domain_Ocases,axiom,
! [A: labeled_graph_b_nat,R4: set_Pr551076371_b_nat] :
( ( member1483953152_b_nat @ A @ ( domain767519072_b_nat @ R4 ) )
=> ~ ! [B3: labeled_graph_b_nat] :
~ ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_341_Domain_Ocases,axiom,
! [A: allegorical_term_b,R4: set_Pr1163220871term_b] :
( ( member93680451term_b @ A @ ( domain859272460term_b @ R4 ) )
=> ~ ! [B3: allegorical_term_b] :
~ ( member516522448term_b @ ( produc1990145943term_b @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_342_Domain__iff,axiom,
! [A: labele935650037_a_nat,R4: set_Pr1987088711_a_nat] :
( ( member964390942_a_nat @ A @ ( domain1068567884_a_nat @ R4 ) )
= ( ? [Y6: labele935650037_a_nat] : ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ Y6 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_343_Domain__iff,axiom,
! [A: labeled_graph_b_nat,R4: set_Pr551076371_b_nat] :
( ( member1483953152_b_nat @ A @ ( domain767519072_b_nat @ R4 ) )
= ( ? [Y6: labeled_graph_b_nat] : ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ Y6 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_344_Domain__iff,axiom,
! [A: allegorical_term_b,R4: set_Pr1163220871term_b] :
( ( member93680451term_b @ A @ ( domain859272460term_b @ R4 ) )
= ( ? [Y6: allegorical_term_b] : ( member516522448term_b @ ( produc1990145943term_b @ A @ Y6 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_345_DomainE,axiom,
! [A: labele935650037_a_nat,R4: set_Pr1987088711_a_nat] :
( ( member964390942_a_nat @ A @ ( domain1068567884_a_nat @ R4 ) )
=> ~ ! [B3: labele935650037_a_nat] :
~ ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_346_DomainE,axiom,
! [A: labeled_graph_b_nat,R4: set_Pr551076371_b_nat] :
( ( member1483953152_b_nat @ A @ ( domain767519072_b_nat @ R4 ) )
=> ~ ! [B3: labeled_graph_b_nat] :
~ ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_347_DomainE,axiom,
! [A: allegorical_term_b,R4: set_Pr1163220871term_b] :
( ( member93680451term_b @ A @ ( domain859272460term_b @ R4 ) )
=> ~ ! [B3: allegorical_term_b] :
~ ( member516522448term_b @ ( produc1990145943term_b @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_348_rev__ImageI,axiom,
! [A: labele935650037_a_nat,A4: set_la1083530965_a_nat,B: labele935650037_a_nat,R4: set_Pr1987088711_a_nat] :
( ( member964390942_a_nat @ A @ A4 )
=> ( ( member832397200_a_nat @ ( produc1676969687_a_nat @ A @ B ) @ R4 )
=> ( member964390942_a_nat @ B @ ( image_1971191571_a_nat @ R4 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_349_rev__ImageI,axiom,
! [A: produc1871334759_a_nat,A4: set_Pr1987088711_a_nat,B: produc1871334759_a_nat,R4: set_Pr924198087_a_nat] :
( ( member832397200_a_nat @ A @ A4 )
=> ( ( member584645392_a_nat @ ( produc1677124439_a_nat @ A @ B ) @ R4 )
=> ( member832397200_a_nat @ B @ ( image_1168831379_a_nat @ R4 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_350_rev__ImageI,axiom,
! [A: labeled_graph_b_nat,A4: set_la1976028319_b_nat,B: labeled_graph_b_nat,R4: set_Pr551076371_b_nat] :
( ( member1483953152_b_nat @ A @ A4 )
=> ( ( member963855452_b_nat @ ( produc951298923_b_nat @ A @ B ) @ R4 )
=> ( member1483953152_b_nat @ B @ ( image_1183964583_b_nat @ R4 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_351_rev__ImageI,axiom,
! [A: allegorical_term_b,A4: set_al1193902458term_b,B: allegorical_term_b,R4: set_Pr1163220871term_b] :
( ( member93680451term_b @ A @ A4 )
=> ( ( member516522448term_b @ ( produc1990145943term_b @ A @ B ) @ R4 )
=> ( member93680451term_b @ B @ ( image_329221075term_b @ R4 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_352_Image__iff,axiom,
! [B: labele935650037_a_nat,R4: set_Pr1987088711_a_nat,A4: set_la1083530965_a_nat] :
( ( member964390942_a_nat @ B @ ( image_1971191571_a_nat @ R4 @ A4 ) )
= ( ? [X5: labele935650037_a_nat] :
( ( member964390942_a_nat @ X5 @ A4 )
& ( member832397200_a_nat @ ( produc1676969687_a_nat @ X5 @ B ) @ R4 ) ) ) ) ).
% Image_iff
thf(fact_353_Image__iff,axiom,
! [B: labeled_graph_b_nat,R4: set_Pr551076371_b_nat,A4: set_la1976028319_b_nat] :
( ( member1483953152_b_nat @ B @ ( image_1183964583_b_nat @ R4 @ A4 ) )
= ( ? [X5: labeled_graph_b_nat] :
( ( member1483953152_b_nat @ X5 @ A4 )
& ( member963855452_b_nat @ ( produc951298923_b_nat @ X5 @ B ) @ R4 ) ) ) ) ).
% Image_iff
thf(fact_354_Image__iff,axiom,
! [B: allegorical_term_b,R4: set_Pr1163220871term_b,A4: set_al1193902458term_b] :
( ( member93680451term_b @ B @ ( image_329221075term_b @ R4 @ A4 ) )
= ( ? [X5: allegorical_term_b] :
( ( member93680451term_b @ X5 @ A4 )
& ( member516522448term_b @ ( produc1990145943term_b @ X5 @ B ) @ R4 ) ) ) ) ).
% Image_iff
% Conjectures (1)
thf(conj_0,conjecture,
( ( graph_529870330at_nat @ ( produc1542243159_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) ) @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) ) @ ( id_on_nat @ ( labele460410879_b_nat @ ( produc1542243159_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) ) ) ) )
& ( ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) )
= ( restrict_b_nat @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) ) ) )
& ( finite_finite_nat @ ( labele460410879_b_nat @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) ) ) )
& ( finite1987068434at_nat @ ( labeled_edges_b_nat @ ( produc194497945_b_nat @ ( produc951298923_b_nat @ ( translation_b @ ( produc854192515term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) @ ( translation_b @ ( produc1223098053term_b @ ( produc1990145943term_b @ u @ ( allegorical_A_Int_b @ u @ v ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------