%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP178^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n022.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:56:11 AM UTC 2026
% Result : Theorem 3.10s 1.76s
% Output : Refutation 3.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : ITP178^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n022.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:28:51 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.45 ---- Original DTF file ---
% 0.26/0.45 thf(spec,logic,$$dhol).
% 0.26/0.45 %------------------------------------------------------------------------------
% 0.26/0.45 % File : ITP178^2 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.45 % Domain : Interactive Theorem Proving
% 0.26/0.45 % Problem : Sledgehammer StandardRules problem prob_429__5391446_1
% 0.26/0.45 % Version : Especial.
% 0.26/0.45 % English :
% 0.26/0.45
% 0.26/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.45 % Source : [Des21]
% 0.26/0.45 % Names : StandardRules/prob_429__5391446_1 [Des21]
% 0.26/0.45
% 0.26/0.45 % Status : Theorem
% 0.26/0.45 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.26/0.45 % Syntax : Number of formulae : 342 ( 107 unt; 63 typ; 0 def)
% 0.26/0.45 % Number of atoms : 800 ( 226 equ; 0 cnn)
% 0.26/0.45 % Maximal formula atoms : 13 ( 2 avg)
% 0.26/0.45 % Number of connectives : 7968 ( 59 ~; 1 |; 88 &;7453 @)
% 0.26/0.45 % ( 0 <=>; 367 =>; 0 <=; 0 <~>)
% 0.26/0.45 % Maximal formula depth : 28 ( 10 avg)
% 0.26/0.45 % Number of types : 4 ( 3 usr)
% 0.26/0.45 % Number of type conns : 172 ( 172 >; 0 *; 0 +; 0 <<)
% 0.26/0.45 % Number of symbols : 62 ( 60 usr; 5 con; 0-10 aty)
% 0.26/0.45 % Number of variables : 1456 ( 68 ^;1263 !; 29 ?;1456 :)
% 0.26/0.45 % ( 96 !>; 0 ?*; 0 @-; 0 @+)
% 0.26/0.45 % SPC : TH1_THM_EQU_NAR
% 0.26/0.45
% 0.26/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:23:24.271
% 0.26/0.45 %------------------------------------------------------------------------------
% 0.26/0.45 % Could-be-implicit typings (9)
% 0.26/0.45 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 0.26/0.45 standard_Constant: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 0.26/0.45 allegorical_term: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 0.26/0.45 labeled_graph: $tType > $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_Product__Type_Oprod,type,
% 0.26/0.45 product_prod: $tType > $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_Set_Oset,type,
% 0.26/0.45 set: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_Nat_Onat,type,
% 0.26/0.45 nat: $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_itself,type,
% 0.26/0.45 itself: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_tf_b,type,
% 0.26/0.45 b: $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_tf_a,type,
% 0.26/0.45 a: $tType ).
% 0.26/0.45
% 0.26/0.45 % Explicit typings (54)
% 0.26/0.45 thf(sy_cl_HOL_Otype,type,
% 0.26/0.45 type:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.26/0.45 ord:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.26/0.45 order:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.26/0.45 finite_finite:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.26/0.45 linorder:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.26/0.45 preorder:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.26/0.45 order_bot:
% 0.26/0.45 !>[A: $tType] : $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Finite__Set_Ofinite,type,
% 0.26/0.45 finite_finite2:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphSemantics_Oallegorical__term_OA__Int,type,
% 0.26/0.45 allegorical_A_Int:
% 0.26/0.45 !>[V: $tType] : ( ( allegorical_term @ V ) > ( allegorical_term @ V ) > ( allegorical_term @ V ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 0.26/0.45 semantics:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 0.26/0.45 graph_homomorphism:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 0.26/0.45 graph_union:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 0.26/0.45 labeled_LG:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 0.26/0.45 labele1974067554_graph:
% 0.26/0.45 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 0.26/0.45 labeled_edges:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 0.26/0.45 labeled_vertices:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Orestrict,type,
% 0.26/0.45 restrict:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_MissingRelation_Ounivalent,type,
% 0.26/0.45 univalent:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.26/0.45 bot_bot:
% 0.26/0.45 !>[A: $tType] : A ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.26/0.45 ord_less_eq:
% 0.26/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Product__Type_OPair,type,
% 0.26/0.45 product_Pair:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.26/0.45 product_rec_prod:
% 0.26/0.45 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.26/0.45 product_fst:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.26/0.45 product_snd:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_ODomain,type,
% 0.26/0.45 domain:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_OId__on,type,
% 0.26/0.45 id_on:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_OImage,type,
% 0.26/0.45 image:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_Orelcomp,type,
% 0.26/0.45 relcomp:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RuleSemanticsConnection_Oinv__translation,type,
% 0.26/0.45 inv_translation: ( set @ nat ) > $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RuleSemanticsConnection_Otranslation,type,
% 0.26/0.45 translation:
% 0.26/0.45 !>[C: $tType] : ( ( allegorical_term @ C ) > ( labeled_graph @ C @ nat ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oagree__on,type,
% 0.26/0.45 agree_on:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Ochain,type,
% 0.26/0.45 chain:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( nat > ( labeled_graph @ L @ V ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Ochain__sup,type,
% 0.26/0.45 chain_sup:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( nat > ( labeled_graph @ L @ V ) ) > ( labeled_graph @ L @ V ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 0.26/0.45 consequence_graph:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oextensible,type,
% 0.26/0.45 extensible:
% 0.26/0.45 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ X @ V ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Ofair__chain,type,
% 0.26/0.45 fair_chain:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( nat > ( labeled_graph @ A @ C ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Ofin__maintained,type,
% 0.26/0.45 fin_maintained:
% 0.26/0.45 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oleast,type,
% 0.26/0.45 least:
% 0.26/0.45 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 0.26/0.45 least_559130134_graph:
% 0.26/0.45 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Omaintained,type,
% 0.26/0.45 maintained:
% 0.26/0.45 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Opushout__step,type,
% 0.26/0.45 pushout_step:
% 0.26/0.45 !>[X: $tType,A: $tType,C: $tType,B: $tType] : ( ( itself @ X ) > ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oset__of__graph__rules,type,
% 0.26/0.45 set_of_graph_rules:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oweak__universal,type,
% 0.26/0.45 weak_universal:
% 0.26/0.45 !>[X: $tType,A: $tType,C: $tType,B: $tType] : ( ( itself @ X ) > ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Set_OCollect,type,
% 0.26/0.45 collect:
% 0.26/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules,type,
% 0.26/0.45 standa1138209853_rules:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
% 0.26/0.45 standa2002409347_rules:
% 0.26/0.45 !>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 0.26/0.45 standa244753842y_rule:
% 0.26/0.45 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule,type,
% 0.26/0.45 standa245363227y_rule:
% 0.26/0.45 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule,type,
% 0.26/0.45 standa400804411p_rule:
% 0.26/0.45 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_member,type,
% 0.26/0.45 member:
% 0.26/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_C,type,
% 0.26/0.45 c: set @ a ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_u,type,
% 0.26/0.45 u: allegorical_term @ b ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_v,type,
% 0.26/0.45 v: allegorical_term @ b ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_x,type,
% 0.26/0.45 x: product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ).
% 0.26/0.45
% 0.26/0.45 % Relevant facts (256)
% 0.26/0.45 thf(fact_0_assms,axiom,
% 0.26/0.45 member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ x @ ( standa1138209853_rules @ a @ c ) ).
% 0.26/0.45
% 0.26/0.45 % assms
% 0.26/0.45 thf(fact_1_graph__rule__translation,axiom,
% 0.26/0.45 ! [A: $tType,X2: allegorical_term @ A,Y: allegorical_term @ A] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ nat @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) @ ( id_on @ nat @ ( labeled_vertices @ A @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) ) )
% 0.26/0.45 & ( ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) )
% 0.26/0.45 = ( restrict @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ nat @ ( labeled_vertices @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_rule_translation
% 0.26/0.45 thf(fact_2_subgraph__refl,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 0.26/0.45 = ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_refl
% 0.26/0.45 thf(fact_3_subgraph__restrict,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 0.26/0.45 = ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_restrict
% 0.26/0.45 thf(fact_4_graph__homomorphism__Id,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A2 ) @ ( restrict @ A @ B @ A2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_homomorphism_Id
% 0.26/0.45 thf(fact_5_verts__in__translation__finite_I2_J,axiom,
% 0.26/0.45 ! [A: $tType,X2: allegorical_term @ A] : ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ A @ nat @ ( translation @ A @ X2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % verts_in_translation_finite(2)
% 0.26/0.45 thf(fact_6_are__rules_I1_J,axiom,
% 0.26/0.45 ! [A: $tType] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ nat @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) @ ( id_on @ nat @ ( labeled_vertices @ A @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) ) )
% 0.26/0.45 & ( ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) )
% 0.26/0.45 = ( restrict @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ nat @ ( labeled_vertices @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % are_rules(1)
% 0.26/0.45 thf(fact_7_are__rules_I2_J,axiom,
% 0.26/0.45 ! [B: $tType,T2: B] :
% 0.26/0.45 ( ( graph_homomorphism @ B @ nat @ nat @ ( product_fst @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) @ ( id_on @ nat @ ( labeled_vertices @ B @ nat @ ( product_fst @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) ) )
% 0.26/0.45 & ( ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) )
% 0.26/0.45 = ( restrict @ B @ nat @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ nat @ ( labeled_vertices @ B @ nat @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ B @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ B @ nat @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % are_rules(2)
% 0.26/0.45 thf(fact_8_verts__in__translation__finite_I1_J,axiom,
% 0.26/0.45 ! [A: $tType,X2: allegorical_term @ A] : ( finite_finite2 @ nat @ ( labeled_vertices @ A @ nat @ ( translation @ A @ X2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % verts_in_translation_finite(1)
% 0.26/0.45 thf(fact_9_are__rules_I3_J,axiom,
% 0.26/0.45 ! [C: $tType,I: C] :
% 0.26/0.45 ( ( graph_homomorphism @ C @ nat @ nat @ ( product_fst @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) @ ( id_on @ nat @ ( labeled_vertices @ C @ nat @ ( product_fst @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) ) )
% 0.26/0.45 & ( ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) )
% 0.26/0.45 = ( restrict @ C @ nat @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ nat @ ( labeled_vertices @ C @ nat @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ C @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ C @ nat @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % are_rules(3)
% 0.26/0.45 thf(fact_10_prod_Ocollapse,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.26/0.45 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.26/0.45 = Prod ) ).
% 0.26/0.45
% 0.26/0.45 % prod.collapse
% 0.26/0.45 thf(fact_11_translation__graph,axiom,
% 0.26/0.45 ! [A: $tType] :
% 0.26/0.45 ( ( translation @ A )
% 0.26/0.45 = ( ^ [X3: allegorical_term @ A] : ( restrict @ A @ nat @ ( translation @ A @ X3 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % translation_graph
% 0.26/0.45 thf(fact_12_vertices__restrict,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
% 0.26/0.45 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
% 0.26/0.45 = ( labeled_vertices @ B @ A @ G ) ) ).
% 0.26/0.45
% 0.26/0.45 % vertices_restrict
% 0.26/0.45 thf(fact_13_old_Oprod_Oinject,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.26/0.45 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.26/0.45 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.26/0.45 = ( ( A2 = A3 )
% 0.26/0.45 & ( B2 = B3 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % old.prod.inject
% 0.26/0.45 thf(fact_14_prod_Oinject,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.26/0.45 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.26/0.45 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.26/0.45 = ( ( X1 = Y1 )
% 0.26/0.45 & ( X22 = Y2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod.inject
% 0.26/0.45 thf(fact_15_restrict__idemp,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X4: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X4 ) )
% 0.26/0.45 = ( restrict @ A @ B @ X4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % restrict_idemp
% 0.26/0.45 thf(fact_16_old_Oprod_Oinducts,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.26/0.45 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.26/0.45 => ( P @ Prod ) ) ).
% 0.26/0.45
% 0.26/0.45 % old.prod.inducts
% 0.26/0.45 thf(fact_17_old_Oprod_Oexhaust,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.26/0.45 ~ ! [A4: A,B4: B] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % old.prod.exhaust
% 0.26/0.45 thf(fact_18_prod__induct7,axiom,
% 0.26/0.45 ! [G2: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 0.26/0.45 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) )
% 0.26/0.45 => ( P @ X4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_induct7
% 0.26/0.45 thf(fact_19_prod__induct6,axiom,
% 0.26/0.45 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.26/0.45 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.26/0.45 => ( P @ X4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_induct6
% 0.26/0.45 thf(fact_20_prod__induct5,axiom,
% 0.26/0.45 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.45 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.26/0.45 => ( P @ X4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_induct5
% 0.26/0.45 thf(fact_21_prod__induct4,axiom,
% 0.26/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.45 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.26/0.45 => ( P @ X4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_induct4
% 0.26/0.45 thf(fact_22_prod__induct3,axiom,
% 0.26/0.45 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.26/0.45 => ( P @ X4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_induct3
% 0.26/0.45 thf(fact_23_prod__cases7,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 0.26/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_cases7
% 0.26/0.45 thf(fact_24_prod__cases6,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.26/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_cases6
% 0.26/0.45 thf(fact_25_prod__cases5,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_cases5
% 0.26/0.45 thf(fact_26_prod__cases4,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_cases4
% 0.26/0.45 thf(fact_27_prod__cases3,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.45 ~ ! [A4: A,B4: B,C2: C] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_cases3
% 0.26/0.45 thf(fact_28_Pair__inject,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.26/0.45 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.26/0.45 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.26/0.45 => ~ ( ( A2 = A3 )
% 0.26/0.45 => ( B2 != B3 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Pair_inject
% 0.26/0.45 thf(fact_29_prod__cases,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.26/0.45 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.26/0.45 => ( P @ P2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_cases
% 0.26/0.45 thf(fact_30_surj__pair,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.26/0.45 ? [X5: A,Y4: B] :
% 0.26/0.45 ( P2
% 0.26/0.45 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % surj_pair
% 0.26/0.45 thf(fact_31_fst__conv,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.26/0.45 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.26/0.45 = X1 ) ).
% 0.26/0.45
% 0.26/0.45 % fst_conv
% 0.26/0.45 thf(fact_32_fst__eqD,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X4: A,Y3: B,A2: A] :
% 0.26/0.45 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X4 @ Y3 ) )
% 0.26/0.45 = A2 )
% 0.26/0.45 => ( X4 = A2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % fst_eqD
% 0.26/0.45 thf(fact_33_snd__conv,axiom,
% 0.26/0.45 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.26/0.45 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.26/0.45 = X22 ) ).
% 0.26/0.45
% 0.26/0.45 % snd_conv
% 0.26/0.45 thf(fact_34_snd__eqD,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X4: B,Y3: A,A2: A] :
% 0.26/0.45 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) )
% 0.26/0.45 = A2 )
% 0.26/0.45 => ( Y3 = A2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % snd_eqD
% 0.26/0.45 thf(fact_35_prod__eq__iff,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType] :
% 0.26/0.45 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 0.26/0.45 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.26/0.45 ( ( ( product_fst @ A @ B @ S )
% 0.26/0.45 = ( product_fst @ A @ B @ T3 ) )
% 0.26/0.45 & ( ( product_snd @ A @ B @ S )
% 0.26/0.45 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_eq_iff
% 0.26/0.45 thf(fact_36_prod_Oexpand,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.26/0.45 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.26/0.45 = ( product_fst @ A @ B @ Prod2 ) )
% 0.26/0.45 & ( ( product_snd @ A @ B @ Prod )
% 0.26/0.45 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.26/0.45 => ( Prod = Prod2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod.expand
% 0.26/0.45 thf(fact_37_prod__eqI,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 0.26/0.45 ( ( ( product_fst @ A @ B @ P2 )
% 0.26/0.45 = ( product_fst @ A @ B @ Q ) )
% 0.26/0.45 => ( ( ( product_snd @ A @ B @ P2 )
% 0.26/0.45 = ( product_snd @ A @ B @ Q ) )
% 0.26/0.45 => ( P2 = Q ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod_eqI
% 0.26/0.45 thf(fact_38_labeled__graph_Oexpand,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 0.26/0.45 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 0.26/0.45 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 0.26/0.45 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 0.26/0.45 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 0.26/0.45 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.expand
% 0.26/0.45 thf(fact_39_surjective__pairing,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.26/0.45 ( T2
% 0.26/0.45 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % surjective_pairing
% 0.26/0.45 thf(fact_40_prod_Oexhaust__sel,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.26/0.45 ( Prod
% 0.26/0.45 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % prod.exhaust_sel
% 0.26/0.45 thf(fact_41_subgraph__preserves__hom,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,X2: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ C @ B @ X2 @ A5 @ H )
% 0.26/0.45 => ( graph_homomorphism @ A @ C @ B @ X2 @ B5 @ H ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_preserves_hom
% 0.26/0.45 thf(fact_42_subgraph__trans,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 0.26/0.45 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_trans
% 0.26/0.45 thf(fact_43_Id__onI,axiom,
% 0.26/0.45 ! [A: $tType,A2: A,A5: set @ A] :
% 0.26/0.45 ( ( member @ A @ A2 @ A5 )
% 0.26/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Id_onI
% 0.26/0.45 thf(fact_44_set__of__graph__rules__def,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType] :
% 0.26/0.45 ( ( set_of_graph_rules @ L @ V )
% 0.26/0.45 = ( ^ [Rs: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) )] :
% 0.26/0.45 ! [X6: product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) @ X6 @ Rs )
% 0.26/0.45 => ( ( graph_homomorphism @ L @ V @ V @ ( product_fst @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) @ ( id_on @ V @ ( labeled_vertices @ L @ V @ ( product_fst @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) ) )
% 0.26/0.45 & ( ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 )
% 0.26/0.45 = ( restrict @ L @ V @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) )
% 0.26/0.45 & ( finite_finite2 @ V @ ( labeled_vertices @ L @ V @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ L @ ( product_prod @ V @ V ) ) @ ( labeled_edges @ L @ V @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % set_of_graph_rules_def
% 0.26/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.26/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.26/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.26/0.45 = ( P @ A2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % mem_Collect_eq
% 0.26/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.26/0.45 ! [A: $tType,A5: set @ A] :
% 0.26/0.45 ( ( collect @ A
% 0.26/0.45 @ ^ [X6: A] : ( member @ A @ X6 @ A5 ) )
% 0.26/0.45 = A5 ) ).
% 0.26/0.45
% 0.26/0.45 % Collect_mem_eq
% 0.26/0.45 thf(fact_47_Collect__cong,axiom,
% 0.26/0.45 ! [A: $tType,P: A > $o,Q2: A > $o] :
% 0.26/0.45 ( ! [X5: A] :
% 0.26/0.45 ( ( P @ X5 )
% 0.26/0.45 = ( Q2 @ X5 ) )
% 0.26/0.45 => ( ( collect @ A @ P )
% 0.26/0.45 = ( collect @ A @ Q2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Collect_cong
% 0.26/0.45 thf(fact_48_verts__in__translation,axiom,
% 0.26/0.45 ! [A: $tType,X2: allegorical_term @ A] : ( inv_translation @ ( labeled_vertices @ A @ nat @ ( translation @ A @ X2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % verts_in_translation
% 0.26/0.45 thf(fact_49_old_Oprod_Orec,axiom,
% 0.26/0.45 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.26/0.45 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.26/0.45 = ( F1 @ A2 @ B2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % old.prod.rec
% 0.26/0.45 thf(fact_50_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,P: A > B > $o,X4: A,Y3: B,A2: product_prod @ A @ B] :
% 0.26/0.45 ( ( P @ X4 @ Y3 )
% 0.26/0.45 => ( ( A2
% 0.26/0.45 = ( product_Pair @ A @ B @ X4 @ Y3 ) )
% 0.26/0.45 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % BNF_Greatest_Fixpoint.subst_Pair
% 0.26/0.45 thf(fact_51_conjI__realizer,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q2: B > $o,Q: B] :
% 0.26/0.45 ( ( P @ P2 )
% 0.26/0.45 => ( ( Q2 @ Q )
% 0.26/0.45 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
% 0.26/0.45 & ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % conjI_realizer
% 0.26/0.45 thf(fact_52_exI__realizer,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,P: A > B > $o,Y3: A,X4: B] :
% 0.26/0.45 ( ( P @ Y3 @ X4 )
% 0.26/0.45 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % exI_realizer
% 0.26/0.45 thf(fact_53_allegorical__term_Oinject_I1_J,axiom,
% 0.26/0.45 ! [V: $tType,X11: allegorical_term @ V,X12: allegorical_term @ V,Y11: allegorical_term @ V,Y12: allegorical_term @ V] :
% 0.26/0.45 ( ( ( allegorical_A_Int @ V @ X11 @ X12 )
% 0.26/0.45 = ( allegorical_A_Int @ V @ Y11 @ Y12 ) )
% 0.26/0.45 = ( ( X11 = Y11 )
% 0.26/0.45 & ( X12 = Y12 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % allegorical_term.inject(1)
% 0.26/0.45 thf(fact_54_fin__maintainedI,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C] :
% 0.26/0.45 ( ! [F3: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ( ( F3
% 0.26/0.45 = ( restrict @ A @ B @ F3 ) )
% 0.26/0.45 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ F3 ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ F3 ) ) )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ B @ F3 @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F3 ) ) )
% 0.26/0.45 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ F3 @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) @ G @ F2 )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ C @ F3 @ G @ F2 )
% 0.26/0.45 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ F3 @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) @ G @ F2 ) ) ) ) )
% 0.26/0.45 => ( fin_maintained @ A @ B @ C @ R @ G ) ) ).
% 0.26/0.45
% 0.26/0.45 % fin_maintainedI
% 0.26/0.45 thf(fact_55_set__of__graph__rulesD_I1_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( set_of_graph_rules @ A @ B @ Rs2 )
% 0.26/0.45 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 0.26/0.45 => ( ( ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R )
% 0.26/0.45 = ( restrict @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 0.26/0.45 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % set_of_graph_rulesD(1)
% 0.26/0.45 thf(fact_56_set__of__graph__rulesD_I2_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( set_of_graph_rules @ A @ B @ Rs2 )
% 0.26/0.45 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 0.26/0.45 => ( ( ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R )
% 0.26/0.45 = ( restrict @ A @ B @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 0.26/0.45 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % set_of_graph_rulesD(2)
% 0.26/0.45 thf(fact_57_extensible__refl,axiom,
% 0.26/0.45 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ C @ R @ G @ F4 )
% 0.26/0.45 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G @ F4 ) ) ).
% 0.26/0.45
% 0.26/0.45 % extensible_refl
% 0.26/0.45 thf(fact_58_set__of__graph__rulesD_I3_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( set_of_graph_rules @ A @ B @ Rs2 )
% 0.26/0.45 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 0.26/0.45 => ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % set_of_graph_rulesD(3)
% 0.26/0.45 thf(fact_59_exE__realizer_H,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 0.26/0.45 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.26/0.45 => ~ ! [X5: B,Y4: A] :
% 0.26/0.45 ~ ( P @ Y4 @ X5 ) ) ).
% 0.26/0.45
% 0.26/0.45 % exE_realizer'
% 0.26/0.45 thf(fact_60_Id__onE,axiom,
% 0.26/0.45 ! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
% 0.26/0.45 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
% 0.26/0.45 => ~ ! [X5: A] :
% 0.26/0.45 ( ( member @ A @ X5 @ A5 )
% 0.26/0.45 => ( C3
% 0.26/0.45 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Id_onE
% 0.26/0.45 thf(fact_61_Id__on__eqI,axiom,
% 0.26/0.45 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.26/0.45 ( ( A2 = B2 )
% 0.26/0.45 => ( ( member @ A @ A2 @ A5 )
% 0.26/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Id_on_eqI
% 0.26/0.45 thf(fact_62_Id__on__iff,axiom,
% 0.26/0.45 ! [A: $tType,X4: A,Y3: A,A5: set @ A] :
% 0.26/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( id_on @ A @ A5 ) )
% 0.26/0.45 = ( ( X4 = Y3 )
% 0.26/0.45 & ( member @ A @ X4 @ A5 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Id_on_iff
% 0.26/0.45 thf(fact_63_fin__maintained__def,axiom,
% 0.26/0.45 ! [V: $tType,X: $tType,L: $tType] :
% 0.26/0.45 ( ( fin_maintained @ L @ X @ V )
% 0.26/0.45 = ( ^ [R2: product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ),G4: labeled_graph @ L @ V] :
% 0.26/0.45 ! [F5: labeled_graph @ L @ X,F6: set @ ( product_prod @ X @ V )] :
% 0.26/0.45 ( ( ( F5
% 0.26/0.45 = ( restrict @ L @ X @ F5 ) )
% 0.26/0.45 & ( finite_finite2 @ X @ ( labeled_vertices @ L @ X @ F5 ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ L @ ( product_prod @ X @ X ) ) @ ( labeled_edges @ L @ X @ F5 ) ) )
% 0.26/0.45 => ( ( graph_homomorphism @ L @ X @ X @ F5 @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ ( id_on @ X @ ( labeled_vertices @ L @ X @ F5 ) ) )
% 0.26/0.45 => ( ( extensible @ L @ X @ V @ ( product_Pair @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ F5 @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) ) @ G4 @ F6 )
% 0.26/0.45 => ( ( graph_homomorphism @ L @ X @ V @ F5 @ G4 @ F6 )
% 0.26/0.45 => ( extensible @ L @ X @ V @ ( product_Pair @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ F5 @ ( product_snd @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) ) @ G4 @ F6 ) ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % fin_maintained_def
% 0.26/0.45 thf(fact_64_fin__maintained__maintained,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C] :
% 0.26/0.45 ( ( ( ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R )
% 0.26/0.45 = ( restrict @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 0.26/0.45 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) )
% 0.26/0.45 => ( ( fin_maintained @ A @ B @ C @ R @ G )
% 0.26/0.45 = ( maintained @ A @ B @ C @ R @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % fin_maintained_maintained
% 0.26/0.45 thf(fact_65_consequence__graphD_I2_J,axiom,
% 0.26/0.45 ! [C: $tType,B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( consequence_graph @ A @ B @ C @ Rs2 @ G )
% 0.26/0.45 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 0.26/0.45 => ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % consequence_graphD(2)
% 0.26/0.45 thf(fact_66_finite__code,axiom,
% 0.26/0.45 ! [A: $tType] :
% 0.26/0.45 ( ( finite_finite @ A )
% 0.26/0.45 => ( ( finite_finite2 @ A )
% 0.26/0.45 = ( ^ [A6: set @ A] : $true ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % finite_code
% 0.26/0.45 thf(fact_67_fair__chainD_I2_J,axiom,
% 0.26/0.45 ! [A: $tType,C: $tType,B: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S2: nat > ( labeled_graph @ A @ C ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),I: nat,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ( fair_chain @ A @ B @ C @ Rs2 @ S2 )
% 0.26/0.45 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ C @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( S2 @ I ) @ F4 )
% 0.26/0.45 => ? [J: nat] : ( extensible @ A @ B @ C @ R @ ( S2 @ J ) @ F4 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % fair_chainD(2)
% 0.26/0.45 thf(fact_68_identity__rules__graph__rule,axiom,
% 0.26/0.45 ! [A: $tType,X4: product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ),L2: set @ ( standard_Constant @ A )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ X4 @ ( standa2002409347_rules @ A @ L2 ) )
% 0.26/0.45 => ( ( graph_homomorphism @ ( standard_Constant @ A ) @ nat @ nat @ ( product_fst @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) @ ( id_on @ nat @ ( labeled_vertices @ ( standard_Constant @ A ) @ nat @ ( product_fst @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) ) )
% 0.26/0.45 & ( ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 )
% 0.26/0.45 = ( restrict @ ( standard_Constant @ A ) @ nat @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) )
% 0.26/0.45 & ( finite_finite2 @ nat @ ( labeled_vertices @ ( standard_Constant @ A ) @ nat @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ ( standard_Constant @ A ) @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ ( standard_Constant @ A ) @ nat @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % identity_rules_graph_rule
% 0.26/0.45 thf(fact_69_maintainedI,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B5: labeled_graph @ C @ A] :
% 0.26/0.45 ( ! [F2: set @ ( product_prod @ A @ B )] :
% 0.26/0.45 ( ( graph_homomorphism @ C @ A @ B @ A5 @ G @ F2 )
% 0.26/0.45 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G @ F2 ) )
% 0.26/0.45 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G ) ) ).
% 0.26/0.45
% 0.26/0.45 % maintainedI
% 0.26/0.45 thf(fact_70_sndI,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,X4: product_prod @ A @ B,Y3: A,Z2: B] :
% 0.26/0.45 ( ( X4
% 0.26/0.45 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 0.26/0.45 => ( ( product_snd @ A @ B @ X4 )
% 0.26/0.45 = Z2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % sndI
% 0.26/0.45 thf(fact_71_finite__identity__rules,axiom,
% 0.26/0.45 ! [A: $tType,L2: set @ ( standard_Constant @ A )] :
% 0.26/0.45 ( ( finite_finite2 @ ( standard_Constant @ A ) @ L2 )
% 0.26/0.45 => ( finite_finite2 @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa2002409347_rules @ A @ L2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % finite_identity_rules
% 0.26/0.45 thf(fact_72_maintained__refl,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,R: labeled_graph @ A @ B,G: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G ) ).
% 0.26/0.45
% 0.26/0.45 % maintained_refl
% 0.26/0.45 thf(fact_73_consequence__graphI,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 0.26/0.45 ( ! [R3: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R3 @ Rs2 )
% 0.26/0.45 => ( maintained @ A @ B @ C @ R3 @ G ) )
% 0.26/0.45 => ( ! [R3: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R3 @ Rs2 )
% 0.26/0.45 => ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) ) ) )
% 0.26/0.45 => ( ( G
% 0.26/0.45 = ( restrict @ A @ C @ G ) )
% 0.26/0.45 => ( consequence_graph @ A @ B @ C @ Rs2 @ G ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % consequence_graphI
% 0.26/0.45 thf(fact_74_consequence__graphD_I1_J,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( consequence_graph @ A @ B @ C @ Rs2 @ G )
% 0.26/0.45 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 0.26/0.45 => ( maintained @ A @ B @ C @ R @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % consequence_graphD(1)
% 0.26/0.45 thf(fact_75_consequence__graphD_I3_J,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 0.26/0.45 ( ( consequence_graph @ A @ B @ C @ Rs2 @ G )
% 0.26/0.45 => ( G
% 0.26/0.45 = ( restrict @ A @ C @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % consequence_graphD(3)
% 0.26/0.45 thf(fact_76_maintained__then__fin__maintained,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C] :
% 0.26/0.45 ( ( maintained @ A @ B @ C @ R @ G )
% 0.26/0.45 => ( fin_maintained @ A @ B @ C @ R @ G ) ) ).
% 0.26/0.45
% 0.26/0.45 % maintained_then_fin_maintained
% 0.26/0.45 thf(fact_77_finite,axiom,
% 0.26/0.45 ! [A: $tType] :
% 0.26/0.45 ( ( finite_finite @ A )
% 0.26/0.45 => ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% 0.26/0.45
% 0.26/0.45 % finite
% 0.26/0.45 thf(fact_78_finite__set__choice,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
% 0.26/0.45 ( ( finite_finite2 @ A @ A5 )
% 0.26/0.45 => ( ! [X5: A] :
% 0.26/0.45 ( ( member @ A @ X5 @ A5 )
% 0.26/0.45 => ? [X_1: B] : ( P @ X5 @ X_1 ) )
% 0.26/0.45 => ? [F2: A > B] :
% 0.26/0.45 ! [X7: A] :
% 0.26/0.45 ( ( member @ A @ X7 @ A5 )
% 0.26/0.45 => ( P @ X7 @ ( F2 @ X7 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % finite_set_choice
% 0.26/0.45 thf(fact_79_consequence__graph__def,axiom,
% 0.26/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.45 ( ( consequence_graph @ A @ B @ C )
% 0.26/0.45 = ( ^ [Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G4: labeled_graph @ A @ C] :
% 0.26/0.45 ( ( G4
% 0.26/0.45 = ( restrict @ A @ C @ G4 ) )
% 0.26/0.45 & ! [X6: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ X6 @ Rs )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ X6 ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ X6 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ X6 ) ) ) )
% 0.26/0.45 & ( maintained @ A @ B @ C @ X6 @ G4 ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % consequence_graph_def
% 0.26/0.45 thf(fact_80_maintained__def,axiom,
% 0.26/0.45 ! [V: $tType,X: $tType,L: $tType] :
% 0.26/0.45 ( ( maintained @ L @ X @ V )
% 0.26/0.45 = ( ^ [R2: product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ),G4: labeled_graph @ L @ V] :
% 0.26/0.45 ! [F6: set @ ( product_prod @ X @ V )] :
% 0.26/0.45 ( ( graph_homomorphism @ L @ X @ V @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ G4 @ F6 )
% 0.26/0.45 => ( extensible @ L @ X @ V @ R2 @ G4 @ F6 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % maintained_def
% 0.26/0.45 thf(fact_81_maintainedD,axiom,
% 0.26/0.45 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F4 )
% 0.26/0.45 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G @ F4 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % maintainedD
% 0.26/0.45 thf(fact_82_fstI,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,Y3: A,Z2: B] :
% 0.26/0.45 ( ( X4
% 0.26/0.45 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 0.26/0.45 => ( ( product_fst @ A @ B @ X4 )
% 0.26/0.45 = Y3 ) ) ).
% 0.26/0.45
% 0.26/0.45 % fstI
% 0.26/0.45 thf(fact_83_least__consequence__graphI,axiom,
% 0.26/0.45 ! [X: $tType,A: $tType,C: $tType,L: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G: labeled_graph @ L @ C,S2: labeled_graph @ L @ C,T2: itself @ X] :
% 0.26/0.45 ( ( consequence_graph @ L @ A @ C @ Rs2 @ G )
% 0.26/0.45 => ( ( graph_homomorphism @ L @ C @ C @ S2 @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 0.26/0.45 => ( ! [C4: labeled_graph @ L @ X] :
% 0.26/0.45 ( ( consequence_graph @ L @ A @ X @ Rs2 @ C4 )
% 0.26/0.45 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G ) @ C4 ) )
% 0.26/0.45 => ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs2 @ S2 @ G ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % least_consequence_graphI
% 0.26/0.45 thf(fact_84_leastI,axiom,
% 0.26/0.45 ! [X: $tType,A: $tType,C: $tType,L: $tType,S2: labeled_graph @ L @ C,G: labeled_graph @ L @ C,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T2: itself @ X] :
% 0.26/0.45 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 0.26/0.45 => ( ! [C4: labeled_graph @ L @ X] :
% 0.26/0.45 ( ( consequence_graph @ L @ A @ X @ Rs2 @ C4 )
% 0.26/0.45 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G ) @ C4 ) )
% 0.26/0.45 => ( least @ X @ L @ A @ C @ T2 @ Rs2 @ S2 @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % leastI
% 0.26/0.45 thf(fact_85_least__def,axiom,
% 0.26/0.45 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.26/0.45 ( ( least @ X @ L @ V @ C )
% 0.26/0.45 = ( ^ [Uu: itself @ X,Rs: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G4: labeled_graph @ L @ C] :
% 0.26/0.45 ( ( graph_homomorphism @ L @ C @ C @ S3 @ G4 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S3 ) ) )
% 0.26/0.45 & ! [C5: labeled_graph @ L @ X] :
% 0.26/0.45 ( ( consequence_graph @ L @ V @ X @ Rs @ C5 )
% 0.26/0.45 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S3 @ G4 ) @ C5 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % least_def
% 0.26/0.45 thf(fact_86_fair__chain__impl__consequence__graph,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S2: nat > ( labeled_graph @ A @ C )] :
% 0.26/0.45 ( ( fair_chain @ A @ B @ C @ Rs2 @ S2 )
% 0.26/0.45 => ( ! [R3: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R3 @ Rs2 )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) ) )
% 0.26/0.45 & ( ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 )
% 0.26/0.45 = ( restrict @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) )
% 0.26/0.45 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) )
% 0.26/0.45 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) ) ) )
% 0.26/0.45 => ( consequence_graph @ A @ B @ C @ Rs2 @ ( chain_sup @ A @ C @ S2 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % fair_chain_impl_consequence_graph
% 0.26/0.45 thf(fact_87_fair__chainI,axiom,
% 0.26/0.45 ! [C: $tType,B: $tType,A: $tType,S2: nat > ( labeled_graph @ A @ B ),Rs2: set @ ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) )] :
% 0.26/0.45 ( ( chain @ A @ B @ S2 )
% 0.26/0.45 => ( ! [R3: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),F2: set @ ( product_prod @ C @ B ),I2: nat] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) ) @ R3 @ Rs2 )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ C @ B @ ( product_fst @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R3 ) @ ( S2 @ I2 ) @ F2 )
% 0.26/0.45 => ? [J2: nat] : ( extensible @ A @ C @ B @ R3 @ ( S2 @ J2 ) @ F2 ) ) )
% 0.26/0.45 => ( fair_chain @ A @ C @ B @ Rs2 @ S2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % fair_chainI
% 0.26/0.45 thf(fact_88_maintained__holds__iff,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,E_L: allegorical_term @ A,E_R: allegorical_term @ A] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( maintained @ A @ nat @ B @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ E_L ) @ ( translation @ A @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) @ G )
% 0.26/0.46 = ( ( semantics @ A @ B @ G @ ( product_fst @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) )
% 0.26/0.46 = ( semantics @ A @ B @ G @ ( product_snd @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintained_holds_iff
% 0.26/0.46 thf(fact_89_fair__chain__def,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.46 ( ( fair_chain @ A @ B @ C )
% 0.26/0.46 = ( ^ [Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S3: nat > ( labeled_graph @ A @ C )] :
% 0.26/0.46 ( ( chain @ A @ C @ S3 )
% 0.26/0.46 & ! [R2: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),F6: set @ ( product_prod @ B @ C ),I3: nat] :
% 0.26/0.46 ( ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
% 0.26/0.46 & ( graph_homomorphism @ A @ B @ C @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 ) @ ( S3 @ I3 ) @ F6 ) )
% 0.26/0.46 => ? [J3: nat] : ( extensible @ A @ B @ C @ R2 @ ( S3 @ J3 ) @ F6 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fair_chain_def
% 0.26/0.46 thf(fact_90_chain__sup__graph,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,S2: nat > ( labeled_graph @ A @ B )] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( ( chain_sup @ A @ B @ S2 )
% 0.26/0.46 = ( restrict @ A @ B @ ( chain_sup @ A @ B @ S2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % chain_sup_graph
% 0.26/0.46 thf(fact_91_extensible__chain__sup,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,S2: nat > ( labeled_graph @ A @ B ),R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),J4: nat,F4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( ( extensible @ A @ C @ B @ R @ ( S2 @ J4 ) @ F4 )
% 0.26/0.46 => ( extensible @ A @ C @ B @ R @ ( chain_sup @ A @ B @ S2 ) @ F4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensible_chain_sup
% 0.26/0.46 thf(fact_92_chain__sup__subgraph,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),J4: nat] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ B @ ( S2 @ J4 ) @ ( chain_sup @ A @ B @ S2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( S2 @ J4 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % chain_sup_subgraph
% 0.26/0.46 thf(fact_93_chain__then__restrict,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),I: nat] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( ( S2 @ I )
% 0.26/0.46 = ( restrict @ A @ B @ ( S2 @ I ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % chain_then_restrict
% 0.26/0.46 thf(fact_94_least__consequence__graph__def,axiom,
% 0.26/0.46 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.26/0.46 ( ( least_559130134_graph @ X @ L @ V @ C )
% 0.26/0.46 = ( ^ [T3: itself @ X,Rs: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G4: labeled_graph @ L @ C] :
% 0.26/0.46 ( ( consequence_graph @ L @ V @ C @ Rs @ G4 )
% 0.26/0.46 & ( least @ X @ L @ V @ C @ T3 @ Rs @ S3 @ G4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % least_consequence_graph_def
% 0.26/0.46 thf(fact_95_graph__homomorphism__semantics,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F4 )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F4 )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F4 )
% 0.26/0.46 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B5 @ E3 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_semantics
% 0.26/0.46 thf(fact_96_fair__chainD_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S2: nat > ( labeled_graph @ A @ C )] :
% 0.26/0.46 ( ( fair_chain @ A @ B @ C @ Rs2 @ S2 )
% 0.26/0.46 => ( chain @ A @ C @ S2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % fair_chainD(1)
% 0.26/0.46 thf(fact_97_semantics__in__vertices_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.26/0.46 ( ( A5
% 0.26/0.46 = ( restrict @ A @ B @ A5 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.26/0.46 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % semantics_in_vertices(2)
% 0.26/0.46 thf(fact_98_semantics__in__vertices_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.26/0.46 ( ( A5
% 0.26/0.46 = ( restrict @ A @ B @ A5 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.26/0.46 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % semantics_in_vertices(1)
% 0.26/0.46 thf(fact_99_subgraph__semantics,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B5 @ E3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_semantics
% 0.26/0.46 thf(fact_100_eq__as__subsets,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A,E_1: allegorical_term @ B,E_2: allegorical_term @ B] :
% 0.26/0.46 ( ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ E_2 ) ) )
% 0.26/0.46 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ E_2 ) ) ) )
% 0.26/0.46 = ( ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) )
% 0.26/0.46 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) ) )
% 0.26/0.46 & ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_2 @ ( allegorical_A_Int @ B @ E_2 @ E_1 ) ) ) )
% 0.26/0.46 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_2 @ ( allegorical_A_Int @ B @ E_2 @ E_1 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_as_subsets
% 0.26/0.46 thf(fact_101_maintained__holds__subset__iff,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,E_L: allegorical_term @ A,E_R: allegorical_term @ A] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( maintained @ A @ nat @ B @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ ( product_fst @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) ) @ ( translation @ A @ ( product_snd @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) ) ) @ G )
% 0.26/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( semantics @ A @ B @ G @ E_L ) @ ( semantics @ A @ B @ G @ E_R ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintained_holds_subset_iff
% 0.26/0.46 thf(fact_102_maintained__holds,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A,E_L: allegorical_term @ B,E_R: allegorical_term @ B] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( semantics @ B @ A @ G @ E_L ) @ ( semantics @ B @ A @ G @ E_R ) )
% 0.26/0.46 => ( maintained @ B @ nat @ A @ ( product_Pair @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( translation @ B @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_L @ ( allegorical_A_Int @ B @ E_L @ E_R ) ) ) ) @ ( translation @ B @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_L @ ( allegorical_A_Int @ B @ E_L @ E_R ) ) ) ) ) @ G ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintained_holds
% 0.26/0.46 thf(fact_103_sentence__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A,E_1: allegorical_term @ B,E_2: allegorical_term @ B] :
% 0.26/0.46 ( ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) )
% 0.26/0.46 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) ) )
% 0.26/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( semantics @ B @ A @ G @ E_1 ) @ ( semantics @ B @ A @ G @ E_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % sentence_iff
% 0.26/0.46 thf(fact_104_find__graph__occurence,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,S2: nat > ( labeled_graph @ A @ B ),E4: set @ ( product_prod @ A @ ( product_prod @ C @ C ) ),V2: set @ C,F4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ E4 )
% 0.26/0.46 => ( ( finite_finite2 @ C @ V2 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ C @ B @ ( labeled_LG @ A @ C @ E4 @ V2 ) @ ( chain_sup @ A @ B @ S2 ) @ F4 )
% 0.26/0.46 => ? [I2: nat] : ( graph_homomorphism @ A @ C @ B @ ( labeled_LG @ A @ C @ E4 @ V2 ) @ ( S2 @ I2 ) @ F4 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % find_graph_occurence
% 0.26/0.46 thf(fact_105_eq__snd__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.26/0.46 ( ( B2
% 0.26/0.46 = ( product_snd @ B @ A @ P2 ) )
% 0.26/0.46 = ( ? [A7: B] :
% 0.26/0.46 ( P2
% 0.26/0.46 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_snd_iff
% 0.26/0.46 thf(fact_106_eq__fst__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( product_fst @ A @ B @ P2 ) )
% 0.26/0.46 = ( ? [B6: B] :
% 0.26/0.46 ( P2
% 0.26/0.46 = ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_fst_iff
% 0.26/0.46 thf(fact_107_labeled__graph_Oinject,axiom,
% 0.26/0.46 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V,Y1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),Y2: set @ V] :
% 0.26/0.46 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 0.26/0.46 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 0.26/0.46 = ( ( X1 = Y1 )
% 0.26/0.46 & ( X22 = Y2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.inject
% 0.26/0.46 thf(fact_108_labeled__graph_Ocollapse,axiom,
% 0.26/0.46 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 0.26/0.46 = Labeled_graph ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.collapse
% 0.26/0.46 thf(fact_109_extensible__refl__concr,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V3: set @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ G @ F4 )
% 0.26/0.46 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ ( labeled_LG @ A @ B @ E_2 @ V3 ) ) @ G @ F4 )
% 0.26/0.46 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V3 ) @ G @ F4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensible_refl_concr
% 0.26/0.46 thf(fact_110_labeled__graph_Oexhaust,axiom,
% 0.26/0.46 ! [L: $tType,V: $tType,Y3: labeled_graph @ L @ V] :
% 0.26/0.46 ~ ! [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( labeled_LG @ L @ V @ X13 @ X23 ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.exhaust
% 0.26/0.46 thf(fact_111_labeled__graph_Oinduct,axiom,
% 0.26/0.46 ! [V: $tType,L: $tType,P: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 0.26/0.46 => ( P @ Labeled_graph ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.induct
% 0.26/0.46 thf(fact_112_subrelI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ! [X5: A,Y4: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R4 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S4 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % subrelI
% 0.26/0.46 thf(fact_113_finite__has__maximal2,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A5: set @ A,A2: A] :
% 0.26/0.46 ( ( finite_finite2 @ A @ A5 )
% 0.26/0.46 => ( ( member @ A @ A2 @ A5 )
% 0.26/0.46 => ? [X5: A] :
% 0.26/0.46 ( ( member @ A @ X5 @ A5 )
% 0.26/0.46 & ( ord_less_eq @ A @ A2 @ X5 )
% 0.26/0.46 & ! [Xa: A] :
% 0.26/0.46 ( ( member @ A @ Xa @ A5 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ X5 @ Xa )
% 0.26/0.46 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_has_maximal2
% 0.26/0.46 thf(fact_114_finite__has__minimal2,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A5: set @ A,A2: A] :
% 0.26/0.46 ( ( finite_finite2 @ A @ A5 )
% 0.26/0.46 => ( ( member @ A @ A2 @ A5 )
% 0.26/0.46 => ? [X5: A] :
% 0.26/0.46 ( ( member @ A @ X5 @ A5 )
% 0.26/0.46 & ( ord_less_eq @ A @ X5 @ A2 )
% 0.26/0.46 & ! [Xa: A] :
% 0.26/0.46 ( ( member @ A @ Xa @ A5 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ Xa @ X5 )
% 0.26/0.46 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_has_minimal2
% 0.26/0.46 thf(fact_115_finite__subset,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ( ( finite_finite2 @ A @ B5 )
% 0.26/0.46 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_subset
% 0.26/0.46 thf(fact_116_infinite__super,axiom,
% 0.26/0.46 ! [A: $tType,S2: set @ A,T4: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
% 0.26/0.46 => ( ~ ( finite_finite2 @ A @ S2 )
% 0.26/0.46 => ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % infinite_super
% 0.26/0.46 thf(fact_117_rev__finite__subset,axiom,
% 0.26/0.46 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 0.26/0.46 ( ( finite_finite2 @ A @ B5 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rev_finite_subset
% 0.26/0.46 thf(fact_118_labeled__graph_Osel_I2_J,axiom,
% 0.26/0.46 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.26/0.46 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.26/0.46 = X22 ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.sel(2)
% 0.26/0.46 thf(fact_119_labeled__graph_Osel_I1_J,axiom,
% 0.26/0.46 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.26/0.46 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.26/0.46 = X1 ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.sel(1)
% 0.26/0.46 thf(fact_120_restrict__subsD,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G ) ) )
% 0.26/0.46 => ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % restrict_subsD
% 0.26/0.46 thf(fact_121_subgraph__def2,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( G_1
% 0.26/0.46 = ( restrict @ A @ B @ G_1 ) )
% 0.26/0.46 => ( ( G_2
% 0.26/0.46 = ( restrict @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 = ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_def2
% 0.26/0.46 thf(fact_122_labeled__graph_Oexhaust__sel,axiom,
% 0.26/0.46 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( Labeled_graph
% 0.26/0.46 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.exhaust_sel
% 0.26/0.46 thf(fact_123_subgraph__subset_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_subset(1)
% 0.26/0.46 thf(fact_124_maintainedD2,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F4 )
% 0.26/0.46 => ~ ! [G3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ B5 @ G @ G3 )
% 0.26/0.46 => ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F4 @ G3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintainedD2
% 0.26/0.46 thf(fact_125_subgraph__subset_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G_1 ) ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_subset(2)
% 0.26/0.46 thf(fact_126_pushout__step__def,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,X: $tType] :
% 0.26/0.46 ( ( pushout_step @ X @ A @ C @ B )
% 0.26/0.46 = ( ^ [T3: itself @ X,R2: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ G_12 @ G_22 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_12 ) ) )
% 0.26/0.46 & ? [F_1: set @ ( product_prod @ C @ B ),F_2: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ C @ B @ ( product_fst @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R2 ) @ G_12 @ F_1 )
% 0.26/0.46 & ( graph_homomorphism @ A @ C @ B @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R2 ) @ G_22 @ F_2 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ C @ B ) ) @ F_1 @ F_2 )
% 0.26/0.46 & ( weak_universal @ X @ A @ C @ B @ T3 @ R2 @ G_12 @ G_22 @ F_1 @ F_2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % pushout_step_def
% 0.26/0.46 thf(fact_127_subsetI,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.26/0.46 ( ! [X5: A] :
% 0.26/0.46 ( ( member @ A @ X5 @ A5 )
% 0.26/0.46 => ( member @ A @ X5 @ B5 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % subsetI
% 0.26/0.46 thf(fact_128_subset__antisym,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.26/0.46 => ( A5 = B5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_antisym
% 0.26/0.46 thf(fact_129_order__refl,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( preorder @ A )
% 0.26/0.46 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % order_refl
% 0.26/0.46 thf(fact_130_labeled__graph_Osplit__sel__asm,axiom,
% 0.26/0.46 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F4 @ Labeled_graph ) )
% 0.26/0.46 = ( ~ ( ( Labeled_graph
% 0.26/0.46 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.26/0.46 & ~ ( P @ ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.split_sel_asm
% 0.26/0.46 thf(fact_131_labeled__graph_Ocase,axiom,
% 0.26/0.46 ! [L: $tType,A: $tType,V: $tType,F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.26/0.46 ( ( labele1974067554_graph @ L @ V @ A @ F4 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.26/0.46 = ( F4 @ X1 @ X22 ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.case
% 0.26/0.46 thf(fact_132_dual__order_Oantisym,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [B2: A,A2: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.26/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % dual_order.antisym
% 0.26/0.46 thf(fact_133_dual__order_Oeq__iff,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.26/0.46 = ( ^ [A7: A,B6: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ B6 @ A7 )
% 0.26/0.46 & ( ord_less_eq @ A @ A7 @ B6 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % dual_order.eq_iff
% 0.26/0.46 thf(fact_134_dual__order_Otrans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [B2: A,A2: A,C3: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.26/0.46 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % dual_order.trans
% 0.26/0.46 thf(fact_135_linorder__wlog,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.26/0.46 ( ! [A4: A,B4: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.26/0.46 => ( P @ A4 @ B4 ) )
% 0.26/0.46 => ( ! [A4: A,B4: A] :
% 0.26/0.46 ( ( P @ B4 @ A4 )
% 0.26/0.46 => ( P @ A4 @ B4 ) )
% 0.26/0.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % linorder_wlog
% 0.26/0.46 thf(fact_136_dual__order_Orefl,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % dual_order.refl
% 0.26/0.46 thf(fact_137_order__trans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( preorder @ A )
% 0.26/0.46 => ! [X4: A,Y3: A,Z2: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.26/0.46 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % order_trans
% 0.26/0.46 thf(fact_138_order__class_Oorder_Oantisym,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A2: A,B2: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.26/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % order_class.order.antisym
% 0.26/0.46 thf(fact_139_ord__le__eq__trans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [A2: A,B2: A,C3: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.26/0.46 => ( ( B2 = C3 )
% 0.26/0.46 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord_le_eq_trans
% 0.26/0.46 thf(fact_140_ord__eq__le__trans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [A2: A,B2: A,C3: A] :
% 0.26/0.46 ( ( A2 = B2 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.26/0.46 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord_eq_le_trans
% 0.26/0.46 thf(fact_141_order__class_Oorder_Oeq__iff,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.26/0.46 = ( ^ [A7: A,B6: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A7 @ B6 )
% 0.26/0.46 & ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % order_class.order.eq_iff
% 0.26/0.46 thf(fact_142_antisym__conv,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [Y3: A,X4: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.26/0.46 = ( X4 = Y3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % antisym_conv
% 0.26/0.46 thf(fact_143_le__cases3,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [X4: A,Y3: A,Z2: A] :
% 0.26/0.46 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.26/0.46 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.26/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.26/0.46 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.26/0.46 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 0.26/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.26/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.26/0.46 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 0.26/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.26/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 0.26/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 0.26/0.46 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % le_cases3
% 0.26/0.46 thf(fact_144_order_Otrans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A2: A,B2: A,C3: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.26/0.46 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % order.trans
% 0.26/0.46 thf(fact_145_le__cases,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [X4: A,Y3: A] :
% 0.26/0.46 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 0.26/0.46 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % le_cases
% 0.26/0.46 thf(fact_146_eq__refl,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( preorder @ A )
% 0.26/0.46 => ! [X4: A,Y3: A] :
% 0.26/0.46 ( ( X4 = Y3 )
% 0.26/0.46 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_refl
% 0.26/0.46 thf(fact_147_linear,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [X4: A,Y3: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.26/0.46 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % linear
% 0.26/0.46 thf(fact_148_antisym,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [X4: A,Y3: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.26/0.46 => ( X4 = Y3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % antisym
% 0.26/0.46 thf(fact_149_eq__iff,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.26/0.46 = ( ^ [X6: A,Y6: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ X6 @ Y6 )
% 0.26/0.46 & ( ord_less_eq @ A @ Y6 @ X6 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_iff
% 0.26/0.46 thf(fact_150_ord__le__eq__subst,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType] :
% 0.26/0.46 ( ( ( ord @ B )
% 0.26/0.46 & ( ord @ A ) )
% 0.26/0.46 => ! [A2: A,B2: A,F4: A > B,C3: B] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.26/0.46 => ( ( ( F4 @ B2 )
% 0.26/0.46 = C3 )
% 0.26/0.46 => ( ! [X5: A,Y4: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 0.26/0.46 => ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 0.26/0.46 => ( ord_less_eq @ B @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord_le_eq_subst
% 0.26/0.46 thf(fact_151_ord__eq__le__subst,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType] :
% 0.26/0.46 ( ( ( ord @ B )
% 0.26/0.46 & ( ord @ A ) )
% 0.26/0.46 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( F4 @ B2 ) )
% 0.26/0.46 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.26/0.46 => ( ! [X5: B,Y4: B] :
% 0.26/0.46 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 0.26/0.46 => ( ord_less_eq @ A @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 0.26/0.46 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord_eq_le_subst
% 0.26/0.46 thf(fact_152_order__subst2,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType] :
% 0.26/0.46 ( ( ( order @ C )
% 0.26/0.46 & ( order @ A ) )
% 0.26/0.46 => ! [A2: A,B2: A,F4: A > C,C3: C] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.26/0.46 => ( ( ord_less_eq @ C @ ( F4 @ B2 ) @ C3 )
% 0.26/0.46 => ( ! [X5: A,Y4: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 0.26/0.46 => ( ord_less_eq @ C @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 0.26/0.46 => ( ord_less_eq @ C @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % order_subst2
% 0.26/0.46 thf(fact_153_order__subst1,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType] :
% 0.26/0.46 ( ( ( order @ B )
% 0.26/0.46 & ( order @ A ) )
% 0.26/0.46 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ ( F4 @ B2 ) )
% 0.26/0.46 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.26/0.46 => ( ! [X5: B,Y4: B] :
% 0.26/0.46 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 0.26/0.46 => ( ord_less_eq @ A @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 0.26/0.46 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % order_subst1
% 0.26/0.46 thf(fact_154_le__fun__def,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( ord @ B )
% 0.26/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.26/0.46 = ( ^ [F6: A > B,G5: A > B] :
% 0.26/0.46 ! [X6: A] : ( ord_less_eq @ B @ ( F6 @ X6 ) @ ( G5 @ X6 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % le_fun_def
% 0.26/0.46 thf(fact_155_le__funI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( ord @ B )
% 0.26/0.46 => ! [F4: A > B,G6: A > B] :
% 0.26/0.46 ( ! [X5: A] : ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( G6 @ X5 ) )
% 0.26/0.46 => ( ord_less_eq @ ( A > B ) @ F4 @ G6 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % le_funI
% 0.26/0.46 thf(fact_156_le__funE,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( ord @ B )
% 0.26/0.46 => ! [F4: A > B,G6: A > B,X4: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( A > B ) @ F4 @ G6 )
% 0.26/0.46 => ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G6 @ X4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % le_funE
% 0.26/0.46 thf(fact_157_le__funD,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( ord @ B )
% 0.26/0.46 => ! [F4: A > B,G6: A > B,X4: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( A > B ) @ F4 @ G6 )
% 0.26/0.46 => ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G6 @ X4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % le_funD
% 0.26/0.46 thf(fact_158_Collect__mono__iff,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,Q2: A > $o] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) )
% 0.26/0.46 = ( ! [X6: A] :
% 0.26/0.46 ( ( P @ X6 )
% 0.26/0.46 => ( Q2 @ X6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_mono_iff
% 0.26/0.46 thf(fact_159_set__eq__subset,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 0.26/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % set_eq_subset
% 0.26/0.46 thf(fact_160_subset__trans,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A,C6: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ C6 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_trans
% 0.26/0.46 thf(fact_161_Collect__mono,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,Q2: A > $o] :
% 0.26/0.46 ( ! [X5: A] :
% 0.26/0.46 ( ( P @ X5 )
% 0.26/0.46 => ( Q2 @ X5 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_mono
% 0.26/0.46 thf(fact_162_subset__refl,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.26/0.46
% 0.26/0.46 % subset_refl
% 0.26/0.46 thf(fact_163_subset__iff,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.26/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.26/0.46 ! [T3: A] :
% 0.26/0.46 ( ( member @ A @ T3 @ A6 )
% 0.26/0.46 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_iff
% 0.26/0.46 thf(fact_164_equalityD2,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.26/0.46 ( ( A5 = B5 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % equalityD2
% 0.26/0.46 thf(fact_165_equalityD1,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.26/0.46 ( ( A5 = B5 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % equalityD1
% 0.26/0.46 thf(fact_166_subset__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.26/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.26/0.46 ! [X6: A] :
% 0.26/0.46 ( ( member @ A @ X6 @ A6 )
% 0.26/0.46 => ( member @ A @ X6 @ B7 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_eq
% 0.26/0.46 thf(fact_167_equalityE,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.26/0.46 ( ( A5 = B5 )
% 0.26/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equalityE
% 0.26/0.46 thf(fact_168_subsetD,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ( ( member @ A @ C3 @ A5 )
% 0.26/0.46 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subsetD
% 0.26/0.46 thf(fact_169_in__mono,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A,B5: set @ A,X4: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.26/0.46 => ( ( member @ A @ X4 @ A5 )
% 0.26/0.46 => ( member @ A @ X4 @ B5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % in_mono
% 0.26/0.46 thf(fact_170_labeled__graph_Ocase__eq__if,axiom,
% 0.26/0.46 ! [A: $tType,V: $tType,L: $tType] :
% 0.26/0.46 ( ( labele1974067554_graph @ L @ V @ A )
% 0.26/0.46 = ( ^ [F6: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F6 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.case_eq_if
% 0.26/0.46 thf(fact_171_chain,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),I: nat,J4: nat] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( ( ord_less_eq @ nat @ I @ J4 )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ B @ ( S2 @ I ) @ ( S2 @ J4 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( S2 @ I ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % chain
% 0.26/0.46 thf(fact_172_chain__def2,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( chain @ A @ B )
% 0.26/0.46 = ( ^ [S3: nat > ( labeled_graph @ A @ B )] :
% 0.26/0.46 ! [I3: nat,J3: nat] :
% 0.26/0.46 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ B @ ( S3 @ I3 ) @ ( S3 @ J3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( S3 @ I3 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % chain_def2
% 0.26/0.46 thf(fact_173_labeled__graph_Osplit__sel,axiom,
% 0.26/0.46 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F4 @ Labeled_graph ) )
% 0.26/0.46 = ( ( Labeled_graph
% 0.26/0.46 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.26/0.46 => ( P @ ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.split_sel
% 0.26/0.46 thf(fact_174_graph__unionI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.46 = G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_unionI
% 0.26/0.46 thf(fact_175_extensibleD,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( extensible @ A @ B @ C @ R @ G @ F4 )
% 0.26/0.46 => ~ ! [G3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ G @ G3 )
% 0.26/0.46 => ~ ( agree_on @ A @ B @ C @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ F4 @ G3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensibleD
% 0.26/0.46 thf(fact_176_extensible__def,axiom,
% 0.26/0.46 ! [V: $tType,X: $tType,L: $tType] :
% 0.26/0.46 ( ( extensible @ L @ X @ V )
% 0.26/0.46 = ( ^ [R2: product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ),G4: labeled_graph @ L @ V,F6: set @ ( product_prod @ X @ V )] :
% 0.26/0.46 ? [G5: set @ ( product_prod @ X @ V )] :
% 0.26/0.46 ( ( graph_homomorphism @ L @ X @ V @ ( product_snd @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ G4 @ G5 )
% 0.26/0.46 & ( agree_on @ L @ X @ V @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ F6 @ G5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensible_def
% 0.26/0.46 thf(fact_177_extensibleI,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G6: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G6 )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ R1 @ F4 @ G6 )
% 0.26/0.46 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensibleI
% 0.26/0.46 thf(fact_178_graph__union__idemp_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_union @ A @ B @ A5 @ A5 )
% 0.26/0.46 = A5 ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_idemp(1)
% 0.26/0.46 thf(fact_179_graph__union__idemp_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B5 ) )
% 0.26/0.46 = ( graph_union @ A @ B @ A5 @ B5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_idemp(2)
% 0.26/0.46 thf(fact_180_graph__union__idemp_I3_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B5 @ A5 ) )
% 0.26/0.46 = ( graph_union @ A @ B @ B5 @ A5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_idemp(3)
% 0.26/0.46 thf(fact_181_agree__on__refl,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R @ F4 @ F4 ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_refl
% 0.26/0.46 thf(fact_182_agree__on__comm,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C )
% 0.26/0.46 = ( ^ [X3: labeled_graph @ A @ B,F6: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X3 @ G5 @ F6 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_comm
% 0.26/0.46 thf(fact_183_graph__union__preserves__restrict,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( G_1
% 0.26/0.46 = ( restrict @ A @ B @ G_1 ) )
% 0.26/0.46 => ( ( G_2
% 0.26/0.46 = ( restrict @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.46 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_preserves_restrict
% 0.26/0.46 thf(fact_184_graph__union__chain__sup,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,S2: nat > ( labeled_graph @ A @ B ),C6: labeled_graph @ A @ B] :
% 0.26/0.46 ( ! [I2: nat] :
% 0.26/0.46 ( ( graph_union @ A @ B @ ( S2 @ I2 ) @ C6 )
% 0.26/0.46 = C6 )
% 0.26/0.46 => ( ( graph_union @ A @ B @ ( chain_sup @ A @ B @ S2 ) @ C6 )
% 0.26/0.46 = C6 ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_chain_sup
% 0.26/0.46 thf(fact_185_agree__on__trans,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,X2: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C @ X2 @ F4 @ G6 )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ X2 @ G6 @ H )
% 0.26/0.46 => ( agree_on @ A @ B @ C @ X2 @ F4 @ H ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_trans
% 0.26/0.46 thf(fact_186_agree__on__subg__compose,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G6: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F7: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C @ R @ G6 @ H )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ F7 @ F4 @ G6 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ B @ F7 @ R @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F7 ) ) )
% 0.26/0.46 => ( agree_on @ A @ B @ C @ F7 @ F4 @ H ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_subg_compose
% 0.26/0.46 thf(fact_187_subgraph__def,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 = ( ( G_1
% 0.26/0.46 = ( restrict @ A @ B @ G_1 ) )
% 0.26/0.46 & ( G_2
% 0.26/0.46 = ( restrict @ A @ B @ G_2 ) )
% 0.26/0.46 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.46 = G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_def
% 0.26/0.46 thf(fact_188_graph__union__iff,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.46 = G_2 )
% 0.26/0.46 = ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.26/0.46 & ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_iff
% 0.26/0.46 thf(fact_189_finite__nat__set__iff__bounded__le,axiom,
% 0.26/0.46 ( ( finite_finite2 @ nat )
% 0.26/0.46 = ( ^ [N: set @ nat] :
% 0.26/0.46 ? [M: nat] :
% 0.26/0.46 ! [X6: nat] :
% 0.26/0.46 ( ( member @ nat @ X6 @ N )
% 0.26/0.46 => ( ord_less_eq @ nat @ X6 @ M ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_nat_set_iff_bounded_le
% 0.26/0.46 thf(fact_190_nonempty__rule,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
% 0.26/0.46 = ( ( labeled_vertices @ A @ B @ G )
% 0.26/0.46 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % nonempty_rule
% 0.26/0.46 thf(fact_191_weak__universalI,axiom,
% 0.26/0.46 ! [X: $tType,A: $tType,B: $tType,C: $tType,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_1: labeled_graph @ A @ B,F_12: set @ ( product_prod @ C @ B ),G_2: labeled_graph @ A @ B,T2: itself @ X,F_22: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ! [H_1: set @ ( product_prod @ C @ X ),H_2: set @ ( product_prod @ B @ X ),G7: labeled_graph @ A @ X] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ C @ X @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R ) @ G7 @ H_1 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ X @ G_1 @ G7 @ H_2 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ X ) ) @ ( relcomp @ C @ B @ X @ F_12 @ H_2 ) @ H_1 )
% 0.26/0.46 => ? [H2: set @ ( product_prod @ B @ X )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ X @ G_2 @ G7 @ H2 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ B @ X ) ) @ H_2 @ H2 ) ) ) ) )
% 0.26/0.46 => ( weak_universal @ X @ A @ C @ B @ T2 @ R @ G_1 @ G_2 @ F_12 @ F_22 ) ) ).
% 0.26/0.46
% 0.26/0.46 % weak_universalI
% 0.26/0.46 thf(fact_192_relcomp__empty1,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_empty1
% 0.26/0.46 thf(fact_193_relcomp__empty2,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] :
% 0.26/0.46 ( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_empty2
% 0.26/0.46 thf(fact_194_empty__subsetI,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 0.26/0.46
% 0.26/0.46 % empty_subsetI
% 0.26/0.46 thf(fact_195_subset__empty,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( A5
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_empty
% 0.26/0.46 thf(fact_196_Id__on__empty,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_empty
% 0.26/0.46 thf(fact_197_graph__homomorphism__composes,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,X4: set @ ( product_prod @ B @ C ),C3: labeled_graph @ A @ D,Y3: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ X4 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ C @ D @ B2 @ C3 @ Y3 )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ D @ A2 @ C3 @ ( relcomp @ B @ C @ D @ X4 @ Y3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_composes
% 0.26/0.46 thf(fact_198_graph__empty__e,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,V3: set @ B] :
% 0.26/0.46 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
% 0.26/0.46 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_empty_e
% 0.26/0.46 thf(fact_199_agree__on__empty,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,F4: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ F4 @ G6 ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_empty
% 0.26/0.46 thf(fact_200_graph__homomorphism__empty,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F4 )
% 0.26/0.46 = ( ( F4
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 0.26/0.46 & ( G
% 0.26/0.46 = ( restrict @ A @ C @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_empty
% 0.26/0.46 thf(fact_201_relcomp__mono,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R5: set @ ( product_prod @ A @ B ),R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C ),S4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ R4 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S5 @ S4 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R5 @ S5 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_mono
% 0.26/0.46 thf(fact_202_bot_Oextremum,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order_bot @ A )
% 0.26/0.46 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % bot.extremum
% 0.26/0.46 thf(fact_203_bot_Oextremum__unique,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order_bot @ A )
% 0.26/0.46 => ! [A2: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.26/0.46 = ( A2
% 0.26/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bot.extremum_unique
% 0.26/0.46 thf(fact_204_bot_Oextremum__uniqueI,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order_bot @ A )
% 0.26/0.46 => ! [A2: A] :
% 0.26/0.46 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.26/0.46 => ( A2
% 0.26/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bot.extremum_uniqueI
% 0.26/0.46 thf(fact_205_agree__on__ext,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,G: labeled_graph @ A @ B,F_12: set @ ( product_prod @ B @ C ),F_22: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C @ G @ F_12 @ F_22 )
% 0.26/0.46 => ( agree_on @ A @ B @ D @ G @ ( relcomp @ B @ C @ D @ F_12 @ G6 ) @ ( relcomp @ B @ C @ D @ F_22 @ G6 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_ext
% 0.26/0.46 thf(fact_206_O__assoc,axiom,
% 0.26/0.46 ! [A: $tType,D: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ D ),S2: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R @ S2 ) @ T4 )
% 0.26/0.46 = ( relcomp @ A @ D @ B @ R @ ( relcomp @ D @ C @ B @ S2 @ T4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % O_assoc
% 0.26/0.46 thf(fact_207_relcomp_OrelcompI,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R4: set @ ( product_prod @ A @ B ),C3: C,S4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C3 ) @ S4 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C3 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.relcompI
% 0.26/0.46 thf(fact_208_relcomp_Oinducts,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C: $tType,X1: A,X22: C,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C ),P: A > C > $o] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X1 @ X22 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) )
% 0.26/0.46 => ( ! [A4: A,B4: B,C2: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R4 )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C2 ) @ S4 )
% 0.26/0.46 => ( P @ A4 @ C2 ) ) )
% 0.26/0.46 => ( P @ X1 @ X22 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.inducts
% 0.26/0.46 thf(fact_209_relcomp_Osimps,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) )
% 0.26/0.46 = ( ? [A7: A,B6: B,C7: C] :
% 0.26/0.46 ( ( A1 = A7 )
% 0.26/0.46 & ( A22 = C7 )
% 0.26/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B6 ) @ R4 )
% 0.26/0.46 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B6 @ C7 ) @ S4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.simps
% 0.26/0.46 thf(fact_210_relcomp_Ocases,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) )
% 0.26/0.46 => ~ ! [B4: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B4 ) @ R4 )
% 0.26/0.46 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A22 ) @ S4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.cases
% 0.26/0.46 thf(fact_211_relcompEpair,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R4: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C3 ) @ ( relcomp @ A @ C @ B @ R4 @ S4 ) )
% 0.26/0.46 => ~ ! [B4: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B4 ) @ R4 )
% 0.26/0.46 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcompEpair
% 0.26/0.46 thf(fact_212_relcompE,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R4: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R4 @ S4 ) )
% 0.26/0.46 => ~ ! [X5: A,Y4: C,Z3: B] :
% 0.26/0.46 ( ( Xz
% 0.26/0.46 = ( product_Pair @ A @ B @ X5 @ Z3 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y4 ) @ R4 )
% 0.26/0.46 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcompE
% 0.26/0.46 thf(fact_213_finite_OemptyI,axiom,
% 0.26/0.46 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite.emptyI
% 0.26/0.46 thf(fact_214_infinite__imp__nonempty,axiom,
% 0.26/0.46 ! [A: $tType,S2: set @ A] :
% 0.26/0.46 ( ~ ( finite_finite2 @ A @ S2 )
% 0.26/0.46 => ( S2
% 0.26/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % infinite_imp_nonempty
% 0.26/0.46 thf(fact_215_finite__has__minimal,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A5: set @ A] :
% 0.26/0.46 ( ( finite_finite2 @ A @ A5 )
% 0.26/0.46 => ( ( A5
% 0.26/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 => ? [X5: A] :
% 0.26/0.46 ( ( member @ A @ X5 @ A5 )
% 0.26/0.46 & ! [Xa: A] :
% 0.26/0.46 ( ( member @ A @ Xa @ A5 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ Xa @ X5 )
% 0.26/0.46 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_has_minimal
% 0.26/0.46 thf(fact_216_finite__has__maximal,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [A5: set @ A] :
% 0.26/0.46 ( ( finite_finite2 @ A @ A5 )
% 0.26/0.46 => ( ( A5
% 0.26/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 => ? [X5: A] :
% 0.26/0.46 ( ( member @ A @ X5 @ A5 )
% 0.26/0.46 & ! [Xa: A] :
% 0.26/0.46 ( ( member @ A @ Xa @ A5 )
% 0.26/0.46 => ( ( ord_less_eq @ A @ X5 @ Xa )
% 0.26/0.46 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_has_maximal
% 0.26/0.46 thf(fact_217_graph__homomorphism__nonempty,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F4 )
% 0.26/0.46 => ( ( ( semantics @ A @ B @ A5 @ E3 )
% 0.26/0.46 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.26/0.46 => ( ( semantics @ A @ C @ B5 @ E3 )
% 0.26/0.46 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_nonempty
% 0.26/0.46 thf(fact_218_bounded__Max__nat,axiom,
% 0.26/0.46 ! [P: nat > $o,X4: nat,M2: nat] :
% 0.26/0.46 ( ( P @ X4 )
% 0.26/0.46 => ( ! [X5: nat] :
% 0.26/0.46 ( ( P @ X5 )
% 0.26/0.46 => ( ord_less_eq @ nat @ X5 @ M2 ) )
% 0.26/0.46 => ~ ! [M3: nat] :
% 0.26/0.46 ( ( P @ M3 )
% 0.26/0.46 => ~ ! [X7: nat] :
% 0.26/0.46 ( ( P @ X7 )
% 0.26/0.46 => ( ord_less_eq @ nat @ X7 @ M3 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bounded_Max_nat
% 0.26/0.46 thf(fact_219_Id__on__vertices__identity_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),Aa2: B,Ba: C] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F4 )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ F4 )
% 0.26/0.46 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ ( relcomp @ B @ C @ C @ F4 @ ( id_on @ C @ ( labeled_vertices @ A @ C @ B2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_vertices_identity(2)
% 0.26/0.46 thf(fact_220_Id__on__vertices__identity_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),Aa2: B,Ba: C] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F4 )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ F4 )
% 0.26/0.46 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) @ F4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_vertices_identity(1)
% 0.26/0.46 thf(fact_221_translation__homomorphism_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,E3: allegorical_term @ A,G: labeled_graph @ A @ B,F4: set @ ( product_prod @ nat @ B )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ nat @ B @ ( translation @ A @ E3 ) @ G @ F4 )
% 0.26/0.46 => ( ( semantics @ A @ B @ G @ E3 )
% 0.26/0.46 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % translation_homomorphism(2)
% 0.26/0.46 thf(fact_222_weak__universalD,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,X: $tType,B: $tType,T2: itself @ X,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,F_12: set @ ( product_prod @ C @ B ),F_22: set @ ( product_prod @ C @ B ),G: labeled_graph @ A @ X,H_12: set @ ( product_prod @ C @ X ),H_22: set @ ( product_prod @ B @ X )] :
% 0.26/0.46 ( ( weak_universal @ X @ A @ C @ B @ T2 @ R @ G_1 @ G_2 @ F_12 @ F_22 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ C @ X @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R ) @ G @ H_12 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ X @ G_1 @ G @ H_22 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ X ) ) @ ( relcomp @ C @ B @ X @ F_12 @ H_22 ) @ H_12 )
% 0.26/0.46 => ? [H3: set @ ( product_prod @ B @ X )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ X @ G_2 @ G @ H3 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ B @ X ) ) @ H_22 @ H3 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % weak_universalD
% 0.26/0.46 thf(fact_223_weak__universal__def,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,X: $tType] :
% 0.26/0.46 ( ( weak_universal @ X @ A @ C @ B )
% 0.26/0.46 = ( ^ [Uu: itself @ X,R2: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B,F_1: set @ ( product_prod @ C @ B ),F_2: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ! [H_13: set @ ( product_prod @ C @ X ),H_23: set @ ( product_prod @ B @ X ),G4: labeled_graph @ A @ X] :
% 0.26/0.46 ( ( ( graph_homomorphism @ A @ C @ X @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R2 ) @ G4 @ H_13 )
% 0.26/0.46 & ( graph_homomorphism @ A @ B @ X @ G_12 @ G4 @ H_23 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ C @ X ) ) @ ( relcomp @ C @ B @ X @ F_1 @ H_23 ) @ H_13 ) )
% 0.26/0.46 => ? [H4: set @ ( product_prod @ B @ X )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ X @ G_22 @ G4 @ H4 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ B @ X ) ) @ H_23 @ H4 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % weak_universal_def
% 0.26/0.46 thf(fact_224_graph__homo__union__id_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A5 @ B5 ) @ G @ F4 )
% 0.26/0.46 => ( ( B5
% 0.26/0.46 = ( restrict @ A @ B @ B5 ) )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ C @ B5 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ B5 ) ) @ F4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homo_union_id(2)
% 0.26/0.46 thf(fact_225_graph__homo__union__id_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A5 @ B5 ) @ G @ F4 )
% 0.26/0.46 => ( ( A5
% 0.26/0.46 = ( restrict @ A @ B @ A5 ) )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ C @ A5 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) @ F4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homo_union_id(1)
% 0.26/0.46 thf(fact_226_finite__relcomp,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 0.26/0.46 => ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
% 0.26/0.46 => ( finite_finite2 @ ( product_prod @ A @ C ) @ ( relcomp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_relcomp
% 0.26/0.46 thf(fact_227_subset__emptyI,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A] :
% 0.26/0.46 ( ! [X5: A] :
% 0.26/0.46 ~ ( member @ A @ X5 @ A5 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_emptyI
% 0.26/0.46 thf(fact_228_bot__empty__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.26/0.46 = ( ^ [X6: A] : ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bot_empty_eq
% 0.26/0.46 thf(fact_229_ssubst__Pair__rhs,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R4: A,S4: B,R: set @ ( product_prod @ A @ B ),S5: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S4 ) @ R )
% 0.26/0.46 => ( ( S5 = S4 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S5 ) @ R ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ssubst_Pair_rhs
% 0.26/0.46 thf(fact_230_less__by__empty,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
% 0.26/0.46 ( ( A5
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % less_by_empty
% 0.26/0.46 thf(fact_231_agree__iff__subset,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,X2: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ G @ X2 @ F4 )
% 0.26/0.46 => ( ( univalent @ B @ C @ G6 )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ G @ F4 @ G6 )
% 0.26/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F4 @ G6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_iff_subset
% 0.26/0.46 thf(fact_232_id__univalent,axiom,
% 0.26/0.46 ! [A: $tType,X4: set @ A] : ( univalent @ A @ A @ ( id_on @ A @ X4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % id_univalent
% 0.26/0.46 thf(fact_233_univalentD,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X4: A,Y3: B,Z2: B] :
% 0.26/0.46 ( ( univalent @ A @ B @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z2 ) @ R )
% 0.26/0.46 => ( Z2 = Y3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % univalentD
% 0.26/0.46 thf(fact_234_univalent__def,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( univalent @ A @ B )
% 0.26/0.46 = ( ^ [R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ! [X6: A,Y6: B,Z4: B] :
% 0.26/0.46 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y6 ) @ R2 )
% 0.26/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Z4 ) @ R2 ) )
% 0.26/0.46 => ( Z4 = Y6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % univalent_def
% 0.26/0.46 thf(fact_235_find__graph__occurence__vertices,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),V2: set @ C,F4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( chain @ A @ B @ S2 )
% 0.26/0.46 => ( ( finite_finite2 @ C @ V2 )
% 0.26/0.46 => ( ( univalent @ C @ B @ F4 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ C @ B @ F4 @ V2 ) @ ( labeled_vertices @ A @ B @ ( chain_sup @ A @ B @ S2 ) ) )
% 0.26/0.46 => ? [I2: nat] : ( ord_less_eq @ ( set @ B ) @ ( image @ C @ B @ F4 @ V2 ) @ ( labeled_vertices @ A @ B @ ( S2 @ I2 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % find_graph_occurence_vertices
% 0.26/0.46 thf(fact_236_agree__on__subset,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,F4: set @ ( product_prod @ A @ B ),G6: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ F4 @ G6 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ ( labeled_vertices @ C @ A @ G ) @ ( domain @ A @ B @ F4 ) )
% 0.26/0.46 => ( ( univalent @ A @ B @ G6 )
% 0.26/0.46 => ( agree_on @ C @ A @ B @ G @ F4 @ G6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_subset
% 0.26/0.46 thf(fact_237_ImageI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: A,B2: B,R4: set @ ( product_prod @ A @ B ),A5: set @ A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 0.26/0.46 => ( ( member @ A @ A2 @ A5 )
% 0.26/0.46 => ( member @ B @ B2 @ ( image @ A @ B @ R4 @ A5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ImageI
% 0.26/0.46 thf(fact_238_Image__empty2,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.26/0.46 ( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_empty2
% 0.26/0.46 thf(fact_239_Domain__Id__on,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A] :
% 0.26/0.46 ( ( domain @ A @ A @ ( id_on @ A @ A5 ) )
% 0.26/0.46 = A5 ) ).
% 0.26/0.46
% 0.26/0.46 % Domain_Id_on
% 0.26/0.46 thf(fact_240_Image__empty1,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,X2: set @ B] :
% 0.26/0.46 ( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X2 )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_empty1
% 0.26/0.46 thf(fact_241_Domain__empty,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain_empty
% 0.26/0.46 thf(fact_242_univalent__finite_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( univalent @ A @ B @ R )
% 0.26/0.46 => ( ( finite_finite2 @ A @ ( domain @ A @ B @ R ) )
% 0.26/0.46 = ( finite_finite2 @ ( product_prod @ A @ B ) @ R ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % univalent_finite(2)
% 0.26/0.46 thf(fact_243_Domain__O,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: set @ A,X4: set @ ( product_prod @ A @ B ),Y3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( domain @ A @ B @ X4 ) )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ X4 @ A2 ) @ ( domain @ B @ C @ Y3 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( domain @ A @ C @ ( relcomp @ A @ B @ C @ X4 @ Y3 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain_O
% 0.26/0.46 thf(fact_244_finite__Domain,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R4 )
% 0.26/0.46 => ( finite_finite2 @ A @ ( domain @ A @ B @ R4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_Domain
% 0.26/0.46 thf(fact_245_finite__Image,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A] :
% 0.26/0.46 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 0.26/0.46 => ( finite_finite2 @ B @ ( image @ A @ B @ R @ A5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % finite_Image
% 0.26/0.46 thf(fact_246_Image__mono,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),R4: set @ ( product_prod @ A @ B ),A8: set @ A,A5: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ R4 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ A5 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R5 @ A8 ) @ ( image @ A @ B @ R4 @ A5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_mono
% 0.26/0.46 thf(fact_247_Domain__mono,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S4 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R4 ) @ ( domain @ A @ B @ S4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain_mono
% 0.26/0.46 thf(fact_248_Domain_Oinducts,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,X4: A,R4: set @ ( product_prod @ A @ B ),P: A > $o] :
% 0.26/0.46 ( ( member @ A @ X4 @ ( domain @ A @ B @ R4 ) )
% 0.26/0.46 => ( ! [A4: A,B4: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R4 )
% 0.26/0.46 => ( P @ A4 ) )
% 0.26/0.46 => ( P @ X4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain.inducts
% 0.26/0.46 thf(fact_249_Domain_ODomainI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: A,B2: B,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 0.26/0.46 => ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain.DomainI
% 0.26/0.46 thf(fact_250_Domain_Osimps,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 0.26/0.46 = ( ? [A7: A,B6: B] :
% 0.26/0.46 ( ( A2 = A7 )
% 0.26/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B6 ) @ R4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain.simps
% 0.26/0.46 thf(fact_251_Domain_Ocases,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 0.26/0.46 => ~ ! [B4: B] :
% 0.26/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain.cases
% 0.26/0.46 thf(fact_252_Domain__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 0.26/0.46 = ( ? [Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y6 ) @ R4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Domain_iff
% 0.26/0.46 thf(fact_253_DomainE,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 0.26/0.46 => ~ ! [B4: B] :
% 0.26/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % DomainE
% 0.26/0.46 thf(fact_254_rev__ImageI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,R4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ A @ A2 @ A5 )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 0.26/0.46 => ( member @ B @ B2 @ ( image @ A @ B @ R4 @ A5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rev_ImageI
% 0.26/0.46 thf(fact_255_Image__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,B2: A,R4: set @ ( product_prod @ B @ A ),A5: set @ B] :
% 0.26/0.46 ( ( member @ A @ B2 @ ( image @ B @ A @ R4 @ A5 ) )
% 0.26/0.46 = ( ? [X6: B] :
% 0.26/0.46 ( ( member @ B @ X6 @ A5 )
% 0.26/0.46 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X6 @ B2 ) @ R4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_iff
% 0.26/0.46
% 0.26/0.46 % Type constructors (22)
% 0.26/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.26/0.46 ! [A9: $tType,A10: $tType] :
% 0.26/0.46 ( ( order_bot @ A10 )
% 0.26/0.46 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.26/0.46 ! [A9: $tType,A10: $tType] :
% 0.26/0.46 ( ( preorder @ A10 )
% 0.26/0.46 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.26/0.46 ! [A9: $tType,A10: $tType] :
% 0.26/0.46 ( ( ( finite_finite @ A9 )
% 0.26/0.46 & ( finite_finite @ A10 ) )
% 0.26/0.46 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.26/0.46 ! [A9: $tType,A10: $tType] :
% 0.26/0.46 ( ( order @ A10 )
% 0.26/0.46 => ( order @ ( A9 > A10 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.26/0.46 ! [A9: $tType,A10: $tType] :
% 0.26/0.46 ( ( ord @ A10 )
% 0.26/0.46 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.26/0.46 order_bot @ nat ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.26/0.46 preorder @ nat ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.26/0.46 linorder @ nat ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.26/0.46 order @ nat ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.26/0.46 ord @ nat ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_5,axiom,
% 0.26/0.46 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
% 0.26/0.46 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_7,axiom,
% 0.26/0.46 ! [A9: $tType] :
% 0.26/0.46 ( ( finite_finite @ A9 )
% 0.26/0.46 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 0.26/0.46 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 0.26/0.46 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_10,axiom,
% 0.26/0.46 order_bot @ $o ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
% 0.26/0.46 preorder @ $o ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_12,axiom,
% 0.26/0.46 linorder @ $o ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_13,axiom,
% 0.26/0.46 finite_finite @ $o ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 0.26/0.46 order @ $o ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.26/0.46 ord @ $o ).
% 0.26/0.46
% 0.26/0.46 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_16,axiom,
% 0.26/0.46 ! [A9: $tType,A10: $tType] :
% 0.26/0.46 ( ( ( finite_finite @ A9 )
% 0.26/0.46 & ( finite_finite @ A10 ) )
% 0.26/0.46 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Conjectures (1)
% 0.26/0.46 thf(conj_0,conjecture,
% 0.26/0.46 ( ( graph_homomorphism @ b @ nat @ nat @ ( product_fst @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) @ ( id_on @ nat @ ( labeled_vertices @ b @ nat @ ( product_fst @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) ) )
% 0.26/0.46 & ( ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) )
% 0.26/0.46 = ( restrict @ b @ nat @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) )
% 0.26/0.46 & ( finite_finite2 @ nat @ ( labeled_vertices @ b @ nat @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) )
% 0.26/0.46 & ( finite_finite2 @ ( product_prod @ b @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ b @ nat @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 ------------------------
% 2.87/1.55 ---- Embedded in THF ---
% 2.87/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.87/1.55 %%% Generated on Tue Feb 24 08:28:52 EST 2026
% 2.87/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.87/1.55 %%% Logic specification used:
% 2.87/1.55 %%% thf(spec, logic, $$dhol).
% 2.87/1.55
% 2.87/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2DTF32651.p
% 2.87/1.55 thf(ty_t_LabeledGraphSemantics_OStandard__Constant, type, standard_Constant: ($tType > $tType)).
% 2.87/1.55 thf(ty_t_LabeledGraphSemantics_Oallegorical__term, type, allegorical_term: ($tType > $tType)).
% 2.87/1.55 thf(ty_t_LabeledGraphs_Olabeled__graph, type, labeled_graph: ($tType > ($tType > $tType))).
% 2.87/1.55 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.87/1.55 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.87/1.55 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.87/1.55 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.87/1.55 thf(ty_tf_b, type, b: $tType).
% 2.87/1.55 thf(ty_tf_a, type, a: $tType).
% 2.87/1.55 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.87/1.55 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.87/1.55 thf(sy_c_LabeledGraphSemantics_Oallegorical__term_OA__Int, type, allegorical_A_Int: (!> [V:$tType]: (((allegorical_term @ V) > ((allegorical_term @ V) > (allegorical_term @ V)))))).
% 2.87/1.55 thf(sy_c_LabeledGraphSemantics_Osemantics, type, semantics: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((allegorical_term @ A) > (set @ ((product_prod @ B) @ B))))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Ograph__homomorphism, type, graph_homomorphism: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ C) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Ograph__union, type, graph_union: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B)))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG, type, labeled_LG: (!> [L:$tType,V:$tType]: (((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > ((labeled_graph @ L) @ V)))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph, type, labele1974067554_graph: (!> [L:$tType,V:$tType,A:$tType]: ((((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)) > (((labeled_graph @ L) @ V) > A))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges, type, labeled_edges: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ ((product_prod @ L) @ ((product_prod @ V) @ V))))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices, type, labeled_vertices: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ V))))).
% 2.87/1.55 thf(sy_c_LabeledGraphs_Orestrict, type, restrict: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B))))).
% 2.87/1.55 thf(sy_c_MissingRelation_Ounivalent, type, univalent: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > $o)))).
% 2.87/1.55 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.87/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.87/1.55 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.87/1.55 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.87/1.55 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.87/1.55 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.87/1.55 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 2.87/1.55 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 2.87/1.55 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 2.87/1.55 thf(sy_c_Relation_Orelcomp, type, relcomp: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ A) @ C))))))).
% 2.87/1.55 thf(sy_c_RuleSemanticsConnection_Oinv__translation, type, inv_translation: ((set @ nat) > $o)).
% 2.87/1.55 thf(sy_c_RuleSemanticsConnection_Otranslation, type, translation: (!> [C:$tType]: (((allegorical_term @ C) > ((labeled_graph @ C) @ nat))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oagree__on, type, agree_on: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > ((set @ ((product_prod @ B) @ C)) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Ochain, type, chain: (!> [L:$tType,V:$tType]: (((nat > ((labeled_graph @ L) @ V)) > $o)))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Ochain__sup, type, chain_sup: (!> [L:$tType,V:$tType]: (((nat > ((labeled_graph @ L) @ V)) > ((labeled_graph @ L) @ V))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oconsequence__graph, type, consequence_graph: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) > (((labeled_graph @ A) @ C) > $o))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oextensible, type, extensible: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > ((set @ ((product_prod @ X) @ V)) > $o)))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Ofair__chain, type, fair_chain: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) > ((nat > ((labeled_graph @ A) @ C)) > $o))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Ofin__maintained, type, fin_maintained: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > $o))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oleast, type, least: (!> [X:$tType,L:$tType,V:$tType,C:$tType]: (((itself @ X) > ((set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))) > (((labeled_graph @ L) @ C) > (((labeled_graph @ L) @ C) > $o))))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oleast__consequence__graph, type, least_559130134_graph: (!> [X:$tType,L:$tType,V:$tType,C:$tType]: (((itself @ X) > ((set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))) > (((labeled_graph @ L) @ C) > (((labeled_graph @ L) @ C) > $o))))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Omaintained, type, maintained: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > $o))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Opushout__step, type, pushout_step: (!> [X:$tType,A:$tType,C:$tType,B:$tType]: (((itself @ X) > (((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) > (((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > $o))))))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oset__of__graph__rules, type, set_of_graph_rules: (!> [L:$tType,V:$tType]: (((set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))) > $o)))).
% 2.87/1.55 thf(sy_c_RulesAndChains_Oweak__universal, type, weak_universal: (!> [X:$tType,A:$tType,C:$tType,B:$tType]: (((itself @ X) > (((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) > (((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > ((set @ ((product_prod @ C) @ B)) > ((set @ ((product_prod @ C) @ B)) > $o))))))))).
% 2.87/1.55 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.87/1.55 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules, type, standa1138209853_rules: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))))))).
% 2.87/1.55 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules, type, standa2002409347_rules: (!> [A:$tType]: (((set @ (standard_Constant @ A)) > (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))))))).
% 2.87/1.55 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule, type, standa244753842y_rule: (!> [L:$tType]: (((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat))))).
% 2.87/1.55 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule, type, standa245363227y_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 2.87/1.55 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule, type, standa400804411p_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 2.87/1.55 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.87/1.55 thf(sy_v_C, type, c: (set @ a)).
% 2.87/1.55 thf(sy_v_u, type, u: (allegorical_term @ b)).
% 2.87/1.55 thf(sy_v_v, type, v: (allegorical_term @ b)).
% 2.87/1.55 thf(sy_v_x, type, x: ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))).
% 2.87/1.55 thf(fact_0_assms, axiom, (((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))) @ x) @ ((standa1138209853_rules @ a) @ c))).
% 2.87/1.55 thf(fact_1_graph__rule__translation, axiom, (! [A:$tType,X2:(allegorical_term @ A),Y:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ nat) @ nat) @ (((product_fst @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y))))) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y))))) @ ((id_on @ nat) @ (((labeled_vertices @ A) @ nat) @ (((product_fst @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y))))))) & (((((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y)))) = (((restrict @ A) @ nat) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y)))))) & (((finite_finite2 @ nat) @ (((labeled_vertices @ A) @ nat) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y)))))) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ A) @ nat) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ X2)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ X2) @ Y)))))))))))).
% 2.87/1.55 thf(fact_2_subgraph__refl, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G) @ G) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G))) = (G = (((restrict @ A) @ B) @ G)))))).
% 2.87/1.55 thf(fact_3_subgraph__restrict, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G) @ (((restrict @ A) @ B) @ G)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G))) = (G = (((restrict @ A) @ B) @ G)))))).
% 2.87/1.55 thf(fact_4_graph__homomorphism__Id, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B)]: (((((((graph_homomorphism @ A) @ B) @ B) @ (((restrict @ A) @ B) @ A2)) @ (((restrict @ A) @ B) @ A2)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2)))))).
% 2.87/1.55 thf(fact_5_verts__in__translation__finite_I2_J, axiom, (! [A:$tType,X2:(allegorical_term @ A)]: (((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ A) @ nat) @ ((translation @ A) @ X2)))))).
% 2.87/1.55 thf(fact_6_are__rules_I1_J, axiom, (! [A:$tType]: ((((((((graph_homomorphism @ A) @ nat) @ nat) @ (((product_fst @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A))) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A))) @ ((id_on @ nat) @ (((labeled_vertices @ A) @ nat) @ (((product_fst @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A))))) & (((((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A)) = (((restrict @ A) @ nat) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A)))) & (((finite_finite2 @ nat) @ (((labeled_vertices @ A) @ nat) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A)))) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ A) @ nat) @ (((product_snd @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ (standa244753842y_rule @ A)))))))))).
% 2.87/1.55 thf(fact_7_are__rules_I2_J, axiom, (! [B:$tType,T2:B]: ((((((((graph_homomorphism @ B) @ nat) @ nat) @ (((product_fst @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2))) @ (((product_snd @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2))) @ ((id_on @ nat) @ (((labeled_vertices @ B) @ nat) @ (((product_fst @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2))))) & (((((product_snd @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2)) = (((restrict @ B) @ nat) @ (((product_snd @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2)))) & (((finite_finite2 @ nat) @ (((labeled_vertices @ B) @ nat) @ (((product_snd @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2)))) & ((finite_finite2 @ ((product_prod @ B) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ B) @ nat) @ (((product_snd @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((standa400804411p_rule @ B) @ T2)))))))))).
% 2.87/1.55 thf(fact_8_verts__in__translation__finite_I1_J, axiom, (! [A:$tType,X2:(allegorical_term @ A)]: (((finite_finite2 @ nat) @ (((labeled_vertices @ A) @ nat) @ ((translation @ A) @ X2)))))).
% 2.87/1.55 thf(fact_9_are__rules_I3_J, axiom, (! [C:$tType,I:C]: ((((((((graph_homomorphism @ C) @ nat) @ nat) @ (((product_fst @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I))) @ (((product_snd @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I))) @ ((id_on @ nat) @ (((labeled_vertices @ C) @ nat) @ (((product_fst @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I))))) & (((((product_snd @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I)) = (((restrict @ C) @ nat) @ (((product_snd @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I)))) & (((finite_finite2 @ nat) @ (((labeled_vertices @ C) @ nat) @ (((product_snd @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I)))) & ((finite_finite2 @ ((product_prod @ C) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ C) @ nat) @ (((product_snd @ ((labeled_graph @ C) @ nat)) @ ((labeled_graph @ C) @ nat)) @ ((standa245363227y_rule @ C) @ I)))))))))).
% 2.87/1.55 thf(fact_10_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.87/1.55 thf(fact_11_translation__graph, axiom, (! [A:$tType]: (((translation @ A) = (^ [X3:(allegorical_term @ A)]: ((((restrict @ A) @ nat) @ ((translation @ A) @ X3)))))))).
% 2.87/1.55 thf(fact_12_vertices__restrict, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ B) @ A)]: (((((labeled_vertices @ B) @ A) @ (((restrict @ B) @ A) @ G)) = (((labeled_vertices @ B) @ A) @ G))))).
% 2.87/1.55 thf(fact_13_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.87/1.55 thf(fact_14_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 2.87/1.55 thf(fact_15_restrict__idemp, axiom, (! [B:$tType,A:$tType,X4:((labeled_graph @ A) @ B)]: (((((restrict @ A) @ B) @ (((restrict @ A) @ B) @ X4)) = (((restrict @ A) @ B) @ X4))))).
% 2.87/1.55 thf(fact_16_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ Prod))))).
% 2.87/1.55 thf(fact_17_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 2.87/1.55 thf(fact_18_prod__induct7, axiom, (! [G2:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G2)) @ E2) @ ((((product_Pair @ F) @ G2) @ F2) @ G3))))))))) => (P @ X4))))).
% 2.87/1.55 thf(fact_19_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X4))))).
% 2.87/1.55 thf(fact_20_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X4))))).
% 2.87/1.55 thf(fact_21_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X4))))).
% 2.87/1.55 thf(fact_22_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P @ X4))))).
% 2.87/1.55 thf(fact_23_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G3:G2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G2)) @ E2) @ ((((product_Pair @ F) @ G2) @ F2) @ G3)))))))))))))).
% 2.87/1.55 thf(fact_24_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.87/1.55 thf(fact_25_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.87/1.55 thf(fact_26_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.87/1.55 thf(fact_27_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C2:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 2.87/1.55 thf(fact_28_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.87/1.55 thf(fact_29_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ P2))))).
% 2.87/1.55 thf(fact_30_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X5:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X5) @ Y4))))))).
% 2.87/1.55 thf(fact_31_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 2.87/1.55 thf(fact_32_fst__eqD, axiom, (! [B:$tType,A:$tType,X4:A,Y3:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) = A2) => (X4 = A2))))).
% 2.87/1.55 thf(fact_33_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 2.87/1.55 thf(fact_34_snd__eqD, axiom, (! [B:$tType,A:$tType,X4:B,Y3:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y3)) = A2) => (Y3 = A2))))).
% 2.87/1.55 thf(fact_35_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y5 = Z))) = (^ [S:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T3))))))))).
% 2.87/1.55 thf(fact_36_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.87/1.55 thf(fact_37_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q)) => (P2 = Q)))))).
% 2.87/1.55 thf(fact_38_labeled__graph_Oexpand, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V),Labeled_graph2:((labeled_graph @ L) @ V)]: (((((((labeled_edges @ L) @ V) @ Labeled_graph) = (((labeled_edges @ L) @ V) @ Labeled_graph2)) & ((((labeled_vertices @ L) @ V) @ Labeled_graph) = (((labeled_vertices @ L) @ V) @ Labeled_graph2))) => (Labeled_graph = Labeled_graph2))))).
% 2.87/1.55 thf(fact_39_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 2.87/1.55 thf(fact_40_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.87/1.55 thf(fact_41_subgraph__preserves__hom, axiom, (! [A:$tType,B:$tType,C:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),X2:((labeled_graph @ A) @ C),H:(set @ ((product_prod @ C) @ B))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B5) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => (((((((graph_homomorphism @ A) @ C) @ B) @ X2) @ A5) @ H) => ((((((graph_homomorphism @ A) @ C) @ B) @ X2) @ B5) @ H)))))).
% 2.87/1.55 thf(fact_42_subgraph__trans, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B),G_3:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) => (((((((graph_homomorphism @ A) @ B) @ B) @ G_2) @ G_3) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_2))) => ((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_3) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1)))))))).
% 2.87/1.55 thf(fact_43_Id__onI, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ A5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ ((id_on @ A) @ A5)))))).
% 2.87/1.55 thf(fact_44_set__of__graph__rules__def, axiom, (! [V:$tType,L:$tType]: ((((set_of_graph_rules @ L) @ V) = (^ [Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)))]: ((! [X6:((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))]: (((((member @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))) @ X6) @ Rs) => (((((((graph_homomorphism @ L) @ V) @ V) @ (((product_fst @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6)) @ (((product_snd @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6)) @ ((id_on @ V) @ (((labeled_vertices @ L) @ V) @ (((product_fst @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6)))) & (((((product_snd @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6) = (((restrict @ L) @ V) @ (((product_snd @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6))) & (((finite_finite2 @ V) @ (((labeled_vertices @ L) @ V) @ (((product_snd @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6))) & ((finite_finite2 @ ((product_prod @ L) @ ((product_prod @ V) @ V))) @ (((labeled_edges @ L) @ V) @ (((product_snd @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V)) @ X6))))))))))))))).
% 2.87/1.55 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.87/1.55 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X6:A]: ((((member @ A) @ X6) @ A5)))) = A5)))).
% 2.87/1.55 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q2:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q2 @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q2)))))).
% 2.87/1.55 thf(fact_48_verts__in__translation, axiom, (! [A:$tType,X2:(allegorical_term @ A)]: ((inv_translation @ (((labeled_vertices @ A) @ nat) @ ((translation @ A) @ X2)))))).
% 2.87/1.55 thf(fact_49_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.87/1.55 thf(fact_50_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X4:A,Y3:B,A2:((product_prod @ A) @ B)]: ((((P @ X4) @ Y3) => ((A2 = ((((product_Pair @ A) @ B) @ X4) @ Y3)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.87/1.55 thf(fact_51_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q2:(B > $o),Q:B]: (((P @ P2) => ((Q2 @ Q) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q))) & (Q2 @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q))))))))).
% 2.87/1.55 thf(fact_52_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y3:A,X4:B]: ((((P @ Y3) @ X4) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y3))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y3))))))).
% 2.87/1.55 thf(fact_53_allegorical__term_Oinject_I1_J, axiom, (! [V:$tType,X11:(allegorical_term @ V),X12:(allegorical_term @ V),Y11:(allegorical_term @ V),Y12:(allegorical_term @ V)]: ((((((allegorical_A_Int @ V) @ X11) @ X12) = (((allegorical_A_Int @ V) @ Y11) @ Y12)) = ((X11 = Y11) & (X12 = Y12)))))).
% 2.87/1.55 thf(fact_54_fin__maintainedI, axiom, (! [B:$tType,C:$tType,A:$tType,R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),G:((labeled_graph @ A) @ C)]: (((! [F3:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C))]: ((((F3 = (((restrict @ A) @ B) @ F3)) & (((finite_finite2 @ B) @ (((labeled_vertices @ A) @ B) @ F3)) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ (((labeled_edges @ A) @ B) @ F3)))) => (((((((graph_homomorphism @ A) @ B) @ B) @ F3) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ F3))) => (((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ F3) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) @ G) @ F2) => (((((((graph_homomorphism @ A) @ B) @ C) @ F3) @ G) @ F2) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ F3) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) @ G) @ F2))))))) => (((((fin_maintained @ A) @ B) @ C) @ R) @ G))))).
% 2.87/1.55 thf(fact_55_set__of__graph__rulesD_I1_J, axiom, (! [B:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((set_of_graph_rules @ A) @ B) @ Rs2) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs2) => (((((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) = (((restrict @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) & (((finite_finite2 @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ (((labeled_edges @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)))))))))).
% 2.87/1.55 thf(fact_56_set__of__graph__rulesD_I2_J, axiom, (! [B:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((set_of_graph_rules @ A) @ B) @ Rs2) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs2) => (((((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) = (((restrict @ A) @ B) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) & (((finite_finite2 @ B) @ (((labeled_vertices @ A) @ B) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ (((labeled_edges @ A) @ B) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)))))))))).
% 2.87/1.55 thf(fact_57_extensible__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R) @ G) @ F4) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) @ R)) @ G) @ F4))))).
% 2.87/1.55 thf(fact_58_set__of__graph__rulesD_I3_J, axiom, (! [B:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((set_of_graph_rules @ A) @ B) @ Rs2) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs2) => ((((((graph_homomorphism @ A) @ B) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))))))))).
% 2.87/1.55 thf(fact_59_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),P2:((product_prod @ B) @ A)]: ((((P @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => (~ ((! [X5:B,Y4:A]: ((~ (((P @ Y4) @ X5))))))))))).
% 2.87/1.55 thf(fact_60_Id__onE, axiom, (! [A:$tType,C3:((product_prod @ A) @ A),A5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C3) @ ((id_on @ A) @ A5)) => (~ ((! [X5:A]: (((((member @ A) @ X5) @ A5) => (C3 != ((((product_Pair @ A) @ A) @ X5) @ X5))))))))))).
% 2.87/1.55 thf(fact_61_Id__on__eqI, axiom, (! [A:$tType,A2:A,B2:A,A5:(set @ A)]: (((A2 = B2) => ((((member @ A) @ A2) @ A5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((id_on @ A) @ A5))))))).
% 2.87/1.55 thf(fact_62_Id__on__iff, axiom, (! [A:$tType,X4:A,Y3:A,A5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ ((id_on @ A) @ A5)) = ((X4 = Y3) & (((member @ A) @ X4) @ A5)))))).
% 2.87/1.55 thf(fact_63_fin__maintained__def, axiom, (! [V:$tType,X:$tType,L:$tType]: (((((fin_maintained @ L) @ X) @ V) = (^ [R2:((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)),G4:((labeled_graph @ L) @ V)]: ((! [F5:((labeled_graph @ L) @ X),F6:(set @ ((product_prod @ X) @ V))]: ((((F5 = (((restrict @ L) @ X) @ F5)) & (((finite_finite2 @ X) @ (((labeled_vertices @ L) @ X) @ F5)) & ((finite_finite2 @ ((product_prod @ L) @ ((product_prod @ X) @ X))) @ (((labeled_edges @ L) @ X) @ F5)))) => (((((((graph_homomorphism @ L) @ X) @ X) @ F5) @ (((product_fst @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ R2)) @ ((id_on @ X) @ (((labeled_vertices @ L) @ X) @ F5))) => (((((((extensible @ L) @ X) @ V) @ ((((product_Pair @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ F5) @ (((product_fst @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ R2))) @ G4) @ F6) => (((((((graph_homomorphism @ L) @ X) @ V) @ F5) @ G4) @ F6) => ((((((extensible @ L) @ X) @ V) @ ((((product_Pair @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ F5) @ (((product_snd @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ R2))) @ G4) @ F6))))))))))))).
% 2.87/1.55 thf(fact_64_fin__maintained__maintained, axiom, (! [B:$tType,C:$tType,A:$tType,R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),G:((labeled_graph @ A) @ C)]: (((((((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) = (((restrict @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) & (((finite_finite2 @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ (((labeled_edges @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))))) => ((((((fin_maintained @ A) @ B) @ C) @ R) @ G) = (((((maintained @ A) @ B) @ C) @ R) @ G)))))).
% 2.87/1.55 thf(fact_65_consequence__graphD_I2_J, axiom, (! [C:$tType,B:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((((consequence_graph @ A) @ B) @ C) @ Rs2) @ G) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs2) => ((((((graph_homomorphism @ A) @ B) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R))))))))).
% 2.87/1.55 thf(fact_66_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A6:(set @ A)]: ($true))))))).
% 2.87/1.55 thf(fact_67_fair__chainD_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),S2:(nat > ((labeled_graph @ A) @ C)),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),I:nat,F4:(set @ ((product_prod @ B) @ C))]: (((((((fair_chain @ A) @ B) @ C) @ Rs2) @ S2) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs2) => (((((((graph_homomorphism @ A) @ B) @ C) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ (S2 @ I)) @ F4) => (? [J:nat]: (((((((extensible @ A) @ B) @ C) @ R) @ (S2 @ J)) @ F4))))))))).
% 2.87/1.55 thf(fact_68_identity__rules__graph__rule, axiom, (! [A:$tType,X4:((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)),L2:(set @ (standard_Constant @ A))]: (((((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))) @ X4) @ ((standa2002409347_rules @ A) @ L2)) => (((((((graph_homomorphism @ (standard_Constant @ A)) @ nat) @ nat) @ (((product_fst @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4)) @ (((product_snd @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4)) @ ((id_on @ nat) @ (((labeled_vertices @ (standard_Constant @ A)) @ nat) @ (((product_fst @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4)))) & (((((product_snd @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4) = (((restrict @ (standard_Constant @ A)) @ nat) @ (((product_snd @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4))) & (((finite_finite2 @ nat) @ (((labeled_vertices @ (standard_Constant @ A)) @ nat) @ (((product_snd @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4))) & ((finite_finite2 @ ((product_prod @ (standard_Constant @ A)) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ (standard_Constant @ A)) @ nat) @ (((product_snd @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ X4)))))))))).
% 2.87/1.55 thf(fact_69_maintainedI, axiom, (! [A:$tType,B:$tType,C:$tType,A5:((labeled_graph @ C) @ A),G:((labeled_graph @ C) @ B),B5:((labeled_graph @ C) @ A)]: (((! [F2:(set @ ((product_prod @ A) @ B))]: ((((((((graph_homomorphism @ C) @ A) @ B) @ A5) @ G) @ F2) => ((((((extensible @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A5) @ B5)) @ G) @ F2)))) => (((((maintained @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A5) @ B5)) @ G))))).
% 2.87/1.55 thf(fact_70_sndI, axiom, (! [A:$tType,B:$tType,X4:((product_prod @ A) @ B),Y3:A,Z2:B]: (((X4 = ((((product_Pair @ A) @ B) @ Y3) @ Z2)) => ((((product_snd @ A) @ B) @ X4) = Z2))))).
% 2.87/1.55 thf(fact_71_finite__identity__rules, axiom, (! [A:$tType,L2:(set @ (standard_Constant @ A))]: ((((finite_finite2 @ (standard_Constant @ A)) @ L2) => ((finite_finite2 @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))) @ ((standa2002409347_rules @ A) @ L2)))))).
% 2.87/1.55 thf(fact_72_maintained__refl, axiom, (! [B:$tType,C:$tType,A:$tType,R:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C)]: ((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) @ R)) @ G)))).
% 2.87/1.55 thf(fact_73_consequence__graphI, axiom, (! [B:$tType,C:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C)]: (((! [R3:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R3) @ Rs2) => (((((maintained @ A) @ B) @ C) @ R3) @ G)))) => ((! [R3:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R3) @ Rs2) => ((((((graph_homomorphism @ A) @ B) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3)) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3))))))) => ((G = (((restrict @ A) @ C) @ G)) => (((((consequence_graph @ A) @ B) @ C) @ Rs2) @ G))))))).
% 2.87/1.55 thf(fact_74_consequence__graphD_I1_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((((consequence_graph @ A) @ B) @ C) @ Rs2) @ G) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs2) => (((((maintained @ A) @ B) @ C) @ R) @ G)))))).
% 2.87/1.55 thf(fact_75_consequence__graphD_I3_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C)]: (((((((consequence_graph @ A) @ B) @ C) @ Rs2) @ G) => (G = (((restrict @ A) @ C) @ G)))))).
% 2.87/1.55 thf(fact_76_maintained__then__fin__maintained, axiom, (! [B:$tType,C:$tType,A:$tType,R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),G:((labeled_graph @ A) @ C)]: (((((((maintained @ A) @ B) @ C) @ R) @ G) => (((((fin_maintained @ A) @ B) @ C) @ R) @ G))))).
% 2.87/1.55 thf(fact_77_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A5:(set @ A)]: (((finite_finite2 @ A) @ A5))))))).
% 2.87/1.55 thf(fact_78_finite__set__choice, axiom, (! [B:$tType,A:$tType,A5:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A5) => ((! [X5:A]: (((((member @ A) @ X5) @ A5) => (? [X_1:B]: (((P @ X5) @ X_1)))))) => (? [F2:(A > B)]: ((! [X7:A]: (((((member @ A) @ X7) @ A5) => ((P @ X7) @ (F2 @ X7)))))))))))).
% 2.87/1.55 thf(fact_79_consequence__graph__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((consequence_graph @ A) @ B) @ C) = (^ [Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G4:((labeled_graph @ A) @ C)]: (((G4 = (((restrict @ A) @ C) @ G4)) & (! [X6:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ X6) @ Rs) => (((((((graph_homomorphism @ A) @ B) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ X6)) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ X6)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ X6)))) & (((((maintained @ A) @ B) @ C) @ X6) @ G4)))))))))))).
% 2.87/1.55 thf(fact_80_maintained__def, axiom, (! [V:$tType,X:$tType,L:$tType]: (((((maintained @ L) @ X) @ V) = (^ [R2:((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)),G4:((labeled_graph @ L) @ V)]: ((! [F6:(set @ ((product_prod @ X) @ V))]: ((((((((graph_homomorphism @ L) @ X) @ V) @ (((product_fst @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ R2)) @ G4) @ F6) => ((((((extensible @ L) @ X) @ V) @ R2) @ G4) @ F6)))))))))).
% 2.87/1.55 thf(fact_81_maintainedD, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B5)) @ G) => (((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ G) @ F4) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B5)) @ G) @ F4)))))).
% 2.87/1.55 thf(fact_82_fstI, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ A) @ B),Y3:A,Z2:B]: (((X4 = ((((product_Pair @ A) @ B) @ Y3) @ Z2)) => ((((product_fst @ A) @ B) @ X4) = Y3))))).
% 2.87/1.55 thf(fact_83_least__consequence__graphI, axiom, (! [X:$tType,A:$tType,C:$tType,L:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ L) @ A)) @ ((labeled_graph @ L) @ A))),G:((labeled_graph @ L) @ C),S2:((labeled_graph @ L) @ C),T2:(itself @ X)]: (((((((consequence_graph @ L) @ A) @ C) @ Rs2) @ G) => (((((((graph_homomorphism @ L) @ C) @ C) @ S2) @ G) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S2))) => ((! [C4:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs2) @ C4) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S2) @ G)) @ C4)))) => ((((((((least_559130134_graph @ X) @ L) @ A) @ C) @ T2) @ Rs2) @ S2) @ G))))))).
% 2.87/1.55 thf(fact_84_leastI, axiom, (! [X:$tType,A:$tType,C:$tType,L:$tType,S2:((labeled_graph @ L) @ C),G:((labeled_graph @ L) @ C),Rs2:(set @ ((product_prod @ ((labeled_graph @ L) @ A)) @ ((labeled_graph @ L) @ A))),T2:(itself @ X)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S2) @ G) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S2))) => ((! [C4:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs2) @ C4) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S2) @ G)) @ C4)))) => ((((((((least @ X) @ L) @ A) @ C) @ T2) @ Rs2) @ S2) @ G)))))).
% 2.87/1.55 thf(fact_85_least__def, axiom, (! [L:$tType,V:$tType,C:$tType,X:$tType]: ((((((least @ X) @ L) @ V) @ C) = (^ [Uu:(itself @ X),Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))),S3:((labeled_graph @ L) @ C),G4:((labeled_graph @ L) @ C)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S3) @ G4) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S3))) & (! [C5:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ V) @ X) @ Rs) @ C5) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S3) @ G4)) @ C5))))))))))).
% 2.87/1.55 thf(fact_86_fair__chain__impl__consequence__graph, axiom, (! [B:$tType,C:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),S2:(nat > ((labeled_graph @ A) @ C))]: (((((((fair_chain @ A) @ B) @ C) @ Rs2) @ S2) => ((! [R3:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R3) @ Rs2) => (((((((graph_homomorphism @ A) @ B) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3)) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3)))) & (((((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3) = (((restrict @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3))) & (((finite_finite2 @ B) @ (((labeled_vertices @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3))) & ((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ (((labeled_edges @ A) @ B) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R3))))))))) => (((((consequence_graph @ A) @ B) @ C) @ Rs2) @ (((chain_sup @ A) @ C) @ S2))))))).
% 2.87/1.55 thf(fact_87_fair__chainI, axiom, (! [C:$tType,B:$tType,A:$tType,S2:(nat > ((labeled_graph @ A) @ B)),Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)))]: (((((chain @ A) @ B) @ S2) => ((! [R3:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)),F2:(set @ ((product_prod @ C) @ B)),I2:nat]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C))) @ R3) @ Rs2) => (((((((graph_homomorphism @ A) @ C) @ B) @ (((product_fst @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) @ R3)) @ (S2 @ I2)) @ F2) => (? [J2:nat]: (((((((extensible @ A) @ C) @ B) @ R3) @ (S2 @ J2)) @ F2))))))) => (((((fair_chain @ A) @ C) @ B) @ Rs2) @ S2)))))).
% 2.87/1.55 thf(fact_88_maintained__holds__iff, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),E_L:(allegorical_term @ A),E_R:(allegorical_term @ A)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((maintained @ A) @ nat) @ B) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ E_L)) @ ((translation @ A) @ (((allegorical_A_Int @ A) @ E_L) @ E_R)))) @ G) = (((((semantics @ A) @ B) @ G) @ (((product_fst @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ ((((product_Pair @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ E_L) @ (((allegorical_A_Int @ A) @ E_L) @ E_R)))) = ((((semantics @ A) @ B) @ G) @ (((product_snd @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ ((((product_Pair @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ E_L) @ (((allegorical_A_Int @ A) @ E_L) @ E_R)))))))))).
% 2.87/1.55 thf(fact_89_fair__chain__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((fair_chain @ A) @ B) @ C) = (^ [Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),S3:(nat > ((labeled_graph @ A) @ C))]: (((((chain @ A) @ C) @ S3) & (! [R2:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),F6:(set @ ((product_prod @ B) @ C)),I3:nat]: ((((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R2) @ Rs) & ((((((graph_homomorphism @ A) @ B) @ C) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R2)) @ (S3 @ I3)) @ F6)) => (? [J3:nat]: (((((((extensible @ A) @ B) @ C) @ R2) @ (S3 @ J3)) @ F6))))))))))))).
% 2.87/1.55 thf(fact_90_chain__sup__graph, axiom, (! [B:$tType,A:$tType,S2:(nat > ((labeled_graph @ A) @ B))]: (((((chain @ A) @ B) @ S2) => ((((chain_sup @ A) @ B) @ S2) = (((restrict @ A) @ B) @ (((chain_sup @ A) @ B) @ S2))))))).
% 2.87/1.55 thf(fact_91_extensible__chain__sup, axiom, (! [A:$tType,B:$tType,C:$tType,S2:(nat > ((labeled_graph @ A) @ B)),R:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)),J4:nat,F4:(set @ ((product_prod @ C) @ B))]: (((((chain @ A) @ B) @ S2) => (((((((extensible @ A) @ C) @ B) @ R) @ (S2 @ J4)) @ F4) => ((((((extensible @ A) @ C) @ B) @ R) @ (((chain_sup @ A) @ B) @ S2)) @ F4)))))).
% 2.87/1.55 thf(fact_92_chain__sup__subgraph, axiom, (! [A:$tType,B:$tType,S2:(nat > ((labeled_graph @ A) @ B)),J4:nat]: (((((chain @ A) @ B) @ S2) => ((((((graph_homomorphism @ A) @ B) @ B) @ (S2 @ J4)) @ (((chain_sup @ A) @ B) @ S2)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (S2 @ J4)))))))).
% 2.87/1.55 thf(fact_93_chain__then__restrict, axiom, (! [A:$tType,B:$tType,S2:(nat > ((labeled_graph @ A) @ B)),I:nat]: (((((chain @ A) @ B) @ S2) => ((S2 @ I) = (((restrict @ A) @ B) @ (S2 @ I))))))).
% 2.87/1.55 thf(fact_94_least__consequence__graph__def, axiom, (! [L:$tType,V:$tType,C:$tType,X:$tType]: ((((((least_559130134_graph @ X) @ L) @ V) @ C) = (^ [T3:(itself @ X),Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))),S3:((labeled_graph @ L) @ C),G4:((labeled_graph @ L) @ C)]: (((((((consequence_graph @ L) @ V) @ C) @ Rs) @ G4) & ((((((((least @ X) @ L) @ V) @ C) @ T3) @ Rs) @ S3) @ G4)))))))).
% 2.87/1.55 thf(fact_95_graph__homomorphism__semantics, axiom, (! [B:$tType,C:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C)),A2:B,B2:B,E3:(allegorical_term @ A),A3:C,B3:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ B5) @ F4) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ A2) @ A3)) @ F4) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B2) @ B3)) @ F4) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ A3) @ B3)) @ ((((semantics @ A) @ C) @ B5) @ E3))))))))).
% 2.87/1.55 thf(fact_96_fair__chainD_I1_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs2:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),S2:(nat > ((labeled_graph @ A) @ C))]: (((((((fair_chain @ A) @ B) @ C) @ Rs2) @ S2) => (((chain @ A) @ C) @ S2))))).
% 2.87/1.55 thf(fact_97_semantics__in__vertices_I2_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: (((A5 = (((restrict @ A) @ B) @ A5)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ B) @ B2) @ (((labeled_vertices @ A) @ B) @ A5))))))).
% 2.87/1.55 thf(fact_98_semantics__in__vertices_I1_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: (((A5 = (((restrict @ A) @ B) @ A5)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ B) @ A2) @ (((labeled_vertices @ A) @ B) @ A5))))))).
% 2.87/1.55 thf(fact_99_subgraph__semantics, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B5) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ B5) @ E3))))))).
% 2.87/1.55 thf(fact_100_eq__as__subsets, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ B) @ A),E_1:(allegorical_term @ B),E_2:(allegorical_term @ B)]: (((((((semantics @ B) @ A) @ G) @ (((product_fst @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_1) @ E_2))) = ((((semantics @ B) @ A) @ G) @ (((product_snd @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_1) @ E_2)))) = ((((((semantics @ B) @ A) @ G) @ (((product_fst @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_1) @ (((allegorical_A_Int @ B) @ E_1) @ E_2)))) = ((((semantics @ B) @ A) @ G) @ (((product_snd @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_1) @ (((allegorical_A_Int @ B) @ E_1) @ E_2))))) & (((((semantics @ B) @ A) @ G) @ (((product_fst @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_2) @ (((allegorical_A_Int @ B) @ E_2) @ E_1)))) = ((((semantics @ B) @ A) @ G) @ (((product_snd @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_2) @ (((allegorical_A_Int @ B) @ E_2) @ E_1)))))))))).
% 2.87/1.55 thf(fact_101_maintained__holds__subset__iff, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),E_L:(allegorical_term @ A),E_R:(allegorical_term @ A)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((maintained @ A) @ nat) @ B) @ ((((product_Pair @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)) @ ((translation @ A) @ (((product_fst @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ ((((product_Pair @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ E_L) @ (((allegorical_A_Int @ A) @ E_L) @ E_R))))) @ ((translation @ A) @ (((product_snd @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ ((((product_Pair @ (allegorical_term @ A)) @ (allegorical_term @ A)) @ E_L) @ (((allegorical_A_Int @ A) @ E_L) @ E_R)))))) @ G) = (((ord_less_eq @ (set @ ((product_prod @ B) @ B))) @ ((((semantics @ A) @ B) @ G) @ E_L)) @ ((((semantics @ A) @ B) @ G) @ E_R))))))).
% 2.87/1.55 thf(fact_102_maintained__holds, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ B) @ A),E_L:(allegorical_term @ B),E_R:(allegorical_term @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((((semantics @ B) @ A) @ G) @ E_L)) @ ((((semantics @ B) @ A) @ G) @ E_R)) => (((((maintained @ B) @ nat) @ A) @ ((((product_Pair @ ((labeled_graph @ B) @ nat)) @ ((labeled_graph @ B) @ nat)) @ ((translation @ B) @ (((product_fst @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_L) @ (((allegorical_A_Int @ B) @ E_L) @ E_R))))) @ ((translation @ B) @ (((product_snd @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_L) @ (((allegorical_A_Int @ B) @ E_L) @ E_R)))))) @ G))))).
% 2.87/1.55 thf(fact_103_sentence__iff, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ B) @ A),E_1:(allegorical_term @ B),E_2:(allegorical_term @ B)]: (((((((semantics @ B) @ A) @ G) @ (((product_fst @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_1) @ (((allegorical_A_Int @ B) @ E_1) @ E_2)))) = ((((semantics @ B) @ A) @ G) @ (((product_snd @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ ((((product_Pair @ (allegorical_term @ B)) @ (allegorical_term @ B)) @ E_1) @ (((allegorical_A_Int @ B) @ E_1) @ E_2))))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((((semantics @ B) @ A) @ G) @ E_1)) @ ((((semantics @ B) @ A) @ G) @ E_2)))))).
% 2.87/1.55 thf(fact_104_find__graph__occurence, axiom, (! [A:$tType,B:$tType,C:$tType,S2:(nat > ((labeled_graph @ A) @ B)),E4:(set @ ((product_prod @ A) @ ((product_prod @ C) @ C))),V2:(set @ C),F4:(set @ ((product_prod @ C) @ B))]: (((((chain @ A) @ B) @ S2) => (((finite_finite2 @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ E4) => (((finite_finite2 @ C) @ V2) => (((((((graph_homomorphism @ A) @ C) @ B) @ ((((labeled_LG @ A) @ C) @ E4) @ V2)) @ (((chain_sup @ A) @ B) @ S2)) @ F4) => (? [I2:nat]: (((((((graph_homomorphism @ A) @ C) @ B) @ ((((labeled_LG @ A) @ C) @ E4) @ V2)) @ (S2 @ I2)) @ F4)))))))))).
% 2.87/1.55 thf(fact_105_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A7:B]: ((P2 = ((((product_Pair @ B) @ A) @ A7) @ B2)))))))).
% 2.87/1.55 thf(fact_106_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B6:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B6)))))))).
% 2.87/1.55 thf(fact_107_labeled__graph_Oinject, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V),Y1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),Y2:(set @ V)]: (((((((labeled_LG @ L) @ V) @ X1) @ X22) = ((((labeled_LG @ L) @ V) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 2.87/1.55 thf(fact_108_labeled__graph_Ocollapse, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V)]: ((((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)) = Labeled_graph)))).
% 2.87/1.55 thf(fact_109_extensible__refl__concr, axiom, (! [A:$tType,C:$tType,B:$tType,E_1:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B))),V3:(set @ B),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C)),E_2:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_1) @ V3)) @ G) @ F4) => (((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ ((((labeled_LG @ A) @ B) @ E_1) @ V3)) @ ((((labeled_LG @ A) @ B) @ E_2) @ V3))) @ G) @ F4) = ((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_2) @ V3)) @ G) @ F4)))))).
% 2.87/1.55 thf(fact_110_labeled__graph_Oexhaust, axiom, (! [L:$tType,V:$tType,Y3:((labeled_graph @ L) @ V)]: ((~ ((! [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X23:(set @ V)]: ((Y3 != ((((labeled_LG @ L) @ V) @ X13) @ X23))))))))).
% 2.87/1.55 thf(fact_111_labeled__graph_Oinduct, axiom, (! [V:$tType,L:$tType,P:(((labeled_graph @ L) @ V) > $o),Labeled_graph:((labeled_graph @ L) @ V)]: (((! [X1a:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X2a:(set @ V)]: ((P @ ((((labeled_LG @ L) @ V) @ X1a) @ X2a)))) => (P @ Labeled_graph))))).
% 2.87/1.55 thf(fact_112_subrelI, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ A) @ B))]: (((! [X5:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y4)) @ R4) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y4)) @ S4)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ S4))))).
% 2.87/1.55 thf(fact_113_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A5) => ((((member @ A) @ A2) @ A5) => (? [X5:A]: (((((member @ A) @ X5) @ A5) & ((((ord_less_eq @ A) @ A2) @ X5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ X5) @ Xa) => (X5 = Xa))))))))))))))))).
% 2.87/1.55 thf(fact_114_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A5) => ((((member @ A) @ A2) @ A5) => (? [X5:A]: (((((member @ A) @ X5) @ A5) & ((((ord_less_eq @ A) @ X5) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ Xa) @ X5) => (X5 = Xa))))))))))))))))).
% 2.87/1.55 thf(fact_115_finite__subset, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((finite_finite2 @ A) @ B5) => ((finite_finite2 @ A) @ A5)))))).
% 2.87/1.55 thf(fact_116_infinite__super, axiom, (! [A:$tType,S2:(set @ A),T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ T4) => ((~ (((finite_finite2 @ A) @ S2))) => (~ (((finite_finite2 @ A) @ T4)))))))).
% 2.87/1.55 thf(fact_117_rev__finite__subset, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A)]: ((((finite_finite2 @ A) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((finite_finite2 @ A) @ A5)))))).
% 2.87/1.55 thf(fact_118_labeled__graph_Osel_I2_J, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((labeled_vertices @ L) @ V) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = X22)))).
% 2.87/1.55 thf(fact_119_labeled__graph_Osel_I1_J, axiom, (! [V:$tType,L:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((labeled_edges @ L) @ V) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = X1)))).
% 2.87/1.55 thf(fact_120_restrict__subsD, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G)) @ (((labeled_edges @ A) @ B) @ (((restrict @ A) @ B) @ G))) => (G = (((restrict @ A) @ B) @ G)))))).
% 2.87/1.55 thf(fact_121_subgraph__def2, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((G_1 = (((restrict @ A) @ B) @ G_1)) => ((G_2 = (((restrict @ A) @ B) @ G_2)) => (((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) = ((((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)) & (((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2))))))))).
% 2.87/1.55 thf(fact_122_labeled__graph_Oexhaust__sel, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V)]: ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))).
% 2.87/1.55 thf(fact_123_subgraph__subset_I1_J, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) => (((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)))))).
% 2.87/1.55 thf(fact_124_maintainedD2, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B5)) @ G) => (((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ G) @ F4) => (~ ((! [G3:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ B5) @ G) @ G3) => (~ ((((ord_less_eq @ (set @ ((product_prod @ B) @ C))) @ F4) @ G3))))))))))))).
% 2.87/1.55 thf(fact_125_subgraph__subset_I2_J, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ (((restrict @ A) @ B) @ G_1))) @ (((labeled_edges @ A) @ B) @ G_2)))))).
% 2.87/1.55 thf(fact_126_pushout__step__def, axiom, (! [A:$tType,C:$tType,B:$tType,X:$tType]: ((((((pushout_step @ X) @ A) @ C) @ B) = (^ [T3:(itself @ X),R2:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)),G_12:((labeled_graph @ A) @ B),G_22:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_12) @ G_22) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_12))) & (? [F_1:(set @ ((product_prod @ C) @ B)),F_2:(set @ ((product_prod @ C) @ B))]: ((((((((graph_homomorphism @ A) @ C) @ B) @ (((product_fst @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) @ R2)) @ G_12) @ F_1) & (((((((graph_homomorphism @ A) @ C) @ B) @ (((product_snd @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) @ R2)) @ G_22) @ F_2) & ((((ord_less_eq @ (set @ ((product_prod @ C) @ B))) @ F_1) @ F_2) & ((((((((((weak_universal @ X) @ A) @ C) @ B) @ T3) @ R2) @ G_12) @ G_22) @ F_1) @ F_2))))))))))))).
% 2.87/1.55 thf(fact_127_subsetI, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A5) => (((member @ A) @ X5) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 2.87/1.55 thf(fact_128_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A5) => (A5 = B5)))))).
% 2.87/1.55 thf(fact_129_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 2.87/1.55 thf(fact_130_labeled__graph_Osplit__sel__asm, axiom, (! [A:$tType,V:$tType,L:$tType,P:(A > $o),F4:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P @ (((((labele1974067554_graph @ L) @ V) @ A) @ F4) @ Labeled_graph)) = (~ (((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) & (~ ((P @ ((F4 @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))))))).
% 2.87/1.55 thf(fact_131_labeled__graph_Ocase, axiom, (! [L:$tType,A:$tType,V:$tType,F4:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((((labele1974067554_graph @ L) @ V) @ A) @ F4) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = ((F4 @ X1) @ X22))))).
% 2.87/1.55 thf(fact_132_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.87/1.55 thf(fact_133_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A7:A,B6:A]: (((((ord_less_eq @ A) @ B6) @ A7) & (((ord_less_eq @ A) @ A7) @ B6))))))))).
% 2.87/1.55 thf(fact_134_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C3) @ B2) => (((ord_less_eq @ A) @ C3) @ A2))))))))).
% 2.87/1.55 thf(fact_135_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.87/1.55 thf(fact_136_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.87/1.55 thf(fact_137_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X4) @ Z2))))))))).
% 2.87/1.55 thf(fact_138_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.87/1.55 thf(fact_139_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.87/1.55 thf(fact_140_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.87/1.55 thf(fact_141_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A7:A,B6:A]: (((((ord_less_eq @ A) @ A7) @ B6) & (((ord_less_eq @ A) @ B6) @ A7))))))))).
% 2.87/1.55 thf(fact_142_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X4:A]: (((((ord_less_eq @ A) @ Y3) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y3) = (X4 = Y3))))))))).
% 2.87/1.55 thf(fact_143_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X4) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z2)))) => (((((ord_less_eq @ A) @ X4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X4)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y3))))))))))))))))).
% 2.87/1.55 thf(fact_144_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.87/1.55 thf(fact_145_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 2.87/1.55 thf(fact_146_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((X4 = Y3) => (((ord_less_eq @ A) @ X4) @ Y3)))))))).
% 2.87/1.55 thf(fact_147_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 2.87/1.55 thf(fact_148_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X4) => (X4 = Y3))))))))).
% 2.87/1.55 thf(fact_149_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X6:A,Y6:A]: (((((ord_less_eq @ A) @ X6) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X6))))))))).
% 2.87/1.55 thf(fact_150_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F4:(A > B),C3:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F4 @ B2) = C3) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F4 @ X5)) @ (F4 @ Y4))))) => (((ord_less_eq @ B) @ (F4 @ A2)) @ C3)))))))))).
% 2.87/1.55 thf(fact_151_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F4:(B > A),B2:B,C3:B]: (((A2 = (F4 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F4 @ X5)) @ (F4 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F4 @ C3))))))))))).
% 2.87/1.55 thf(fact_152_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F4:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F4 @ B2)) @ C3) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ C) @ (F4 @ X5)) @ (F4 @ Y4))))) => (((ord_less_eq @ C) @ (F4 @ A2)) @ C3)))))))))).
% 2.87/1.55 thf(fact_153_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F4:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F4 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F4 @ X5)) @ (F4 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F4 @ C3))))))))))).
% 2.87/1.55 thf(fact_154_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F6:(A > B),G5:(A > B)]: ((! [X6:A]: ((((ord_less_eq @ B) @ (F6 @ X6)) @ (G5 @ X6))))))))))).
% 2.87/1.55 thf(fact_155_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F4:(A > B),G6:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F4 @ X5)) @ (G6 @ X5)))) => (((ord_less_eq @ (A > B)) @ F4) @ G6)))))))).
% 2.87/1.55 thf(fact_156_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F4:(A > B),G6:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F4) @ G6) => (((ord_less_eq @ B) @ (F4 @ X4)) @ (G6 @ X4))))))))).
% 2.87/1.55 thf(fact_157_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F4:(A > B),G6:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F4) @ G6) => (((ord_less_eq @ B) @ (F4 @ X4)) @ (G6 @ X4))))))))).
% 2.87/1.55 thf(fact_158_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q2)) = (! [X6:A]: (((P @ X6) => (Q2 @ X6)))))))).
% 2.87/1.55 thf(fact_159_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z:(set @ A)]: ((Y5 = Z))) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 2.87/1.55 thf(fact_160_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C6) => (((ord_less_eq @ (set @ A)) @ A5) @ C6)))))).
% 2.87/1.55 thf(fact_161_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q2:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q2 @ X5)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q2)))))).
% 2.87/1.55 thf(fact_162_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 2.87/1.55 thf(fact_163_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A6) => (((member @ A) @ T3) @ B7)))))))))).
% 2.87/1.55 thf(fact_164_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A5))))).
% 2.87/1.55 thf(fact_165_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 2.87/1.55 thf(fact_166_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [X6:A]: (((((member @ A) @ X6) @ A6) => (((member @ A) @ X6) @ B7)))))))))).
% 2.87/1.55 thf(fact_167_equalityE, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 2.87/1.55 thf(fact_168_subsetD, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B5)))))).
% 2.87/1.55 thf(fact_169_in__mono, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B5)))))).
% 2.87/1.55 thf(fact_170_labeled__graph_Ocase__eq__if, axiom, (! [A:$tType,V:$tType,L:$tType]: (((((labele1974067554_graph @ L) @ V) @ A) = (^ [F6:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph3:((labeled_graph @ L) @ V)]: (((F6 @ (((labeled_edges @ L) @ V) @ Labeled_graph3)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph3)))))))).
% 2.87/1.55 thf(fact_171_chain, axiom, (! [A:$tType,B:$tType,S2:(nat > ((labeled_graph @ A) @ B)),I:nat,J4:nat]: (((((chain @ A) @ B) @ S2) => ((((ord_less_eq @ nat) @ I) @ J4) => ((((((graph_homomorphism @ A) @ B) @ B) @ (S2 @ I)) @ (S2 @ J4)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (S2 @ I))))))))).
% 2.87/1.55 thf(fact_172_chain__def2, axiom, (! [B:$tType,A:$tType]: ((((chain @ A) @ B) = (^ [S3:(nat > ((labeled_graph @ A) @ B))]: ((! [I3:nat,J3:nat]: (((((ord_less_eq @ nat) @ I3) @ J3) => ((((((graph_homomorphism @ A) @ B) @ B) @ (S3 @ I3)) @ (S3 @ J3)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (S3 @ I3))))))))))))).
% 2.87/1.55 thf(fact_173_labeled__graph_Osplit__sel, axiom, (! [A:$tType,V:$tType,L:$tType,P:(A > $o),F4:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P @ (((((labele1974067554_graph @ L) @ V) @ A) @ F4) @ Labeled_graph)) = ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) => (P @ ((F4 @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))).
% 2.87/1.55 thf(fact_174_graph__unionI, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2)) => ((((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)) => (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2)))))).
% 2.87/1.55 thf(fact_175_extensibleD, axiom, (! [A:$tType,C:$tType,B:$tType,R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: ((((((((extensible @ A) @ B) @ C) @ R) @ G) @ F4) => (~ ((! [G3:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ (((product_snd @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ G) @ G3) => (~ (((((((agree_on @ A) @ B) @ C) @ (((product_fst @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R)) @ F4) @ G3)))))))))))).
% 2.87/1.55 thf(fact_176_extensible__def, axiom, (! [V:$tType,X:$tType,L:$tType]: (((((extensible @ L) @ X) @ V) = (^ [R2:((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)),G4:((labeled_graph @ L) @ V),F6:(set @ ((product_prod @ X) @ V))]: ((? [G5:(set @ ((product_prod @ X) @ V))]: ((((((((graph_homomorphism @ L) @ X) @ V) @ (((product_snd @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ R2)) @ G4) @ G5) & ((((((agree_on @ L) @ X) @ V) @ (((product_fst @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) @ R2)) @ F6) @ G5)))))))))).
% 2.87/1.55 thf(fact_177_extensibleI, axiom, (! [A:$tType,C:$tType,B:$tType,R22:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),G6:(set @ ((product_prod @ B) @ C)),R1:((labeled_graph @ A) @ B),F4:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R22) @ G) @ G6) => (((((((agree_on @ A) @ B) @ C) @ R1) @ F4) @ G6) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R1) @ R22)) @ G) @ F4)))))).
% 2.87/1.55 thf(fact_178_graph__union__idemp_I1_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ A5) = A5)))).
% 2.87/1.55 thf(fact_179_graph__union__idemp_I2_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ ((((graph_union @ A) @ B) @ A5) @ B5)) = ((((graph_union @ A) @ B) @ A5) @ B5))))).
% 2.87/1.55 thf(fact_180_graph__union__idemp_I3_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ ((((graph_union @ A) @ B) @ B5) @ A5)) = ((((graph_union @ A) @ B) @ B5) @ A5))))).
% 2.87/1.55 thf(fact_181_agree__on__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R:((labeled_graph @ A) @ B),F4:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ R) @ F4) @ F4)))).
% 2.87/1.55 thf(fact_182_agree__on__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((agree_on @ A) @ B) @ C) = (^ [X3:((labeled_graph @ A) @ B),F6:(set @ ((product_prod @ B) @ C)),G5:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ X3) @ G5) @ F6))))))).
% 2.87/1.55 thf(fact_183_graph__union__preserves__restrict, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((G_1 = (((restrict @ A) @ B) @ G_1)) => ((G_2 = (((restrict @ A) @ B) @ G_2)) => (((((graph_union @ A) @ B) @ G_1) @ G_2) = (((restrict @ A) @ B) @ ((((graph_union @ A) @ B) @ G_1) @ G_2)))))))).
% 2.87/1.55 thf(fact_184_graph__union__chain__sup, axiom, (! [B:$tType,A:$tType,S2:(nat > ((labeled_graph @ A) @ B)),C6:((labeled_graph @ A) @ B)]: (((! [I2:nat]: ((((((graph_union @ A) @ B) @ (S2 @ I2)) @ C6) = C6))) => (((((graph_union @ A) @ B) @ (((chain_sup @ A) @ B) @ S2)) @ C6) = C6))))).
% 2.87/1.55 thf(fact_185_agree__on__trans, axiom, (! [A:$tType,C:$tType,B:$tType,X2:((labeled_graph @ A) @ B),F4:(set @ ((product_prod @ B) @ C)),G6:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ X2) @ F4) @ G6) => (((((((agree_on @ A) @ B) @ C) @ X2) @ G6) @ H) => ((((((agree_on @ A) @ B) @ C) @ X2) @ F4) @ H)))))).
% 2.87/1.55 thf(fact_186_agree__on__subg__compose, axiom, (! [A:$tType,C:$tType,B:$tType,R:((labeled_graph @ A) @ B),G6:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C)),F7:((labeled_graph @ A) @ B),F4:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ R) @ G6) @ H) => (((((((agree_on @ A) @ B) @ C) @ F7) @ F4) @ G6) => (((((((graph_homomorphism @ A) @ B) @ B) @ F7) @ R) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ F7))) => ((((((agree_on @ A) @ B) @ C) @ F7) @ F4) @ H))))))).
% 2.87/1.55 thf(fact_187_subgraph__def, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) = ((G_1 = (((restrict @ A) @ B) @ G_1)) & ((G_2 = (((restrict @ A) @ B) @ G_2)) & (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2))))))).
% 2.87/1.55 thf(fact_188_graph__union__iff, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2) = ((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2)) & (((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2))))))).
% 2.87/1.55 thf(fact_189_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N:(set @ nat)]: ((? [M:nat]: ((! [X6:nat]: (((((member @ nat) @ X6) @ N) => (((ord_less_eq @ nat) @ X6) @ M)))))))))).
% 2.87/1.55 thf(fact_190_nonempty__rule, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ A) @ B)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((maintained @ A) @ nat) @ B) @ (standa244753842y_rule @ A)) @ G) = ((((labeled_vertices @ A) @ B) @ G) != (bot_bot @ (set @ B)))))))).
% 2.87/1.55 thf(fact_191_weak__universalI, axiom, (! [X:$tType,A:$tType,B:$tType,C:$tType,R:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)),G_1:((labeled_graph @ A) @ B),F_12:(set @ ((product_prod @ C) @ B)),G_2:((labeled_graph @ A) @ B),T2:(itself @ X),F_22:(set @ ((product_prod @ C) @ B))]: (((! [H_1:(set @ ((product_prod @ C) @ X)),H_2:(set @ ((product_prod @ B) @ X)),G7:((labeled_graph @ A) @ X)]: ((((((((graph_homomorphism @ A) @ C) @ X) @ (((product_snd @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) @ R)) @ G7) @ H_1) => (((((((graph_homomorphism @ A) @ B) @ X) @ G_1) @ G7) @ H_2) => ((((ord_less_eq @ (set @ ((product_prod @ C) @ X))) @ (((((relcomp @ C) @ B) @ X) @ F_12) @ H_2)) @ H_1) => (? [H2:(set @ ((product_prod @ B) @ X))]: ((((((((graph_homomorphism @ A) @ B) @ X) @ G_2) @ G7) @ H2) & (((ord_less_eq @ (set @ ((product_prod @ B) @ X))) @ H_2) @ H2))))))))) => ((((((((((weak_universal @ X) @ A) @ C) @ B) @ T2) @ R) @ G_1) @ G_2) @ F_12) @ F_22))))).
% 2.87/1.55 thf(fact_192_relcomp__empty1, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ C)))) @ R) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.87/1.55 thf(fact_193_relcomp__empty2, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ C))]: (((((((relcomp @ A) @ C) @ B) @ R) @ (bot_bot @ (set @ ((product_prod @ C) @ B)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.87/1.55 thf(fact_194_empty__subsetI, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A5)))).
% 2.87/1.55 thf(fact_195_subset__empty, axiom, (! [A:$tType,A5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.87/1.55 thf(fact_196_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.87/1.55 thf(fact_197_graph__homomorphism__composes, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ C),X4:(set @ ((product_prod @ B) @ C)),C3:((labeled_graph @ A) @ D),Y3:(set @ ((product_prod @ C) @ D))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ B2) @ X4) => (((((((graph_homomorphism @ A) @ C) @ D) @ B2) @ C3) @ Y3) => ((((((graph_homomorphism @ A) @ B) @ D) @ A2) @ C3) @ (((((relcomp @ B) @ C) @ D) @ X4) @ Y3))))))).
% 2.87/1.55 thf(fact_198_graph__empty__e, axiom, (! [A:$tType,B:$tType,V3:(set @ B)]: ((((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V3) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V3)))))).
% 2.87/1.55 thf(fact_199_agree__on__empty, axiom, (! [A:$tType,C:$tType,B:$tType,F4:(set @ ((product_prod @ B) @ C)),G6:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ (bot_bot @ (set @ B)))) @ F4) @ G6)))).
% 2.87/1.55 thf(fact_200_graph__homomorphism__empty, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ (bot_bot @ (set @ B)))) @ G) @ F4) = ((F4 = (bot_bot @ (set @ ((product_prod @ B) @ C)))) & (G = (((restrict @ A) @ C) @ G))))))).
% 2.87/1.55 thf(fact_201_relcomp__mono, axiom, (! [A:$tType,C:$tType,B:$tType,R5:(set @ ((product_prod @ A) @ B)),R4:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ B) @ C)),S4:(set @ ((product_prod @ B) @ C))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R5) @ R4) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ C))) @ S5) @ S4) => (((ord_less_eq @ (set @ ((product_prod @ A) @ C))) @ (((((relcomp @ A) @ B) @ C) @ R5) @ S5)) @ (((((relcomp @ A) @ B) @ C) @ R4) @ S4))))))).
% 2.87/1.55 thf(fact_202_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.87/1.55 thf(fact_203_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.87/1.55 thf(fact_204_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.87/1.55 thf(fact_205_agree__on__ext, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,G:((labeled_graph @ A) @ B),F_12:(set @ ((product_prod @ B) @ C)),F_22:(set @ ((product_prod @ B) @ C)),G6:(set @ ((product_prod @ C) @ D))]: ((((((((agree_on @ A) @ B) @ C) @ G) @ F_12) @ F_22) => ((((((agree_on @ A) @ B) @ D) @ G) @ (((((relcomp @ B) @ C) @ D) @ F_12) @ G6)) @ (((((relcomp @ B) @ C) @ D) @ F_22) @ G6)))))).
% 2.87/1.55 thf(fact_206_O__assoc, axiom, (! [A:$tType,D:$tType,B:$tType,C:$tType,R:(set @ ((product_prod @ A) @ D)),S2:(set @ ((product_prod @ D) @ C)),T4:(set @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ (((((relcomp @ A) @ D) @ C) @ R) @ S2)) @ T4) = (((((relcomp @ A) @ D) @ B) @ R) @ (((((relcomp @ D) @ C) @ B) @ S2) @ T4)))))).
% 2.87/1.55 thf(fact_207_relcomp_OrelcompI, axiom, (! [A:$tType,C:$tType,B:$tType,A2:A,B2:B,R4:(set @ ((product_prod @ A) @ B)),C3:C,S4:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R4) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B2) @ C3)) @ S4) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A2) @ C3)) @ (((((relcomp @ A) @ B) @ C) @ R4) @ S4))))))).
% 2.87/1.55 thf(fact_208_relcomp_Oinducts, axiom, (! [B:$tType,A:$tType,C:$tType,X1:A,X22:C,R4:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ B) @ C)),P:(A > (C > $o))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X1) @ X22)) @ (((((relcomp @ A) @ B) @ C) @ R4) @ S4)) => ((! [A4:A,B4:B,C2:C]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R4) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B4) @ C2)) @ S4) => ((P @ A4) @ C2))))) => ((P @ X1) @ X22)))))).
% 2.87/1.55 thf(fact_209_relcomp_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,A1:A,A22:C,R4:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A1) @ A22)) @ (((((relcomp @ A) @ B) @ C) @ R4) @ S4)) = (? [A7:A,B6:B,C7:C]: (((A1 = A7) & ((A22 = C7) & ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B6)) @ R4) & (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B6) @ C7)) @ S4)))))))))).
% 2.87/1.55 thf(fact_210_relcomp_Ocases, axiom, (! [A:$tType,C:$tType,B:$tType,A1:A,A22:C,R4:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A1) @ A22)) @ (((((relcomp @ A) @ B) @ C) @ R4) @ S4)) => (~ ((! [B4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A1) @ B4)) @ R4) => (~ ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B4) @ A22)) @ S4)))))))))))).
% 2.87/1.55 thf(fact_211_relcompEpair, axiom, (! [A:$tType,B:$tType,C:$tType,A2:A,C3:B,R4:(set @ ((product_prod @ A) @ C)),S4:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ C3)) @ (((((relcomp @ A) @ C) @ B) @ R4) @ S4)) => (~ ((! [B4:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A2) @ B4)) @ R4) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ B4) @ C3)) @ S4)))))))))))).
% 2.87/1.55 thf(fact_212_relcompE, axiom, (! [A:$tType,B:$tType,C:$tType,Xz:((product_prod @ A) @ B),R4:(set @ ((product_prod @ A) @ C)),S4:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ Xz) @ (((((relcomp @ A) @ C) @ B) @ R4) @ S4)) => (~ ((! [X5:A,Y4:C,Z3:B]: (((Xz = ((((product_Pair @ A) @ B) @ X5) @ Z3)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X5) @ Y4)) @ R4) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ Y4) @ Z3)) @ S4))))))))))))).
% 2.87/1.55 thf(fact_213_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 2.87/1.55 thf(fact_214_infinite__imp__nonempty, axiom, (! [A:$tType,S2:(set @ A)]: (((~ (((finite_finite2 @ A) @ S2))) => (S2 != (bot_bot @ (set @ A))))))).
% 2.87/1.55 thf(fact_215_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((A5 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ A5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ Xa) @ X5) => (X5 = Xa)))))))))))))))).
% 2.87/1.55 thf(fact_216_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((A5 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ A5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ X5) @ Xa) => (X5 = Xa)))))))))))))))).
% 2.87/1.55 thf(fact_217_graph__homomorphism__nonempty, axiom, (! [B:$tType,A:$tType,C:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C)),E3:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ B5) @ F4) => ((((((semantics @ A) @ B) @ A5) @ E3) != (bot_bot @ (set @ ((product_prod @ B) @ B)))) => (((((semantics @ A) @ C) @ B5) @ E3) != (bot_bot @ (set @ ((product_prod @ C) @ C))))))))).
% 2.87/1.55 thf(fact_218_bounded__Max__nat, axiom, (! [P:(nat > $o),X4:nat,M2:nat]: (((P @ X4) => ((! [X5:nat]: (((P @ X5) => (((ord_less_eq @ nat) @ X5) @ M2)))) => (~ ((! [M3:nat]: (((P @ M3) => (~ ((! [X7:nat]: (((P @ X7) => (((ord_less_eq @ nat) @ X7) @ M3)))))))))))))))).
% 2.87/1.55 thf(fact_219_Id__on__vertices__identity_I2_J, axiom, (! [B:$tType,C:$tType,A:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C)),Aa2:B,Ba:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ B2) @ F4) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa2) @ Ba)) @ F4) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa2) @ Ba)) @ (((((relcomp @ B) @ C) @ C) @ F4) @ ((id_on @ C) @ (((labeled_vertices @ A) @ C) @ B2))))))))).
% 2.87/1.55 thf(fact_220_Id__on__vertices__identity_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C)),Aa2:B,Ba:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ B2) @ F4) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa2) @ Ba)) @ F4) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa2) @ Ba)) @ (((((relcomp @ B) @ B) @ C) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2))) @ F4))))))).
% 2.87/1.55 thf(fact_221_translation__homomorphism_I2_J, axiom, (! [A:$tType,B:$tType,E3:(allegorical_term @ A),G:((labeled_graph @ A) @ B),F4:(set @ ((product_prod @ nat) @ B))]: ((((((((graph_homomorphism @ A) @ nat) @ B) @ ((translation @ A) @ E3)) @ G) @ F4) => (((((semantics @ A) @ B) @ G) @ E3) != (bot_bot @ (set @ ((product_prod @ B) @ B)))))))).
% 2.87/1.55 thf(fact_222_weak__universalD, axiom, (! [A:$tType,C:$tType,X:$tType,B:$tType,T2:(itself @ X),R:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)),G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B),F_12:(set @ ((product_prod @ C) @ B)),F_22:(set @ ((product_prod @ C) @ B)),G:((labeled_graph @ A) @ X),H_12:(set @ ((product_prod @ C) @ X)),H_22:(set @ ((product_prod @ B) @ X))]: ((((((((((((weak_universal @ X) @ A) @ C) @ B) @ T2) @ R) @ G_1) @ G_2) @ F_12) @ F_22) => (((((((graph_homomorphism @ A) @ C) @ X) @ (((product_snd @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) @ R)) @ G) @ H_12) => (((((((graph_homomorphism @ A) @ B) @ X) @ G_1) @ G) @ H_22) => ((((ord_less_eq @ (set @ ((product_prod @ C) @ X))) @ (((((relcomp @ C) @ B) @ X) @ F_12) @ H_22)) @ H_12) => (? [H3:(set @ ((product_prod @ B) @ X))]: ((((((((graph_homomorphism @ A) @ B) @ X) @ G_2) @ G) @ H3) & (((ord_less_eq @ (set @ ((product_prod @ B) @ X))) @ H_22) @ H3))))))))))).
% 2.87/1.55 thf(fact_223_weak__universal__def, axiom, (! [A:$tType,C:$tType,B:$tType,X:$tType]: ((((((weak_universal @ X) @ A) @ C) @ B) = (^ [Uu:(itself @ X),R2:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)),G_12:((labeled_graph @ A) @ B),G_22:((labeled_graph @ A) @ B),F_1:(set @ ((product_prod @ C) @ B)),F_2:(set @ ((product_prod @ C) @ B))]: ((! [H_13:(set @ ((product_prod @ C) @ X)),H_23:(set @ ((product_prod @ B) @ X)),G4:((labeled_graph @ A) @ X)]: (((((((((graph_homomorphism @ A) @ C) @ X) @ (((product_snd @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C)) @ R2)) @ G4) @ H_13) & (((((((graph_homomorphism @ A) @ B) @ X) @ G_12) @ G4) @ H_23) & (((ord_less_eq @ (set @ ((product_prod @ C) @ X))) @ (((((relcomp @ C) @ B) @ X) @ F_1) @ H_23)) @ H_13))) => (? [H4:(set @ ((product_prod @ B) @ X))]: ((((((((graph_homomorphism @ A) @ B) @ X) @ G_22) @ G4) @ H4) & (((ord_less_eq @ (set @ ((product_prod @ B) @ X))) @ H_23) @ H4))))))))))))).
% 2.87/1.55 thf(fact_224_graph__homo__union__id_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((graph_union @ A) @ B) @ A5) @ B5)) @ G) @ F4) => ((B5 = (((restrict @ A) @ B) @ B5)) => ((((((graph_homomorphism @ A) @ B) @ C) @ B5) @ G) @ (((((relcomp @ B) @ B) @ C) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ B5))) @ F4))))))).
% 2.87/1.55 thf(fact_225_graph__homo__union__id_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((graph_union @ A) @ B) @ A5) @ B5)) @ G) @ F4) => ((A5 = (((restrict @ A) @ B) @ A5)) => ((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ G) @ (((((relcomp @ B) @ B) @ C) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) @ F4))))))).
% 2.87/1.55 thf(fact_226_finite__relcomp, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ B) @ C))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R) => (((finite_finite2 @ ((product_prod @ B) @ C)) @ S2) => ((finite_finite2 @ ((product_prod @ A) @ C)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S2))))))).
% 2.87/1.55 thf(fact_227_subset__emptyI, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A5))))) => (((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))))))).
% 2.87/1.55 thf(fact_228_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X6:A]: ((((member @ A) @ X6) @ (bot_bot @ (set @ A))))))))).
% 2.87/1.55 thf(fact_229_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R4:A,S4:B,R:(set @ ((product_prod @ A) @ B)),S5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R4) @ S4)) @ R) => ((S5 = S4) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R4) @ S5)) @ R)))))).
% 2.87/1.55 thf(fact_230_less__by__empty, axiom, (! [A:$tType,A5:(set @ ((product_prod @ A) @ A)),B5:(set @ ((product_prod @ A) @ A))]: (((A5 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A5) @ B5))))).
% 2.87/1.55 thf(fact_231_agree__iff__subset, axiom, (! [A:$tType,C:$tType,B:$tType,G:((labeled_graph @ A) @ B),X2:((labeled_graph @ A) @ C),F4:(set @ ((product_prod @ B) @ C)),G6:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ G) @ X2) @ F4) => ((((univalent @ B) @ C) @ G6) => (((((((agree_on @ A) @ B) @ C) @ G) @ F4) @ G6) = (((ord_less_eq @ (set @ ((product_prod @ B) @ C))) @ F4) @ G6))))))).
% 2.87/1.55 thf(fact_232_id__univalent, axiom, (! [A:$tType,X4:(set @ A)]: ((((univalent @ A) @ A) @ ((id_on @ A) @ X4))))).
% 2.87/1.55 thf(fact_233_univalentD, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),X4:A,Y3:B,Z2:B]: (((((univalent @ A) @ B) @ R) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ R) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Z2)) @ R) => (Z2 = Y3))))))).
% 2.87/1.55 thf(fact_234_univalent__def, axiom, (! [B:$tType,A:$tType]: ((((univalent @ A) @ B) = (^ [R2:(set @ ((product_prod @ A) @ B))]: ((! [X6:A,Y6:B,Z4:B]: ((((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Y6)) @ R2) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Z4)) @ R2)) => (Z4 = Y6)))))))))).
% 2.87/1.55 thf(fact_235_find__graph__occurence__vertices, axiom, (! [C:$tType,A:$tType,B:$tType,S2:(nat > ((labeled_graph @ A) @ B)),V2:(set @ C),F4:(set @ ((product_prod @ C) @ B))]: (((((chain @ A) @ B) @ S2) => (((finite_finite2 @ C) @ V2) => ((((univalent @ C) @ B) @ F4) => ((((ord_less_eq @ (set @ B)) @ ((((image @ C) @ B) @ F4) @ V2)) @ (((labeled_vertices @ A) @ B) @ (((chain_sup @ A) @ B) @ S2))) => (? [I2:nat]: ((((ord_less_eq @ (set @ B)) @ ((((image @ C) @ B) @ F4) @ V2)) @ (((labeled_vertices @ A) @ B) @ (S2 @ I2)))))))))))).
% 2.87/1.55 thf(fact_236_agree__on__subset, axiom, (! [C:$tType,B:$tType,A:$tType,F4:(set @ ((product_prod @ A) @ B)),G6:(set @ ((product_prod @ A) @ B)),G:((labeled_graph @ C) @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ F4) @ G6) => ((((ord_less_eq @ (set @ A)) @ (((labeled_vertices @ C) @ A) @ G)) @ (((domain @ A) @ B) @ F4)) => ((((univalent @ A) @ B) @ G6) => ((((((agree_on @ C) @ A) @ B) @ G) @ F4) @ G6))))))).
% 2.87/1.55 thf(fact_237_ImageI, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R4:(set @ ((product_prod @ A) @ B)),A5:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R4) => ((((member @ A) @ A2) @ A5) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R4) @ A5))))))).
% 2.87/1.55 thf(fact_238_Image__empty2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((image @ B) @ A) @ R) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.87/1.55 thf(fact_239_Domain__Id__on, axiom, (! [A:$tType,A5:(set @ A)]: (((((domain @ A) @ A) @ ((id_on @ A) @ A5)) = A5)))).
% 2.87/1.55 thf(fact_240_Image__empty1, axiom, (! [B:$tType,A:$tType,X2:(set @ B)]: ((((((image @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) @ X2) = (bot_bot @ (set @ A)))))).
% 2.87/1.55 thf(fact_241_Domain__empty, axiom, (! [B:$tType,A:$tType]: (((((domain @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (bot_bot @ (set @ A)))))).
% 2.87/1.55 thf(fact_242_univalent__finite_I2_J, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((univalent @ A) @ B) @ R) => (((finite_finite2 @ A) @ (((domain @ A) @ B) @ R)) = ((finite_finite2 @ ((product_prod @ A) @ B)) @ R)))))).
% 2.87/1.55 thf(fact_243_Domain__O, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(set @ A),X4:(set @ ((product_prod @ A) @ B)),Y3:(set @ ((product_prod @ B) @ C))]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((domain @ A) @ B) @ X4)) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ X4) @ A2)) @ (((domain @ B) @ C) @ Y3)) => (((ord_less_eq @ (set @ A)) @ A2) @ (((domain @ A) @ C) @ (((((relcomp @ A) @ B) @ C) @ X4) @ Y3)))))))).
% 2.87/1.55 thf(fact_244_finite__Domain, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R4) => ((finite_finite2 @ A) @ (((domain @ A) @ B) @ R4)))))).
% 2.87/1.55 thf(fact_245_finite__Image, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),A5:(set @ A)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ R) @ A5)))))).
% 2.87/1.55 thf(fact_246_Image__mono, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),R4:(set @ ((product_prod @ A) @ B)),A8:(set @ A),A5:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R5) @ R4) => ((((ord_less_eq @ (set @ A)) @ A8) @ A5) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R5) @ A8)) @ ((((image @ A) @ B) @ R4) @ A5))))))).
% 2.87/1.55 thf(fact_247_Domain__mono, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ S4) => (((ord_less_eq @ (set @ A)) @ (((domain @ A) @ B) @ R4)) @ (((domain @ A) @ B) @ S4)))))).
% 2.87/1.55 thf(fact_248_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X4:A,R4:(set @ ((product_prod @ A) @ B)),P:(A > $o)]: (((((member @ A) @ X4) @ (((domain @ A) @ B) @ R4)) => ((! [A4:A,B4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R4) => (P @ A4)))) => (P @ X4)))))).
% 2.87/1.55 thf(fact_249_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R4:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R4) => (((member @ A) @ A2) @ (((domain @ A) @ B) @ R4)))))).
% 2.87/1.55 thf(fact_250_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A2:A,R4:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R4)) = (? [A7:A,B6:B]: (((A2 = A7) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B6)) @ R4)))))))).
% 2.87/1.55 thf(fact_251_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A2:A,R4:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R4)) => (~ ((! [B4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B4)) @ R4))))))))))).
% 2.87/1.55 thf(fact_252_Domain__iff, axiom, (! [A:$tType,B:$tType,A2:A,R4:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R4)) = (? [Y6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ Y6)) @ R4))))))).
% 2.87/1.55 thf(fact_253_DomainE, axiom, (! [B:$tType,A:$tType,A2:A,R4:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R4)) => (~ ((! [B4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B4)) @ R4))))))))))).
% 2.87/1.55 thf(fact_254_rev__ImageI, axiom, (! [B:$tType,A:$tType,A2:A,A5:(set @ A),B2:B,R4:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ A5) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R4) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R4) @ A5))))))).
% 2.87/1.55 thf(fact_255_Image__iff, axiom, (! [A:$tType,B:$tType,B2:A,R4:(set @ ((product_prod @ B) @ A)),A5:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R4) @ A5)) = (? [X6:B]: (((((member @ B) @ X6) @ A5) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X6) @ B2)) @ R4)))))))).
% 2.87/1.55 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A9:$tType,A10:$tType]: (((order_bot @ A10) => (order_bot @ (A9 > A10)))))).
% 2.87/1.55 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.87/1.55 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 2.87/1.55 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.87/1.55 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.87/1.55 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1, axiom, (order_bot @ nat)).
% 2.87/1.55 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, (preorder @ nat)).
% 2.87/1.55 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.87/1.55 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, (order @ nat)).
% 2.87/1.55 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, (ord @ nat)).
% 2.87/1.55 thf(tcon_Set_Oset___Orderings_Oorder__bot_5, axiom, (! [A9:$tType]: ((order_bot @ (set @ A9))))).
% 2.87/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_6, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.87/1.55 thf(tcon_Set_Oset___Finite__Set_Ofinite_7, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 2.87/1.55 thf(tcon_Set_Oset___Orderings_Oorder_8, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.87/1.55 thf(tcon_Set_Oset___Orderings_Oord_9, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.87/1.55 thf(tcon_HOL_Obool___Orderings_Oorder__bot_10, axiom, (order_bot @ $o)).
% 2.87/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_11, axiom, (preorder @ $o)).
% 2.87/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder_12, axiom, (linorder @ $o)).
% 2.87/1.55 thf(tcon_HOL_Obool___Finite__Set_Ofinite_13, axiom, (finite_finite @ $o)).
% 2.87/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_14, axiom, (order @ $o)).
% 2.87/1.55 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, (ord @ $o)).
% 2.87/1.55 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_16, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((product_prod @ A9) @ A10)))))).
% 2.87/1.55 thf(conj_0, conjecture, (((((((graph_homomorphism @ b) @ nat) @ nat) @ (((product_fst @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))))) @ (((product_snd @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))))) @ ((id_on @ nat) @ (((labeled_vertices @ b) @ nat) @ (((product_fst @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))))))) & (((((product_snd @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v)))))) = (((restrict @ b) @ nat) @ (((product_snd @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v)))))))) & (((finite_finite2 @ nat) @ (((labeled_vertices @ b) @ nat) @ (((product_snd @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v)))))))) & ((finite_finite2 @ ((product_prod @ b) @ ((product_prod @ nat) @ nat))) @ (((labeled_edges @ b) @ nat) @ (((product_snd @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((((product_Pair @ ((labeled_graph @ b) @ nat)) @ ((labeled_graph @ b) @ nat)) @ ((translation @ b) @ (((product_fst @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v))))) @ ((translation @ b) @ (((product_snd @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ ((((product_Pair @ (allegorical_term @ b)) @ (allegorical_term @ b)) @ u) @ (((allegorical_A_Int @ b) @ u) @ v)))))))))))).
% 2.87/1.55 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2DTF32651.p
% 2.87/1.55 ------------------------
% 2.87/1.59 ---- Cleaned THF ---
% 2.87/1.59 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 2.87/1.59 standard_Constant: $tType > $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 2.87/1.59 allegorical_term: $tType > $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 2.87/1.59 labeled_graph: $tType > $tType > $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_t_Product__Type_Oprod,type,
% 2.87/1.59 product_prod: $tType > $tType > $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_t_Set_Oset,type,
% 2.87/1.59 set: $tType > $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_t_Nat_Onat,type,
% 2.87/1.59 nat: $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_t_itself,type,
% 2.87/1.59 itself: $tType > $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_tf_b,type,
% 2.87/1.59 b: $tType ).
% 2.87/1.59
% 2.87/1.59 thf(ty_tf_a,type,
% 2.87/1.59 a: $tType ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_HOL_Otype,type,
% 2.87/1.59 type:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_Orderings_Oord,type,
% 2.87/1.59 ord:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_Orderings_Oorder,type,
% 2.87/1.59 order:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.87/1.59 finite_finite:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_Orderings_Olinorder,type,
% 2.87/1.59 linorder:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_Orderings_Opreorder,type,
% 2.87/1.59 preorder:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.87/1.59 order_bot:
% 2.87/1.59 !>[A: $tType] : $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Finite__Set_Ofinite,type,
% 2.87/1.59 finite_finite2:
% 2.87/1.59 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphSemantics_Oallegorical__term_OA__Int,type,
% 2.87/1.59 allegorical_A_Int:
% 2.87/1.59 !>[V: $tType] : ( ( allegorical_term @ V ) > ( allegorical_term @ V ) > ( allegorical_term @ V ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 2.87/1.59 semantics:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 2.87/1.59 graph_homomorphism:
% 2.87/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 2.87/1.59 graph_union:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 2.87/1.59 labeled_LG:
% 2.87/1.59 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 2.87/1.59 labele1974067554_graph:
% 2.87/1.59 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 2.87/1.59 labeled_edges:
% 2.87/1.59 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 2.87/1.59 labeled_vertices:
% 2.87/1.59 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_LabeledGraphs_Orestrict,type,
% 2.87/1.59 restrict:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_MissingRelation_Ounivalent,type,
% 2.87/1.59 univalent:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.87/1.59 bot_bot:
% 2.87/1.59 !>[A: $tType] : A ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.87/1.59 ord_less_eq:
% 2.87/1.59 !>[A: $tType] : ( A > A > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Product__Type_OPair,type,
% 2.87/1.59 product_Pair:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.87/1.59 product_rec_prod:
% 2.87/1.59 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.87/1.59 product_fst:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.87/1.59 product_snd:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Relation_ODomain,type,
% 2.87/1.59 domain:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Relation_OId__on,type,
% 2.87/1.59 id_on:
% 2.87/1.59 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Relation_OImage,type,
% 2.87/1.59 image:
% 2.87/1.59 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Relation_Orelcomp,type,
% 2.87/1.59 relcomp:
% 2.87/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RuleSemanticsConnection_Oinv__translation,type,
% 2.87/1.59 inv_translation: ( set @ nat ) > $o ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RuleSemanticsConnection_Otranslation,type,
% 2.87/1.59 translation:
% 2.87/1.59 !>[C: $tType] : ( ( allegorical_term @ C ) > ( labeled_graph @ C @ nat ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oagree__on,type,
% 2.87/1.59 agree_on:
% 2.87/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Ochain,type,
% 2.87/1.59 chain:
% 2.87/1.59 !>[L: $tType,V: $tType] : ( ( nat > ( labeled_graph @ L @ V ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Ochain__sup,type,
% 2.87/1.59 chain_sup:
% 2.87/1.59 !>[L: $tType,V: $tType] : ( ( nat > ( labeled_graph @ L @ V ) ) > ( labeled_graph @ L @ V ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 2.87/1.59 consequence_graph:
% 2.87/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oextensible,type,
% 2.87/1.59 extensible:
% 2.87/1.59 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ X @ V ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Ofair__chain,type,
% 2.87/1.59 fair_chain:
% 2.87/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( nat > ( labeled_graph @ A @ C ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Ofin__maintained,type,
% 2.87/1.59 fin_maintained:
% 2.87/1.59 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oleast,type,
% 2.87/1.59 least:
% 2.87/1.59 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 2.87/1.59 least_559130134_graph:
% 2.87/1.59 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Omaintained,type,
% 2.87/1.59 maintained:
% 2.87/1.59 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Opushout__step,type,
% 2.87/1.59 pushout_step:
% 2.87/1.59 !>[X: $tType,A: $tType,C: $tType,B: $tType] : ( ( itself @ X ) > ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oset__of__graph__rules,type,
% 2.87/1.59 set_of_graph_rules:
% 2.87/1.59 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_RulesAndChains_Oweak__universal,type,
% 2.87/1.59 weak_universal:
% 2.87/1.59 !>[X: $tType,A: $tType,C: $tType,B: $tType] : ( ( itself @ X ) > ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_Set_OCollect,type,
% 2.87/1.59 collect:
% 2.87/1.59 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules,type,
% 2.87/1.59 standa1138209853_rules:
% 2.87/1.59 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
% 2.87/1.59 standa2002409347_rules:
% 2.87/1.59 !>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 2.87/1.59 standa244753842y_rule:
% 2.87/1.59 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule,type,
% 2.87/1.59 standa245363227y_rule:
% 2.87/1.59 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule,type,
% 2.87/1.59 standa400804411p_rule:
% 2.87/1.59 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_c_member,type,
% 2.87/1.59 member:
% 2.87/1.59 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.87/1.59
% 2.87/1.59 thf(sy_v_C,type,
% 2.87/1.59 c: set @ a ).
% 2.87/1.59
% 2.87/1.59 thf(sy_v_u,type,
% 2.87/1.59 u: allegorical_term @ b ).
% 2.87/1.59
% 2.87/1.59 thf(sy_v_v,type,
% 2.87/1.59 v: allegorical_term @ b ).
% 2.87/1.59
% 2.87/1.59 thf(sy_v_x,type,
% 2.87/1.59 x: product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_0_assms,axiom,
% 2.87/1.59 member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ x @ ( standa1138209853_rules @ a @ c ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_1_graph__rule__translation,axiom,
% 2.87/1.59 ! [A: $tType,X2: allegorical_term @ A,Y: allegorical_term @ A] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ nat @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) @ ( id_on @ nat @ ( labeled_vertices @ A @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) ) )
% 2.87/1.59 & ( ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) )
% 2.87/1.59 = ( restrict @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ nat @ ( labeled_vertices @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ X2 ) @ ( translation @ A @ ( allegorical_A_Int @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_2_subgraph__refl,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 2.87/1.59 = ( G
% 2.87/1.59 = ( restrict @ A @ B @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_3_subgraph__restrict,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 2.87/1.59 = ( G
% 2.87/1.59 = ( restrict @ A @ B @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_4_graph__homomorphism__Id,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A2 ) @ ( restrict @ A @ B @ A2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_5_verts__in__translation__finite_I2_J,axiom,
% 2.87/1.59 ! [A: $tType,X2: allegorical_term @ A] : ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ A @ nat @ ( translation @ A @ X2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_6_are__rules_I1_J,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ nat @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) @ ( id_on @ nat @ ( labeled_vertices @ A @ nat @ ( product_fst @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) ) )
% 2.87/1.59 & ( ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) )
% 2.87/1.59 = ( restrict @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ nat @ ( labeled_vertices @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ A @ nat @ ( product_snd @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( standa244753842y_rule @ A ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_7_are__rules_I2_J,axiom,
% 2.87/1.59 ! [B: $tType,T2: B] :
% 2.87/1.59 ( ( graph_homomorphism @ B @ nat @ nat @ ( product_fst @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) @ ( id_on @ nat @ ( labeled_vertices @ B @ nat @ ( product_fst @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) ) )
% 2.87/1.59 & ( ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) )
% 2.87/1.59 = ( restrict @ B @ nat @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ nat @ ( labeled_vertices @ B @ nat @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ B @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ B @ nat @ ( product_snd @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( standa400804411p_rule @ B @ T2 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_8_verts__in__translation__finite_I1_J,axiom,
% 2.87/1.59 ! [A: $tType,X2: allegorical_term @ A] : ( finite_finite2 @ nat @ ( labeled_vertices @ A @ nat @ ( translation @ A @ X2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_9_are__rules_I3_J,axiom,
% 2.87/1.59 ! [C: $tType,I: C] :
% 2.87/1.59 ( ( graph_homomorphism @ C @ nat @ nat @ ( product_fst @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) @ ( id_on @ nat @ ( labeled_vertices @ C @ nat @ ( product_fst @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) ) )
% 2.87/1.59 & ( ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) )
% 2.87/1.59 = ( restrict @ C @ nat @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ nat @ ( labeled_vertices @ C @ nat @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ C @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ C @ nat @ ( product_snd @ ( labeled_graph @ C @ nat ) @ ( labeled_graph @ C @ nat ) @ ( standa245363227y_rule @ C @ I ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_10_prod_Ocollapse,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.87/1.59 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.87/1.59 = Prod ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_11_translation__graph,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( translation @ A )
% 2.87/1.59 = ( ^ [X3: allegorical_term @ A] : ( restrict @ A @ nat @ ( translation @ A @ X3 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_12_vertices__restrict,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
% 2.87/1.59 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
% 2.87/1.59 = ( labeled_vertices @ B @ A @ G ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_13_old_Oprod_Oinject,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.87/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.87/1.59 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.87/1.59 = ( ( A2 = A3 )
% 2.87/1.59 & ( B2 = B3 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_14_prod_Oinject,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.87/1.59 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.87/1.59 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.87/1.59 = ( ( X1 = Y1 )
% 2.87/1.59 & ( X22 = Y2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_15_restrict__idemp,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,X4: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X4 ) )
% 2.87/1.59 = ( restrict @ A @ B @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_16_old_Oprod_Oinducts,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.87/1.59 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.87/1.59 => ( P @ Prod ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_17_old_Oprod_Oexhaust,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.87/1.59 ~ ! [A4: A,B4: B] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_18_prod__induct7,axiom,
% 2.87/1.59 ! [G2: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 2.87/1.59 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) )
% 2.87/1.59 => ( P @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_19_prod__induct6,axiom,
% 2.87/1.59 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.87/1.59 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.87/1.59 => ( P @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_20_prod__induct5,axiom,
% 2.87/1.59 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.87/1.59 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.87/1.59 => ( P @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_21_prod__induct4,axiom,
% 2.87/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.87/1.59 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.87/1.59 => ( P @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_22_prod__induct3,axiom,
% 2.87/1.59 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 2.87/1.59 => ( P @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_23_prod__cases7,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
% 2.87/1.59 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_24_prod__cases6,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.87/1.59 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_25_prod__cases5,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.87/1.59 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_26_prod__cases4,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.87/1.59 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_27_prod__cases3,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.87/1.59 ~ ! [A4: A,B4: B,C2: C] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_28_Pair__inject,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.87/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.87/1.59 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.87/1.59 => ~ ( ( A2 = A3 )
% 2.87/1.59 => ( B2 != B3 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_29_prod__cases,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.87/1.59 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.87/1.59 => ( P @ P2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_30_surj__pair,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.87/1.59 ? [X5: A,Y4: B] :
% 2.87/1.59 ( P2
% 2.87/1.59 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_31_fst__conv,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 2.87/1.59 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.87/1.59 = X1 ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_32_fst__eqD,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,X4: A,Y3: B,A2: A] :
% 2.87/1.59 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X4 @ Y3 ) )
% 2.87/1.59 = A2 )
% 2.87/1.59 => ( X4 = A2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_33_snd__conv,axiom,
% 2.87/1.59 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 2.87/1.59 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 2.87/1.59 = X22 ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_34_snd__eqD,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,X4: B,Y3: A,A2: A] :
% 2.87/1.59 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) )
% 2.87/1.59 = A2 )
% 2.87/1.59 => ( Y3 = A2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_35_prod__eq__iff,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType] :
% 2.87/1.59 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 2.87/1.59 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 2.87/1.59 ( ( ( product_fst @ A @ B @ S )
% 2.87/1.59 = ( product_fst @ A @ B @ T3 ) )
% 2.87/1.59 & ( ( product_snd @ A @ B @ S )
% 2.87/1.59 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_36_prod_Oexpand,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.87/1.59 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.87/1.59 = ( product_fst @ A @ B @ Prod2 ) )
% 2.87/1.59 & ( ( product_snd @ A @ B @ Prod )
% 2.87/1.59 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.87/1.59 => ( Prod = Prod2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_37_prod__eqI,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 2.87/1.59 ( ( ( product_fst @ A @ B @ P2 )
% 2.87/1.59 = ( product_fst @ A @ B @ Q ) )
% 2.87/1.59 => ( ( ( product_snd @ A @ B @ P2 )
% 2.87/1.59 = ( product_snd @ A @ B @ Q ) )
% 2.87/1.59 => ( P2 = Q ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_38_labeled__graph_Oexpand,axiom,
% 2.87/1.59 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 2.87/1.59 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 2.87/1.59 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 2.87/1.59 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 2.87/1.59 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 2.87/1.59 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_39_surjective__pairing,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.87/1.59 ( T2
% 2.87/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_40_prod_Oexhaust__sel,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.87/1.59 ( Prod
% 2.87/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_41_subgraph__preserves__hom,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,X2: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ C @ B @ X2 @ A5 @ H )
% 2.87/1.59 => ( graph_homomorphism @ A @ C @ B @ X2 @ B5 @ H ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_42_subgraph__trans,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 2.87/1.59 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_43_Id__onI,axiom,
% 2.87/1.59 ! [A: $tType,A2: A,A5: set @ A] :
% 2.87/1.59 ( ( member @ A @ A2 @ A5 )
% 2.87/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_44_set__of__graph__rules__def,axiom,
% 2.87/1.59 ! [V: $tType,L: $tType] :
% 2.87/1.59 ( ( set_of_graph_rules @ L @ V )
% 2.87/1.59 = ( ^ [Rs: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) )] :
% 2.87/1.59 ! [X6: product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V )] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) @ X6 @ Rs )
% 2.87/1.59 => ( ( graph_homomorphism @ L @ V @ V @ ( product_fst @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) @ ( id_on @ V @ ( labeled_vertices @ L @ V @ ( product_fst @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) ) )
% 2.87/1.59 & ( ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 )
% 2.87/1.59 = ( restrict @ L @ V @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) )
% 2.87/1.59 & ( finite_finite2 @ V @ ( labeled_vertices @ L @ V @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ L @ ( product_prod @ V @ V ) ) @ ( labeled_edges @ L @ V @ ( product_snd @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) @ X6 ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.87/1.59 ! [A: $tType,A2: A,P: A > $o] :
% 2.87/1.59 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.87/1.59 = ( P @ A2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.87/1.59 ! [A: $tType,A5: set @ A] :
% 2.87/1.59 ( ( collect @ A
% 2.87/1.59 @ ^ [X6: A] : ( member @ A @ X6 @ A5 ) )
% 2.87/1.59 = A5 ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_47_Collect__cong,axiom,
% 2.87/1.59 ! [A: $tType,P: A > $o,Q2: A > $o] :
% 2.87/1.59 ( ! [X5: A] :
% 2.87/1.59 ( ( P @ X5 )
% 2.87/1.59 = ( Q2 @ X5 ) )
% 2.87/1.59 => ( ( collect @ A @ P )
% 2.87/1.59 = ( collect @ A @ Q2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_48_verts__in__translation,axiom,
% 2.87/1.59 ! [A: $tType,X2: allegorical_term @ A] : ( inv_translation @ ( labeled_vertices @ A @ nat @ ( translation @ A @ X2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_49_old_Oprod_Orec,axiom,
% 2.87/1.59 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.87/1.59 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.87/1.59 = ( F1 @ A2 @ B2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_50_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,P: A > B > $o,X4: A,Y3: B,A2: product_prod @ A @ B] :
% 2.87/1.59 ( ( P @ X4 @ Y3 )
% 2.87/1.59 => ( ( A2
% 2.87/1.59 = ( product_Pair @ A @ B @ X4 @ Y3 ) )
% 2.87/1.59 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_51_conjI__realizer,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q2: B > $o,Q: B] :
% 2.87/1.59 ( ( P @ P2 )
% 2.87/1.59 => ( ( Q2 @ Q )
% 2.87/1.59 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
% 2.87/1.59 & ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_52_exI__realizer,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,P: A > B > $o,Y3: A,X4: B] :
% 2.87/1.59 ( ( P @ Y3 @ X4 )
% 2.87/1.59 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_53_allegorical__term_Oinject_I1_J,axiom,
% 2.87/1.59 ! [V: $tType,X11: allegorical_term @ V,X12: allegorical_term @ V,Y11: allegorical_term @ V,Y12: allegorical_term @ V] :
% 2.87/1.59 ( ( ( allegorical_A_Int @ V @ X11 @ X12 )
% 2.87/1.59 = ( allegorical_A_Int @ V @ Y11 @ Y12 ) )
% 2.87/1.59 = ( ( X11 = Y11 )
% 2.87/1.59 & ( X12 = Y12 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_54_fin__maintainedI,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C] :
% 2.87/1.59 ( ! [F3: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ( ( F3
% 2.87/1.59 = ( restrict @ A @ B @ F3 ) )
% 2.87/1.59 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ F3 ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ F3 ) ) )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ B @ F3 @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F3 ) ) )
% 2.87/1.59 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ F3 @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) @ G @ F2 )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ C @ F3 @ G @ F2 )
% 2.87/1.59 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ F3 @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) @ G @ F2 ) ) ) ) )
% 2.87/1.59 => ( fin_maintained @ A @ B @ C @ R @ G ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_55_set__of__graph__rulesD_I1_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( set_of_graph_rules @ A @ B @ Rs2 )
% 2.87/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 2.87/1.59 => ( ( ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R )
% 2.87/1.59 = ( restrict @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 2.87/1.59 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_56_set__of__graph__rulesD_I2_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( set_of_graph_rules @ A @ B @ Rs2 )
% 2.87/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 2.87/1.59 => ( ( ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R )
% 2.87/1.59 = ( restrict @ A @ B @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 2.87/1.59 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_57_extensible__refl,axiom,
% 2.87/1.59 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ C @ R @ G @ F4 )
% 2.87/1.59 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G @ F4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_58_set__of__graph__rulesD_I3_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( set_of_graph_rules @ A @ B @ Rs2 )
% 2.87/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 2.87/1.59 => ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_59_exE__realizer_H,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 2.87/1.59 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 2.87/1.59 => ~ ! [X5: B,Y4: A] :
% 2.87/1.59 ~ ( P @ Y4 @ X5 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_60_Id__onE,axiom,
% 2.87/1.59 ! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
% 2.87/1.59 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
% 2.87/1.59 => ~ ! [X5: A] :
% 2.87/1.59 ( ( member @ A @ X5 @ A5 )
% 2.87/1.59 => ( C3
% 2.87/1.59 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_61_Id__on__eqI,axiom,
% 2.87/1.59 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 2.87/1.59 ( ( A2 = B2 )
% 2.87/1.59 => ( ( member @ A @ A2 @ A5 )
% 2.87/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_62_Id__on__iff,axiom,
% 2.87/1.59 ! [A: $tType,X4: A,Y3: A,A5: set @ A] :
% 2.87/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( id_on @ A @ A5 ) )
% 2.87/1.59 = ( ( X4 = Y3 )
% 2.87/1.59 & ( member @ A @ X4 @ A5 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_63_fin__maintained__def,axiom,
% 2.87/1.59 ! [V: $tType,X: $tType,L: $tType] :
% 2.87/1.59 ( ( fin_maintained @ L @ X @ V )
% 2.87/1.59 = ( ^ [R2: product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ),G4: labeled_graph @ L @ V] :
% 2.87/1.59 ! [F5: labeled_graph @ L @ X,F6: set @ ( product_prod @ X @ V )] :
% 2.87/1.59 ( ( ( F5
% 2.87/1.59 = ( restrict @ L @ X @ F5 ) )
% 2.87/1.59 & ( finite_finite2 @ X @ ( labeled_vertices @ L @ X @ F5 ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ L @ ( product_prod @ X @ X ) ) @ ( labeled_edges @ L @ X @ F5 ) ) )
% 2.87/1.59 => ( ( graph_homomorphism @ L @ X @ X @ F5 @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ ( id_on @ X @ ( labeled_vertices @ L @ X @ F5 ) ) )
% 2.87/1.59 => ( ( extensible @ L @ X @ V @ ( product_Pair @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ F5 @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) ) @ G4 @ F6 )
% 2.87/1.59 => ( ( graph_homomorphism @ L @ X @ V @ F5 @ G4 @ F6 )
% 2.87/1.59 => ( extensible @ L @ X @ V @ ( product_Pair @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ F5 @ ( product_snd @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) ) @ G4 @ F6 ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_64_fin__maintained__maintained,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C] :
% 2.87/1.59 ( ( ( ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R )
% 2.87/1.59 = ( restrict @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 2.87/1.59 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) )
% 2.87/1.59 => ( ( fin_maintained @ A @ B @ C @ R @ G )
% 2.87/1.59 = ( maintained @ A @ B @ C @ R @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_65_consequence__graphD_I2_J,axiom,
% 2.87/1.59 ! [C: $tType,B: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( consequence_graph @ A @ B @ C @ Rs2 @ G )
% 2.87/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 2.87/1.59 => ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_66_finite__code,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( finite_finite @ A )
% 2.87/1.59 => ( ( finite_finite2 @ A )
% 2.87/1.59 = ( ^ [A6: set @ A] : $true ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_67_fair__chainD_I2_J,axiom,
% 2.87/1.59 ! [A: $tType,C: $tType,B: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S2: nat > ( labeled_graph @ A @ C ),R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),I: nat,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ( fair_chain @ A @ B @ C @ Rs2 @ S2 )
% 2.87/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ C @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ ( S2 @ I ) @ F4 )
% 2.87/1.59 => ? [J: nat] : ( extensible @ A @ B @ C @ R @ ( S2 @ J ) @ F4 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_68_identity__rules__graph__rule,axiom,
% 2.87/1.59 ! [A: $tType,X4: product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ),L2: set @ ( standard_Constant @ A )] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ X4 @ ( standa2002409347_rules @ A @ L2 ) )
% 2.87/1.59 => ( ( graph_homomorphism @ ( standard_Constant @ A ) @ nat @ nat @ ( product_fst @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) @ ( id_on @ nat @ ( labeled_vertices @ ( standard_Constant @ A ) @ nat @ ( product_fst @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) ) )
% 2.87/1.59 & ( ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 )
% 2.87/1.59 = ( restrict @ ( standard_Constant @ A ) @ nat @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) )
% 2.87/1.59 & ( finite_finite2 @ nat @ ( labeled_vertices @ ( standard_Constant @ A ) @ nat @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ ( standard_Constant @ A ) @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ ( standard_Constant @ A ) @ nat @ ( product_snd @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ X4 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_69_maintainedI,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B5: labeled_graph @ C @ A] :
% 2.87/1.59 ( ! [F2: set @ ( product_prod @ A @ B )] :
% 2.87/1.59 ( ( graph_homomorphism @ C @ A @ B @ A5 @ G @ F2 )
% 2.87/1.59 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G @ F2 ) )
% 2.87/1.59 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_70_sndI,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,X4: product_prod @ A @ B,Y3: A,Z2: B] :
% 2.87/1.59 ( ( X4
% 2.87/1.59 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 2.87/1.59 => ( ( product_snd @ A @ B @ X4 )
% 2.87/1.59 = Z2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_71_finite__identity__rules,axiom,
% 2.87/1.59 ! [A: $tType,L2: set @ ( standard_Constant @ A )] :
% 2.87/1.59 ( ( finite_finite2 @ ( standard_Constant @ A ) @ L2 )
% 2.87/1.59 => ( finite_finite2 @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa2002409347_rules @ A @ L2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_72_maintained__refl,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,R: labeled_graph @ A @ B,G: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_73_consequence__graphI,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 2.87/1.59 ( ! [R3: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R3 @ Rs2 )
% 2.87/1.59 => ( maintained @ A @ B @ C @ R3 @ G ) )
% 2.87/1.59 => ( ! [R3: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R3 @ Rs2 )
% 2.87/1.59 => ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) ) ) )
% 2.87/1.59 => ( ( G
% 2.87/1.59 = ( restrict @ A @ C @ G ) )
% 2.87/1.59 => ( consequence_graph @ A @ B @ C @ Rs2 @ G ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_74_consequence__graphD_I1_J,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( consequence_graph @ A @ B @ C @ Rs2 @ G )
% 2.87/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs2 )
% 2.87/1.59 => ( maintained @ A @ B @ C @ R @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_75_consequence__graphD_I3_J,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 2.87/1.59 ( ( consequence_graph @ A @ B @ C @ Rs2 @ G )
% 2.87/1.59 => ( G
% 2.87/1.59 = ( restrict @ A @ C @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_76_maintained__then__fin__maintained,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C] :
% 2.87/1.59 ( ( maintained @ A @ B @ C @ R @ G )
% 2.87/1.59 => ( fin_maintained @ A @ B @ C @ R @ G ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_77_finite,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( finite_finite @ A )
% 2.87/1.59 => ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_78_finite__set__choice,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
% 2.87/1.59 ( ( finite_finite2 @ A @ A5 )
% 2.87/1.59 => ( ! [X5: A] :
% 2.87/1.59 ( ( member @ A @ X5 @ A5 )
% 2.87/1.59 => ? [X_1: B] : ( P @ X5 @ X_1 ) )
% 2.87/1.59 => ? [F2: A > B] :
% 2.87/1.59 ! [X7: A] :
% 2.87/1.59 ( ( member @ A @ X7 @ A5 )
% 2.87/1.59 => ( P @ X7 @ ( F2 @ X7 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_79_consequence__graph__def,axiom,
% 2.87/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.87/1.59 ( ( consequence_graph @ A @ B @ C )
% 2.87/1.59 = ( ^ [Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G4: labeled_graph @ A @ C] :
% 2.87/1.59 ( ( G4
% 2.87/1.59 = ( restrict @ A @ C @ G4 ) )
% 2.87/1.59 & ! [X6: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ X6 @ Rs )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ X6 ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ X6 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ X6 ) ) ) )
% 2.87/1.59 & ( maintained @ A @ B @ C @ X6 @ G4 ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_80_maintained__def,axiom,
% 2.87/1.59 ! [V: $tType,X: $tType,L: $tType] :
% 2.87/1.59 ( ( maintained @ L @ X @ V )
% 2.87/1.59 = ( ^ [R2: product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ),G4: labeled_graph @ L @ V] :
% 2.87/1.59 ! [F6: set @ ( product_prod @ X @ V )] :
% 2.87/1.59 ( ( graph_homomorphism @ L @ X @ V @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ G4 @ F6 )
% 2.87/1.59 => ( extensible @ L @ X @ V @ R2 @ G4 @ F6 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_81_maintainedD,axiom,
% 2.87/1.59 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F4 )
% 2.87/1.59 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G @ F4 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_82_fstI,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,Y3: A,Z2: B] :
% 2.87/1.59 ( ( X4
% 2.87/1.59 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 2.87/1.59 => ( ( product_fst @ A @ B @ X4 )
% 2.87/1.59 = Y3 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_83_least__consequence__graphI,axiom,
% 2.87/1.59 ! [X: $tType,A: $tType,C: $tType,L: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G: labeled_graph @ L @ C,S2: labeled_graph @ L @ C,T2: itself @ X] :
% 2.87/1.59 ( ( consequence_graph @ L @ A @ C @ Rs2 @ G )
% 2.87/1.59 => ( ( graph_homomorphism @ L @ C @ C @ S2 @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 2.87/1.59 => ( ! [C4: labeled_graph @ L @ X] :
% 2.87/1.59 ( ( consequence_graph @ L @ A @ X @ Rs2 @ C4 )
% 2.87/1.59 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G ) @ C4 ) )
% 2.87/1.59 => ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs2 @ S2 @ G ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_84_leastI,axiom,
% 2.87/1.59 ! [X: $tType,A: $tType,C: $tType,L: $tType,S2: labeled_graph @ L @ C,G: labeled_graph @ L @ C,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T2: itself @ X] :
% 2.87/1.59 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 2.87/1.59 => ( ! [C4: labeled_graph @ L @ X] :
% 2.87/1.59 ( ( consequence_graph @ L @ A @ X @ Rs2 @ C4 )
% 2.87/1.59 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G ) @ C4 ) )
% 2.87/1.59 => ( least @ X @ L @ A @ C @ T2 @ Rs2 @ S2 @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_85_least__def,axiom,
% 2.87/1.59 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 2.87/1.59 ( ( least @ X @ L @ V @ C )
% 2.87/1.59 = ( ^ [Uu: itself @ X,Rs: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G4: labeled_graph @ L @ C] :
% 2.87/1.59 ( ( graph_homomorphism @ L @ C @ C @ S3 @ G4 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S3 ) ) )
% 2.87/1.59 & ! [C5: labeled_graph @ L @ X] :
% 2.87/1.59 ( ( consequence_graph @ L @ V @ X @ Rs @ C5 )
% 2.87/1.59 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S3 @ G4 ) @ C5 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_86_fair__chain__impl__consequence__graph,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S2: nat > ( labeled_graph @ A @ C )] :
% 2.87/1.59 ( ( fair_chain @ A @ B @ C @ Rs2 @ S2 )
% 2.87/1.59 => ( ! [R3: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R3 @ Rs2 )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) ) )
% 2.87/1.59 & ( ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 )
% 2.87/1.59 = ( restrict @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) )
% 2.87/1.59 & ( finite_finite2 @ B @ ( labeled_vertices @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) )
% 2.87/1.59 & ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( labeled_edges @ A @ B @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R3 ) ) ) ) )
% 2.87/1.59 => ( consequence_graph @ A @ B @ C @ Rs2 @ ( chain_sup @ A @ C @ S2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_87_fair__chainI,axiom,
% 2.87/1.59 ! [C: $tType,B: $tType,A: $tType,S2: nat > ( labeled_graph @ A @ B ),Rs2: set @ ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) )] :
% 2.87/1.59 ( ( chain @ A @ B @ S2 )
% 2.87/1.59 => ( ! [R3: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),F2: set @ ( product_prod @ C @ B ),I2: nat] :
% 2.87/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) ) @ R3 @ Rs2 )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ C @ B @ ( product_fst @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R3 ) @ ( S2 @ I2 ) @ F2 )
% 2.87/1.59 => ? [J2: nat] : ( extensible @ A @ C @ B @ R3 @ ( S2 @ J2 ) @ F2 ) ) )
% 2.87/1.59 => ( fair_chain @ A @ C @ B @ Rs2 @ S2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_88_maintained__holds__iff,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,E_L: allegorical_term @ A,E_R: allegorical_term @ A] :
% 2.87/1.59 ( ( G
% 2.87/1.59 = ( restrict @ A @ B @ G ) )
% 2.87/1.59 => ( ( maintained @ A @ nat @ B @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ E_L ) @ ( translation @ A @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) @ G )
% 2.87/1.59 = ( ( semantics @ A @ B @ G @ ( product_fst @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) )
% 2.87/1.59 = ( semantics @ A @ B @ G @ ( product_snd @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_89_fair__chain__def,axiom,
% 2.87/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.87/1.59 ( ( fair_chain @ A @ B @ C )
% 2.87/1.59 = ( ^ [Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S3: nat > ( labeled_graph @ A @ C )] :
% 2.87/1.59 ( ( chain @ A @ C @ S3 )
% 2.87/1.59 & ! [R2: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),F6: set @ ( product_prod @ B @ C ),I3: nat] :
% 2.87/1.59 ( ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
% 2.87/1.59 & ( graph_homomorphism @ A @ B @ C @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 ) @ ( S3 @ I3 ) @ F6 ) )
% 2.87/1.59 => ? [J3: nat] : ( extensible @ A @ B @ C @ R2 @ ( S3 @ J3 ) @ F6 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_90_chain__sup__graph,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,S2: nat > ( labeled_graph @ A @ B )] :
% 2.87/1.59 ( ( chain @ A @ B @ S2 )
% 2.87/1.59 => ( ( chain_sup @ A @ B @ S2 )
% 2.87/1.59 = ( restrict @ A @ B @ ( chain_sup @ A @ B @ S2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_91_extensible__chain__sup,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,S2: nat > ( labeled_graph @ A @ B ),R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),J4: nat,F4: set @ ( product_prod @ C @ B )] :
% 2.87/1.59 ( ( chain @ A @ B @ S2 )
% 2.87/1.59 => ( ( extensible @ A @ C @ B @ R @ ( S2 @ J4 ) @ F4 )
% 2.87/1.59 => ( extensible @ A @ C @ B @ R @ ( chain_sup @ A @ B @ S2 ) @ F4 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_92_chain__sup__subgraph,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),J4: nat] :
% 2.87/1.59 ( ( chain @ A @ B @ S2 )
% 2.87/1.59 => ( graph_homomorphism @ A @ B @ B @ ( S2 @ J4 ) @ ( chain_sup @ A @ B @ S2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( S2 @ J4 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_93_chain__then__restrict,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),I: nat] :
% 2.87/1.59 ( ( chain @ A @ B @ S2 )
% 2.87/1.59 => ( ( S2 @ I )
% 2.87/1.59 = ( restrict @ A @ B @ ( S2 @ I ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_94_least__consequence__graph__def,axiom,
% 2.87/1.59 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 2.87/1.59 ( ( least_559130134_graph @ X @ L @ V @ C )
% 2.87/1.59 = ( ^ [T3: itself @ X,Rs: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G4: labeled_graph @ L @ C] :
% 2.87/1.59 ( ( consequence_graph @ L @ V @ C @ Rs @ G4 )
% 2.87/1.59 & ( least @ X @ L @ V @ C @ T3 @ Rs @ S3 @ G4 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_95_graph__homomorphism__semantics,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F4 )
% 2.87/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.87/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F4 )
% 2.87/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F4 )
% 2.87/1.59 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B5 @ E3 ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_96_fair__chainD_I1_J,axiom,
% 2.87/1.59 ! [B: $tType,C: $tType,A: $tType,Rs2: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),S2: nat > ( labeled_graph @ A @ C )] :
% 2.87/1.59 ( ( fair_chain @ A @ B @ C @ Rs2 @ S2 )
% 2.87/1.59 => ( chain @ A @ C @ S2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_97_semantics__in__vertices_I2_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 2.87/1.59 ( ( A5
% 2.87/1.59 = ( restrict @ A @ B @ A5 ) )
% 2.87/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.87/1.59 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_98_semantics__in__vertices_I1_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 2.87/1.59 ( ( A5
% 2.87/1.59 = ( restrict @ A @ B @ A5 ) )
% 2.87/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.87/1.59 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_99_subgraph__semantics,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 2.87/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.87/1.59 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B5 @ E3 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_100_eq__as__subsets,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A,E_1: allegorical_term @ B,E_2: allegorical_term @ B] :
% 2.87/1.59 ( ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ E_2 ) ) )
% 2.87/1.59 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ E_2 ) ) ) )
% 2.87/1.59 = ( ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) )
% 2.87/1.59 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) ) )
% 2.87/1.59 & ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_2 @ ( allegorical_A_Int @ B @ E_2 @ E_1 ) ) ) )
% 2.87/1.59 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_2 @ ( allegorical_A_Int @ B @ E_2 @ E_1 ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_101_maintained__holds__subset__iff,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,E_L: allegorical_term @ A,E_R: allegorical_term @ A] :
% 2.87/1.59 ( ( G
% 2.87/1.59 = ( restrict @ A @ B @ G ) )
% 2.87/1.59 => ( ( maintained @ A @ nat @ B @ ( product_Pair @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) @ ( translation @ A @ ( product_fst @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) ) @ ( translation @ A @ ( product_snd @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ ( product_Pair @ ( allegorical_term @ A ) @ ( allegorical_term @ A ) @ E_L @ ( allegorical_A_Int @ A @ E_L @ E_R ) ) ) ) ) @ G )
% 2.87/1.59 = ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( semantics @ A @ B @ G @ E_L ) @ ( semantics @ A @ B @ G @ E_R ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_102_maintained__holds,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A,E_L: allegorical_term @ B,E_R: allegorical_term @ B] :
% 2.87/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( semantics @ B @ A @ G @ E_L ) @ ( semantics @ B @ A @ G @ E_R ) )
% 2.87/1.59 => ( maintained @ B @ nat @ A @ ( product_Pair @ ( labeled_graph @ B @ nat ) @ ( labeled_graph @ B @ nat ) @ ( translation @ B @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_L @ ( allegorical_A_Int @ B @ E_L @ E_R ) ) ) ) @ ( translation @ B @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_L @ ( allegorical_A_Int @ B @ E_L @ E_R ) ) ) ) ) @ G ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_103_sentence__iff,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A,E_1: allegorical_term @ B,E_2: allegorical_term @ B] :
% 2.87/1.59 ( ( ( semantics @ B @ A @ G @ ( product_fst @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) )
% 2.87/1.59 = ( semantics @ B @ A @ G @ ( product_snd @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ ( product_Pair @ ( allegorical_term @ B ) @ ( allegorical_term @ B ) @ E_1 @ ( allegorical_A_Int @ B @ E_1 @ E_2 ) ) ) ) )
% 2.87/1.59 = ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( semantics @ B @ A @ G @ E_1 ) @ ( semantics @ B @ A @ G @ E_2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_104_find__graph__occurence,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,C: $tType,S2: nat > ( labeled_graph @ A @ B ),E4: set @ ( product_prod @ A @ ( product_prod @ C @ C ) ),V2: set @ C,F4: set @ ( product_prod @ C @ B )] :
% 2.87/1.59 ( ( chain @ A @ B @ S2 )
% 2.87/1.59 => ( ( finite_finite2 @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ E4 )
% 2.87/1.59 => ( ( finite_finite2 @ C @ V2 )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ C @ B @ ( labeled_LG @ A @ C @ E4 @ V2 ) @ ( chain_sup @ A @ B @ S2 ) @ F4 )
% 2.87/1.59 => ? [I2: nat] : ( graph_homomorphism @ A @ C @ B @ ( labeled_LG @ A @ C @ E4 @ V2 ) @ ( S2 @ I2 ) @ F4 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_105_eq__snd__iff,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 2.87/1.59 ( ( B2
% 2.87/1.59 = ( product_snd @ B @ A @ P2 ) )
% 2.87/1.59 = ( ? [A7: B] :
% 2.87/1.59 ( P2
% 2.87/1.59 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_106_eq__fst__iff,axiom,
% 2.87/1.59 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 2.87/1.59 ( ( A2
% 2.87/1.59 = ( product_fst @ A @ B @ P2 ) )
% 2.87/1.59 = ( ? [B6: B] :
% 2.87/1.59 ( P2
% 2.87/1.59 = ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_107_labeled__graph_Oinject,axiom,
% 2.87/1.59 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V,Y1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),Y2: set @ V] :
% 2.87/1.59 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 2.87/1.59 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 2.87/1.59 = ( ( X1 = Y1 )
% 2.87/1.59 & ( X22 = Y2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_108_labeled__graph_Ocollapse,axiom,
% 2.87/1.59 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.87/1.59 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 2.87/1.59 = Labeled_graph ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_109_extensible__refl__concr,axiom,
% 2.87/1.59 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V3: set @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ G @ F4 )
% 2.87/1.59 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ ( labeled_LG @ A @ B @ E_2 @ V3 ) ) @ G @ F4 )
% 2.87/1.59 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V3 ) @ G @ F4 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_110_labeled__graph_Oexhaust,axiom,
% 2.87/1.59 ! [L: $tType,V: $tType,Y3: labeled_graph @ L @ V] :
% 2.87/1.59 ~ ! [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 2.87/1.59 ( Y3
% 2.87/1.59 != ( labeled_LG @ L @ V @ X13 @ X23 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_111_labeled__graph_Oinduct,axiom,
% 2.87/1.59 ! [V: $tType,L: $tType,P: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 2.87/1.59 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 2.87/1.59 => ( P @ Labeled_graph ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_112_subrelI,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 2.87/1.59 ( ! [X5: A,Y4: B] :
% 2.87/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R4 )
% 2.87/1.59 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S4 ) )
% 2.87/1.59 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_113_finite__has__maximal2,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ! [A5: set @ A,A2: A] :
% 2.87/1.59 ( ( finite_finite2 @ A @ A5 )
% 2.87/1.59 => ( ( member @ A @ A2 @ A5 )
% 2.87/1.59 => ? [X5: A] :
% 2.87/1.59 ( ( member @ A @ X5 @ A5 )
% 2.87/1.59 & ( ord_less_eq @ A @ A2 @ X5 )
% 2.87/1.59 & ! [Xa: A] :
% 2.87/1.59 ( ( member @ A @ Xa @ A5 )
% 2.87/1.59 => ( ( ord_less_eq @ A @ X5 @ Xa )
% 2.87/1.59 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_114_finite__has__minimal2,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ! [A5: set @ A,A2: A] :
% 2.87/1.59 ( ( finite_finite2 @ A @ A5 )
% 2.87/1.59 => ( ( member @ A @ A2 @ A5 )
% 2.87/1.59 => ? [X5: A] :
% 2.87/1.59 ( ( member @ A @ X5 @ A5 )
% 2.87/1.59 & ( ord_less_eq @ A @ X5 @ A2 )
% 2.87/1.59 & ! [Xa: A] :
% 2.87/1.59 ( ( member @ A @ Xa @ A5 )
% 2.87/1.59 => ( ( ord_less_eq @ A @ Xa @ X5 )
% 2.87/1.59 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_115_finite__subset,axiom,
% 2.87/1.59 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.87/1.59 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.59 => ( ( finite_finite2 @ A @ B5 )
% 2.87/1.59 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_116_infinite__super,axiom,
% 2.87/1.59 ! [A: $tType,S2: set @ A,T4: set @ A] :
% 2.87/1.59 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
% 2.87/1.59 => ( ~ ( finite_finite2 @ A @ S2 )
% 2.87/1.59 => ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_117_rev__finite__subset,axiom,
% 2.87/1.59 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 2.87/1.59 ( ( finite_finite2 @ A @ B5 )
% 2.87/1.59 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.59 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_118_labeled__graph_Osel_I2_J,axiom,
% 2.87/1.59 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.87/1.59 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.87/1.59 = X22 ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_119_labeled__graph_Osel_I1_J,axiom,
% 2.87/1.59 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.87/1.59 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.87/1.59 = X1 ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_120_restrict__subsD,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G ) ) )
% 2.87/1.59 => ( G
% 2.87/1.59 = ( restrict @ A @ B @ G ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_121_subgraph__def2,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( G_1
% 2.87/1.59 = ( restrict @ A @ B @ G_1 ) )
% 2.87/1.59 => ( ( G_2
% 2.87/1.59 = ( restrict @ A @ B @ G_2 ) )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.87/1.59 = ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 2.87/1.59 & ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_122_labeled__graph_Oexhaust__sel,axiom,
% 2.87/1.59 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.87/1.59 ( Labeled_graph
% 2.87/1.59 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_123_subgraph__subset_I1_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.87/1.59 => ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_124_maintainedD2,axiom,
% 2.87/1.59 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G )
% 2.87/1.59 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F4 )
% 2.87/1.59 => ~ ! [G3: set @ ( product_prod @ B @ C )] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ C @ B5 @ G @ G3 )
% 2.87/1.59 => ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F4 @ G3 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_125_subgraph__subset_I2_J,axiom,
% 2.87/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.87/1.59 => ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G_1 ) ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_126_pushout__step__def,axiom,
% 2.87/1.59 ! [A: $tType,C: $tType,B: $tType,X: $tType] :
% 2.87/1.59 ( ( pushout_step @ X @ A @ C @ B )
% 2.87/1.59 = ( ^ [T3: itself @ X,R2: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_12 @ G_22 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_12 ) ) )
% 2.87/1.59 & ? [F_1: set @ ( product_prod @ C @ B ),F_2: set @ ( product_prod @ C @ B )] :
% 2.87/1.59 ( ( graph_homomorphism @ A @ C @ B @ ( product_fst @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R2 ) @ G_12 @ F_1 )
% 2.87/1.59 & ( graph_homomorphism @ A @ C @ B @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R2 ) @ G_22 @ F_2 )
% 2.87/1.59 & ( ord_less_eq @ ( set @ ( product_prod @ C @ B ) ) @ F_1 @ F_2 )
% 2.87/1.59 & ( weak_universal @ X @ A @ C @ B @ T3 @ R2 @ G_12 @ G_22 @ F_1 @ F_2 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_127_subsetI,axiom,
% 2.87/1.59 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.87/1.59 ( ! [X5: A] :
% 2.87/1.59 ( ( member @ A @ X5 @ A5 )
% 2.87/1.59 => ( member @ A @ X5 @ B5 ) )
% 2.87/1.59 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_128_subset__antisym,axiom,
% 2.87/1.59 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.87/1.59 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.59 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 2.87/1.59 => ( A5 = B5 ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_129_order__refl,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( preorder @ A )
% 2.87/1.59 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_130_labeled__graph_Osplit__sel__asm,axiom,
% 2.87/1.59 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.87/1.59 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F4 @ Labeled_graph ) )
% 2.87/1.59 = ( ~ ( ( Labeled_graph
% 2.87/1.59 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.87/1.59 & ~ ( P @ ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_131_labeled__graph_Ocase,axiom,
% 2.87/1.59 ! [L: $tType,A: $tType,V: $tType,F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.87/1.59 ( ( labele1974067554_graph @ L @ V @ A @ F4 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.87/1.59 = ( F4 @ X1 @ X22 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_132_dual__order_Oantisym,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ! [B2: A,A2: A] :
% 2.87/1.59 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.87/1.59 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.59 => ( A2 = B2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_133_dual__order_Oeq__iff,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.87/1.59 = ( ^ [A7: A,B6: A] :
% 2.87/1.59 ( ( ord_less_eq @ A @ B6 @ A7 )
% 2.87/1.59 & ( ord_less_eq @ A @ A7 @ B6 ) ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_134_dual__order_Otrans,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ! [B2: A,A2: A,C3: A] :
% 2.87/1.59 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.87/1.59 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 2.87/1.59 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_135_linorder__wlog,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( linorder @ A )
% 2.87/1.59 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.87/1.59 ( ! [A4: A,B4: A] :
% 2.87/1.59 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.87/1.59 => ( P @ A4 @ B4 ) )
% 2.87/1.59 => ( ! [A4: A,B4: A] :
% 2.87/1.59 ( ( P @ B4 @ A4 )
% 2.87/1.59 => ( P @ A4 @ B4 ) )
% 2.87/1.59 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_136_dual__order_Orefl,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_137_order__trans,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( preorder @ A )
% 2.87/1.59 => ! [X4: A,Y3: A,Z2: A] :
% 2.87/1.59 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.87/1.59 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.87/1.59 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_138_order__class_Oorder_Oantisym,axiom,
% 2.87/1.59 ! [A: $tType] :
% 2.87/1.59 ( ( order @ A )
% 2.87/1.59 => ! [A2: A,B2: A] :
% 2.87/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.59 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.87/1.59 => ( A2 = B2 ) ) ) ) ).
% 2.87/1.59
% 2.87/1.59 thf(fact_139_ord__le__eq__trans,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( ord @ A )
% 2.87/1.60 => ! [A2: A,B2: A,C3: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.60 => ( ( B2 = C3 )
% 2.87/1.60 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_140_ord__eq__le__trans,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( ord @ A )
% 2.87/1.60 => ! [A2: A,B2: A,C3: A] :
% 2.87/1.60 ( ( A2 = B2 )
% 2.87/1.60 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.87/1.60 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_141_order__class_Oorder_Oeq__iff,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order @ A )
% 2.87/1.60 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.87/1.60 = ( ^ [A7: A,B6: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A7 @ B6 )
% 2.87/1.60 & ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_142_antisym__conv,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order @ A )
% 2.87/1.60 => ! [Y3: A,X4: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.87/1.60 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.87/1.60 = ( X4 = Y3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_143_le__cases3,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( linorder @ A )
% 2.87/1.60 => ! [X4: A,Y3: A,Z2: A] :
% 2.87/1.60 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.87/1.60 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.87/1.60 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.87/1.60 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.87/1.60 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 2.87/1.60 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.87/1.60 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.87/1.60 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 2.87/1.60 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.87/1.60 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 2.87/1.60 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 2.87/1.60 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_144_order_Otrans,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order @ A )
% 2.87/1.60 => ! [A2: A,B2: A,C3: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.60 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.87/1.60 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_145_le__cases,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( linorder @ A )
% 2.87/1.60 => ! [X4: A,Y3: A] :
% 2.87/1.60 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 2.87/1.60 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_146_eq__refl,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( preorder @ A )
% 2.87/1.60 => ! [X4: A,Y3: A] :
% 2.87/1.60 ( ( X4 = Y3 )
% 2.87/1.60 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_147_linear,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( linorder @ A )
% 2.87/1.60 => ! [X4: A,Y3: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.87/1.60 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_148_antisym,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order @ A )
% 2.87/1.60 => ! [X4: A,Y3: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.87/1.60 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.87/1.60 => ( X4 = Y3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_149_eq__iff,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order @ A )
% 2.87/1.60 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.87/1.60 = ( ^ [X6: A,Y6: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ X6 @ Y6 )
% 2.87/1.60 & ( ord_less_eq @ A @ Y6 @ X6 ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_150_ord__le__eq__subst,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType] :
% 2.87/1.60 ( ( ( ord @ B )
% 2.87/1.60 & ( ord @ A ) )
% 2.87/1.60 => ! [A2: A,B2: A,F4: A > B,C3: B] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.60 => ( ( ( F4 @ B2 )
% 2.87/1.60 = C3 )
% 2.87/1.60 => ( ! [X5: A,Y4: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 2.87/1.60 => ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 2.87/1.60 => ( ord_less_eq @ B @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_151_ord__eq__le__subst,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType] :
% 2.87/1.60 ( ( ( ord @ B )
% 2.87/1.60 & ( ord @ A ) )
% 2.87/1.60 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 2.87/1.60 ( ( A2
% 2.87/1.60 = ( F4 @ B2 ) )
% 2.87/1.60 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.87/1.60 => ( ! [X5: B,Y4: B] :
% 2.87/1.60 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 2.87/1.60 => ( ord_less_eq @ A @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 2.87/1.60 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_152_order__subst2,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType] :
% 2.87/1.60 ( ( ( order @ C )
% 2.87/1.60 & ( order @ A ) )
% 2.87/1.60 => ! [A2: A,B2: A,F4: A > C,C3: C] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.60 => ( ( ord_less_eq @ C @ ( F4 @ B2 ) @ C3 )
% 2.87/1.60 => ( ! [X5: A,Y4: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 2.87/1.60 => ( ord_less_eq @ C @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 2.87/1.60 => ( ord_less_eq @ C @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_153_order__subst1,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType] :
% 2.87/1.60 ( ( ( order @ B )
% 2.87/1.60 & ( order @ A ) )
% 2.87/1.60 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A2 @ ( F4 @ B2 ) )
% 2.87/1.60 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.87/1.60 => ( ! [X5: B,Y4: B] :
% 2.87/1.60 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 2.87/1.60 => ( ord_less_eq @ A @ ( F4 @ X5 ) @ ( F4 @ Y4 ) ) )
% 2.87/1.60 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_154_le__fun__def,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType] :
% 2.87/1.60 ( ( ord @ B )
% 2.87/1.60 => ( ( ord_less_eq @ ( A > B ) )
% 2.87/1.60 = ( ^ [F6: A > B,G5: A > B] :
% 2.87/1.60 ! [X6: A] : ( ord_less_eq @ B @ ( F6 @ X6 ) @ ( G5 @ X6 ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_155_le__funI,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType] :
% 2.87/1.60 ( ( ord @ B )
% 2.87/1.60 => ! [F4: A > B,G6: A > B] :
% 2.87/1.60 ( ! [X5: A] : ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( G6 @ X5 ) )
% 2.87/1.60 => ( ord_less_eq @ ( A > B ) @ F4 @ G6 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_156_le__funE,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType] :
% 2.87/1.60 ( ( ord @ B )
% 2.87/1.60 => ! [F4: A > B,G6: A > B,X4: A] :
% 2.87/1.60 ( ( ord_less_eq @ ( A > B ) @ F4 @ G6 )
% 2.87/1.60 => ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G6 @ X4 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_157_le__funD,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType] :
% 2.87/1.60 ( ( ord @ B )
% 2.87/1.60 => ! [F4: A > B,G6: A > B,X4: A] :
% 2.87/1.60 ( ( ord_less_eq @ ( A > B ) @ F4 @ G6 )
% 2.87/1.60 => ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G6 @ X4 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_158_Collect__mono__iff,axiom,
% 2.87/1.60 ! [A: $tType,P: A > $o,Q2: A > $o] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) )
% 2.87/1.60 = ( ! [X6: A] :
% 2.87/1.60 ( ( P @ X6 )
% 2.87/1.60 => ( Q2 @ X6 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_159_set__eq__subset,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 2.87/1.60 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 2.87/1.60 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_160_subset__trans,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A,B5: set @ A,C6: set @ A] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.60 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
% 2.87/1.60 => ( ord_less_eq @ ( set @ A ) @ A5 @ C6 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_161_Collect__mono,axiom,
% 2.87/1.60 ! [A: $tType,P: A > $o,Q2: A > $o] :
% 2.87/1.60 ( ! [X5: A] :
% 2.87/1.60 ( ( P @ X5 )
% 2.87/1.60 => ( Q2 @ X5 ) )
% 2.87/1.60 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_162_subset__refl,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_163_subset__iff,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) )
% 2.87/1.60 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.87/1.60 ! [T3: A] :
% 2.87/1.60 ( ( member @ A @ T3 @ A6 )
% 2.87/1.60 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_164_equalityD2,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.87/1.60 ( ( A5 = B5 )
% 2.87/1.60 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_165_equalityD1,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.87/1.60 ( ( A5 = B5 )
% 2.87/1.60 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_166_subset__eq,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) )
% 2.87/1.60 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.87/1.60 ! [X6: A] :
% 2.87/1.60 ( ( member @ A @ X6 @ A6 )
% 2.87/1.60 => ( member @ A @ X6 @ B7 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_167_equalityE,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.87/1.60 ( ( A5 = B5 )
% 2.87/1.60 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.60 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_168_subsetD,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.60 => ( ( member @ A @ C3 @ A5 )
% 2.87/1.60 => ( member @ A @ C3 @ B5 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_169_in__mono,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A,B5: set @ A,X4: A] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.87/1.60 => ( ( member @ A @ X4 @ A5 )
% 2.87/1.60 => ( member @ A @ X4 @ B5 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_170_labeled__graph_Ocase__eq__if,axiom,
% 2.87/1.60 ! [A: $tType,V: $tType,L: $tType] :
% 2.87/1.60 ( ( labele1974067554_graph @ L @ V @ A )
% 2.87/1.60 = ( ^ [F6: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F6 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_171_chain,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),I: nat,J4: nat] :
% 2.87/1.60 ( ( chain @ A @ B @ S2 )
% 2.87/1.60 => ( ( ord_less_eq @ nat @ I @ J4 )
% 2.87/1.60 => ( graph_homomorphism @ A @ B @ B @ ( S2 @ I ) @ ( S2 @ J4 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( S2 @ I ) ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_172_chain__def2,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType] :
% 2.87/1.60 ( ( chain @ A @ B )
% 2.87/1.60 = ( ^ [S3: nat > ( labeled_graph @ A @ B )] :
% 2.87/1.60 ! [I3: nat,J3: nat] :
% 2.87/1.60 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 2.87/1.60 => ( graph_homomorphism @ A @ B @ B @ ( S3 @ I3 ) @ ( S3 @ J3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( S3 @ I3 ) ) ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_173_labeled__graph_Osplit__sel,axiom,
% 2.87/1.60 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.87/1.60 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F4 @ Labeled_graph ) )
% 2.87/1.60 = ( ( Labeled_graph
% 2.87/1.60 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.87/1.60 => ( P @ ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_174_graph__unionI,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 2.87/1.60 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 2.87/1.60 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.87/1.60 = G_2 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_175_extensibleD,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( extensible @ A @ B @ C @ R @ G @ F4 )
% 2.87/1.60 => ~ ! [G3: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ B @ C @ ( product_snd @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ G @ G3 )
% 2.87/1.60 => ~ ( agree_on @ A @ B @ C @ ( product_fst @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R ) @ F4 @ G3 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_176_extensible__def,axiom,
% 2.87/1.60 ! [V: $tType,X: $tType,L: $tType] :
% 2.87/1.60 ( ( extensible @ L @ X @ V )
% 2.87/1.60 = ( ^ [R2: product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ),G4: labeled_graph @ L @ V,F6: set @ ( product_prod @ X @ V )] :
% 2.87/1.60 ? [G5: set @ ( product_prod @ X @ V )] :
% 2.87/1.60 ( ( graph_homomorphism @ L @ X @ V @ ( product_snd @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ G4 @ G5 )
% 2.87/1.60 & ( agree_on @ L @ X @ V @ ( product_fst @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) @ R2 ) @ F6 @ G5 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_177_extensibleI,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G6: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G6 )
% 2.87/1.60 => ( ( agree_on @ A @ B @ C @ R1 @ F4 @ G6 )
% 2.87/1.60 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F4 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_178_graph__union__idemp_I1_J,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( graph_union @ A @ B @ A5 @ A5 )
% 2.87/1.60 = A5 ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_179_graph__union__idemp_I2_J,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B5 ) )
% 2.87/1.60 = ( graph_union @ A @ B @ A5 @ B5 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_180_graph__union__idemp_I3_J,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B5 @ A5 ) )
% 2.87/1.60 = ( graph_union @ A @ B @ B5 @ A5 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_181_agree__on__refl,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R @ F4 @ F4 ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_182_agree__on__comm,axiom,
% 2.87/1.60 ! [C: $tType,B: $tType,A: $tType] :
% 2.87/1.60 ( ( agree_on @ A @ B @ C )
% 2.87/1.60 = ( ^ [X3: labeled_graph @ A @ B,F6: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X3 @ G5 @ F6 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_183_graph__union__preserves__restrict,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( G_1
% 2.87/1.60 = ( restrict @ A @ B @ G_1 ) )
% 2.87/1.60 => ( ( G_2
% 2.87/1.60 = ( restrict @ A @ B @ G_2 ) )
% 2.87/1.60 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.87/1.60 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_184_graph__union__chain__sup,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,S2: nat > ( labeled_graph @ A @ B ),C6: labeled_graph @ A @ B] :
% 2.87/1.60 ( ! [I2: nat] :
% 2.87/1.60 ( ( graph_union @ A @ B @ ( S2 @ I2 ) @ C6 )
% 2.87/1.60 = C6 )
% 2.87/1.60 => ( ( graph_union @ A @ B @ ( chain_sup @ A @ B @ S2 ) @ C6 )
% 2.87/1.60 = C6 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_185_agree__on__trans,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,X2: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( agree_on @ A @ B @ C @ X2 @ F4 @ G6 )
% 2.87/1.60 => ( ( agree_on @ A @ B @ C @ X2 @ G6 @ H )
% 2.87/1.60 => ( agree_on @ A @ B @ C @ X2 @ F4 @ H ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_186_agree__on__subg__compose,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G6: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F7: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( agree_on @ A @ B @ C @ R @ G6 @ H )
% 2.87/1.60 => ( ( agree_on @ A @ B @ C @ F7 @ F4 @ G6 )
% 2.87/1.60 => ( ( graph_homomorphism @ A @ B @ B @ F7 @ R @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F7 ) ) )
% 2.87/1.60 => ( agree_on @ A @ B @ C @ F7 @ F4 @ H ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_187_subgraph__def,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.87/1.60 = ( ( G_1
% 2.87/1.60 = ( restrict @ A @ B @ G_1 ) )
% 2.87/1.60 & ( G_2
% 2.87/1.60 = ( restrict @ A @ B @ G_2 ) )
% 2.87/1.60 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.87/1.60 = G_2 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_188_graph__union__iff,axiom,
% 2.87/1.60 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.87/1.60 = G_2 )
% 2.87/1.60 = ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 2.87/1.60 & ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_189_finite__nat__set__iff__bounded__le,axiom,
% 2.87/1.60 ( ( finite_finite2 @ nat )
% 2.87/1.60 = ( ^ [N: set @ nat] :
% 2.87/1.60 ? [M: nat] :
% 2.87/1.60 ! [X6: nat] :
% 2.87/1.60 ( ( member @ nat @ X6 @ N )
% 2.87/1.60 => ( ord_less_eq @ nat @ X6 @ M ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_190_nonempty__rule,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
% 2.87/1.60 ( ( G
% 2.87/1.60 = ( restrict @ A @ B @ G ) )
% 2.87/1.60 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
% 2.87/1.60 = ( ( labeled_vertices @ A @ B @ G )
% 2.87/1.60 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_191_weak__universalI,axiom,
% 2.87/1.60 ! [X: $tType,A: $tType,B: $tType,C: $tType,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_1: labeled_graph @ A @ B,F_12: set @ ( product_prod @ C @ B ),G_2: labeled_graph @ A @ B,T2: itself @ X,F_22: set @ ( product_prod @ C @ B )] :
% 2.87/1.60 ( ! [H_1: set @ ( product_prod @ C @ X ),H_2: set @ ( product_prod @ B @ X ),G7: labeled_graph @ A @ X] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ C @ X @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R ) @ G7 @ H_1 )
% 2.87/1.60 => ( ( graph_homomorphism @ A @ B @ X @ G_1 @ G7 @ H_2 )
% 2.87/1.60 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ X ) ) @ ( relcomp @ C @ B @ X @ F_12 @ H_2 ) @ H_1 )
% 2.87/1.60 => ? [H2: set @ ( product_prod @ B @ X )] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ B @ X @ G_2 @ G7 @ H2 )
% 2.87/1.60 & ( ord_less_eq @ ( set @ ( product_prod @ B @ X ) ) @ H_2 @ H2 ) ) ) ) )
% 2.87/1.60 => ( weak_universal @ X @ A @ C @ B @ T2 @ R @ G_1 @ G_2 @ F_12 @ F_22 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_192_relcomp__empty1,axiom,
% 2.87/1.60 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] :
% 2.87/1.60 ( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R )
% 2.87/1.60 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_193_relcomp__empty2,axiom,
% 2.87/1.60 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] :
% 2.87/1.60 ( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
% 2.87/1.60 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_194_empty__subsetI,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_195_subset__empty,axiom,
% 2.87/1.60 ! [A: $tType,A5: set @ A] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 2.87/1.60 = ( A5
% 2.87/1.60 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_196_Id__on__empty,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.87/1.60 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_197_graph__homomorphism__composes,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,X4: set @ ( product_prod @ B @ C ),C3: labeled_graph @ A @ D,Y3: set @ ( product_prod @ C @ D )] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ X4 )
% 2.87/1.60 => ( ( graph_homomorphism @ A @ C @ D @ B2 @ C3 @ Y3 )
% 2.87/1.60 => ( graph_homomorphism @ A @ B @ D @ A2 @ C3 @ ( relcomp @ B @ C @ D @ X4 @ Y3 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_198_graph__empty__e,axiom,
% 2.87/1.60 ! [A: $tType,B: $tType,V3: set @ B] :
% 2.87/1.60 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
% 2.87/1.60 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_199_agree__on__empty,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,F4: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ F4 @ G6 ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_200_graph__homomorphism__empty,axiom,
% 2.87/1.60 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F4 )
% 2.87/1.60 = ( ( F4
% 2.87/1.60 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 2.87/1.60 & ( G
% 2.87/1.60 = ( restrict @ A @ C @ G ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_201_relcomp__mono,axiom,
% 2.87/1.60 ! [A: $tType,C: $tType,B: $tType,R5: set @ ( product_prod @ A @ B ),R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C ),S4: set @ ( product_prod @ B @ C )] :
% 2.87/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ R4 )
% 2.87/1.60 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S5 @ S4 )
% 2.87/1.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R5 @ S5 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) ) ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_202_bot_Oextremum,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order_bot @ A )
% 2.87/1.60 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.87/1.60
% 2.87/1.60 thf(fact_203_bot_Oextremum__unique,axiom,
% 2.87/1.60 ! [A: $tType] :
% 2.87/1.60 ( ( order_bot @ A )
% 2.87/1.60 => ! [A2: A] :
% 2.87/1.60 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 3.10/1.60 = ( A2
% 3.10/1.60 = ( bot_bot @ A ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_204_bot_Oextremum__uniqueI,axiom,
% 3.10/1.60 ! [A: $tType] :
% 3.10/1.60 ( ( order_bot @ A )
% 3.10/1.60 => ! [A2: A] :
% 3.10/1.60 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 3.10/1.60 => ( A2
% 3.10/1.60 = ( bot_bot @ A ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_205_agree__on__ext,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,G: labeled_graph @ A @ B,F_12: set @ ( product_prod @ B @ C ),F_22: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ C @ D )] :
% 3.10/1.60 ( ( agree_on @ A @ B @ C @ G @ F_12 @ F_22 )
% 3.10/1.60 => ( agree_on @ A @ B @ D @ G @ ( relcomp @ B @ C @ D @ F_12 @ G6 ) @ ( relcomp @ B @ C @ D @ F_22 @ G6 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_206_O__assoc,axiom,
% 3.10/1.60 ! [A: $tType,D: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ D ),S2: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
% 3.10/1.60 ( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R @ S2 ) @ T4 )
% 3.10/1.60 = ( relcomp @ A @ D @ B @ R @ ( relcomp @ D @ C @ B @ S2 @ T4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_207_relcomp_OrelcompI,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R4: set @ ( product_prod @ A @ B ),C3: C,S4: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 3.10/1.60 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C3 ) @ S4 )
% 3.10/1.60 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C3 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_208_relcomp_Oinducts,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,C: $tType,X1: A,X22: C,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C ),P: A > C > $o] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X1 @ X22 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) )
% 3.10/1.60 => ( ! [A4: A,B4: B,C2: C] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R4 )
% 3.10/1.60 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C2 ) @ S4 )
% 3.10/1.60 => ( P @ A4 @ C2 ) ) )
% 3.10/1.60 => ( P @ X1 @ X22 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_209_relcomp_Osimps,axiom,
% 3.10/1.60 ! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) )
% 3.10/1.60 = ( ? [A7: A,B6: B,C7: C] :
% 3.10/1.60 ( ( A1 = A7 )
% 3.10/1.60 & ( A22 = C7 )
% 3.10/1.60 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B6 ) @ R4 )
% 3.10/1.60 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B6 @ C7 ) @ S4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_210_relcomp_Ocases,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R4 @ S4 ) )
% 3.10/1.60 => ~ ! [B4: B] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B4 ) @ R4 )
% 3.10/1.60 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A22 ) @ S4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_211_relcompEpair,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R4: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C3 ) @ ( relcomp @ A @ C @ B @ R4 @ S4 ) )
% 3.10/1.60 => ~ ! [B4: C] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B4 ) @ R4 )
% 3.10/1.60 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_212_relcompE,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R4: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R4 @ S4 ) )
% 3.10/1.60 => ~ ! [X5: A,Y4: C,Z3: B] :
% 3.10/1.60 ( ( Xz
% 3.10/1.60 = ( product_Pair @ A @ B @ X5 @ Z3 ) )
% 3.10/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y4 ) @ R4 )
% 3.10/1.60 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_213_finite_OemptyI,axiom,
% 3.10/1.60 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_214_infinite__imp__nonempty,axiom,
% 3.10/1.60 ! [A: $tType,S2: set @ A] :
% 3.10/1.60 ( ~ ( finite_finite2 @ A @ S2 )
% 3.10/1.60 => ( S2
% 3.10/1.60 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_215_finite__has__minimal,axiom,
% 3.10/1.60 ! [A: $tType] :
% 3.10/1.60 ( ( order @ A )
% 3.10/1.60 => ! [A5: set @ A] :
% 3.10/1.60 ( ( finite_finite2 @ A @ A5 )
% 3.10/1.60 => ( ( A5
% 3.10/1.60 != ( bot_bot @ ( set @ A ) ) )
% 3.10/1.60 => ? [X5: A] :
% 3.10/1.60 ( ( member @ A @ X5 @ A5 )
% 3.10/1.60 & ! [Xa: A] :
% 3.10/1.60 ( ( member @ A @ Xa @ A5 )
% 3.10/1.60 => ( ( ord_less_eq @ A @ Xa @ X5 )
% 3.10/1.60 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_216_finite__has__maximal,axiom,
% 3.10/1.60 ! [A: $tType] :
% 3.10/1.60 ( ( order @ A )
% 3.10/1.60 => ! [A5: set @ A] :
% 3.10/1.60 ( ( finite_finite2 @ A @ A5 )
% 3.10/1.60 => ( ( A5
% 3.10/1.60 != ( bot_bot @ ( set @ A ) ) )
% 3.10/1.60 => ? [X5: A] :
% 3.10/1.60 ( ( member @ A @ X5 @ A5 )
% 3.10/1.60 & ! [Xa: A] :
% 3.10/1.60 ( ( member @ A @ Xa @ A5 )
% 3.10/1.60 => ( ( ord_less_eq @ A @ X5 @ Xa )
% 3.10/1.60 => ( X5 = Xa ) ) ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_217_graph__homomorphism__nonempty,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F4 )
% 3.10/1.60 => ( ( ( semantics @ A @ B @ A5 @ E3 )
% 3.10/1.60 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 3.10/1.60 => ( ( semantics @ A @ C @ B5 @ E3 )
% 3.10/1.60 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_218_bounded__Max__nat,axiom,
% 3.10/1.60 ! [P: nat > $o,X4: nat,M2: nat] :
% 3.10/1.60 ( ( P @ X4 )
% 3.10/1.60 => ( ! [X5: nat] :
% 3.10/1.60 ( ( P @ X5 )
% 3.10/1.60 => ( ord_less_eq @ nat @ X5 @ M2 ) )
% 3.10/1.60 => ~ ! [M3: nat] :
% 3.10/1.60 ( ( P @ M3 )
% 3.10/1.60 => ~ ! [X7: nat] :
% 3.10/1.60 ( ( P @ X7 )
% 3.10/1.60 => ( ord_less_eq @ nat @ X7 @ M3 ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_219_Id__on__vertices__identity_I2_J,axiom,
% 3.10/1.60 ! [B: $tType,C: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),Aa2: B,Ba: C] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F4 )
% 3.10/1.60 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ F4 )
% 3.10/1.60 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ ( relcomp @ B @ C @ C @ F4 @ ( id_on @ C @ ( labeled_vertices @ A @ C @ B2 ) ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_220_Id__on__vertices__identity_I1_J,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),Aa2: B,Ba: C] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F4 )
% 3.10/1.60 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ F4 )
% 3.10/1.60 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa2 @ Ba ) @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) @ F4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_221_translation__homomorphism_I2_J,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,E3: allegorical_term @ A,G: labeled_graph @ A @ B,F4: set @ ( product_prod @ nat @ B )] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ nat @ B @ ( translation @ A @ E3 ) @ G @ F4 )
% 3.10/1.60 => ( ( semantics @ A @ B @ G @ E3 )
% 3.10/1.60 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_222_weak__universalD,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,X: $tType,B: $tType,T2: itself @ X,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,F_12: set @ ( product_prod @ C @ B ),F_22: set @ ( product_prod @ C @ B ),G: labeled_graph @ A @ X,H_12: set @ ( product_prod @ C @ X ),H_22: set @ ( product_prod @ B @ X )] :
% 3.10/1.60 ( ( weak_universal @ X @ A @ C @ B @ T2 @ R @ G_1 @ G_2 @ F_12 @ F_22 )
% 3.10/1.60 => ( ( graph_homomorphism @ A @ C @ X @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R ) @ G @ H_12 )
% 3.10/1.60 => ( ( graph_homomorphism @ A @ B @ X @ G_1 @ G @ H_22 )
% 3.10/1.60 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ X ) ) @ ( relcomp @ C @ B @ X @ F_12 @ H_22 ) @ H_12 )
% 3.10/1.60 => ? [H3: set @ ( product_prod @ B @ X )] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ X @ G_2 @ G @ H3 )
% 3.10/1.60 & ( ord_less_eq @ ( set @ ( product_prod @ B @ X ) ) @ H_22 @ H3 ) ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_223_weak__universal__def,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,X: $tType] :
% 3.10/1.60 ( ( weak_universal @ X @ A @ C @ B )
% 3.10/1.60 = ( ^ [Uu: itself @ X,R2: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ),G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B,F_1: set @ ( product_prod @ C @ B ),F_2: set @ ( product_prod @ C @ B )] :
% 3.10/1.60 ! [H_13: set @ ( product_prod @ C @ X ),H_23: set @ ( product_prod @ B @ X ),G4: labeled_graph @ A @ X] :
% 3.10/1.60 ( ( ( graph_homomorphism @ A @ C @ X @ ( product_snd @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C ) @ R2 ) @ G4 @ H_13 )
% 3.10/1.60 & ( graph_homomorphism @ A @ B @ X @ G_12 @ G4 @ H_23 )
% 3.10/1.60 & ( ord_less_eq @ ( set @ ( product_prod @ C @ X ) ) @ ( relcomp @ C @ B @ X @ F_1 @ H_23 ) @ H_13 ) )
% 3.10/1.60 => ? [H4: set @ ( product_prod @ B @ X )] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ X @ G_22 @ G4 @ H4 )
% 3.10/1.60 & ( ord_less_eq @ ( set @ ( product_prod @ B @ X ) ) @ H_23 @ H4 ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_224_graph__homo__union__id_I2_J,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A5 @ B5 ) @ G @ F4 )
% 3.10/1.60 => ( ( B5
% 3.10/1.60 = ( restrict @ A @ B @ B5 ) )
% 3.10/1.60 => ( graph_homomorphism @ A @ B @ C @ B5 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ B5 ) ) @ F4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_225_graph__homo__union__id_I1_J,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A5 @ B5 ) @ G @ F4 )
% 3.10/1.60 => ( ( A5
% 3.10/1.60 = ( restrict @ A @ B @ A5 ) )
% 3.10/1.60 => ( graph_homomorphism @ A @ B @ C @ A5 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) @ F4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_226_finite__relcomp,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 3.10/1.60 => ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
% 3.10/1.60 => ( finite_finite2 @ ( product_prod @ A @ C ) @ ( relcomp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_227_subset__emptyI,axiom,
% 3.10/1.60 ! [A: $tType,A5: set @ A] :
% 3.10/1.60 ( ! [X5: A] :
% 3.10/1.60 ~ ( member @ A @ X5 @ A5 )
% 3.10/1.60 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_228_bot__empty__eq,axiom,
% 3.10/1.60 ! [A: $tType] :
% 3.10/1.60 ( ( bot_bot @ ( A > $o ) )
% 3.10/1.60 = ( ^ [X6: A] : ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_229_ssubst__Pair__rhs,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R4: A,S4: B,R: set @ ( product_prod @ A @ B ),S5: B] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S4 ) @ R )
% 3.10/1.60 => ( ( S5 = S4 )
% 3.10/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S5 ) @ R ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_230_less__by__empty,axiom,
% 3.10/1.60 ! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
% 3.10/1.60 ( ( A5
% 3.10/1.60 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 3.10/1.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_231_agree__iff__subset,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,X2: labeled_graph @ A @ C,F4: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( graph_homomorphism @ A @ B @ C @ G @ X2 @ F4 )
% 3.10/1.60 => ( ( univalent @ B @ C @ G6 )
% 3.10/1.60 => ( ( agree_on @ A @ B @ C @ G @ F4 @ G6 )
% 3.10/1.60 = ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F4 @ G6 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_232_id__univalent,axiom,
% 3.10/1.60 ! [A: $tType,X4: set @ A] : ( univalent @ A @ A @ ( id_on @ A @ X4 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_233_univalentD,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X4: A,Y3: B,Z2: B] :
% 3.10/1.60 ( ( univalent @ A @ B @ R )
% 3.10/1.60 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 3.10/1.60 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z2 ) @ R )
% 3.10/1.60 => ( Z2 = Y3 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_234_univalent__def,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType] :
% 3.10/1.60 ( ( univalent @ A @ B )
% 3.10/1.60 = ( ^ [R2: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ! [X6: A,Y6: B,Z4: B] :
% 3.10/1.60 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y6 ) @ R2 )
% 3.10/1.60 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Z4 ) @ R2 ) )
% 3.10/1.60 => ( Z4 = Y6 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_235_find__graph__occurence__vertices,axiom,
% 3.10/1.60 ! [C: $tType,A: $tType,B: $tType,S2: nat > ( labeled_graph @ A @ B ),V2: set @ C,F4: set @ ( product_prod @ C @ B )] :
% 3.10/1.60 ( ( chain @ A @ B @ S2 )
% 3.10/1.60 => ( ( finite_finite2 @ C @ V2 )
% 3.10/1.60 => ( ( univalent @ C @ B @ F4 )
% 3.10/1.60 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ C @ B @ F4 @ V2 ) @ ( labeled_vertices @ A @ B @ ( chain_sup @ A @ B @ S2 ) ) )
% 3.10/1.60 => ? [I2: nat] : ( ord_less_eq @ ( set @ B ) @ ( image @ C @ B @ F4 @ V2 ) @ ( labeled_vertices @ A @ B @ ( S2 @ I2 ) ) ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_236_agree__on__subset,axiom,
% 3.10/1.60 ! [C: $tType,B: $tType,A: $tType,F4: set @ ( product_prod @ A @ B ),G6: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] :
% 3.10/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ F4 @ G6 )
% 3.10/1.60 => ( ( ord_less_eq @ ( set @ A ) @ ( labeled_vertices @ C @ A @ G ) @ ( domain @ A @ B @ F4 ) )
% 3.10/1.60 => ( ( univalent @ A @ B @ G6 )
% 3.10/1.60 => ( agree_on @ C @ A @ B @ G @ F4 @ G6 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_237_ImageI,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,A2: A,B2: B,R4: set @ ( product_prod @ A @ B ),A5: set @ A] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 3.10/1.60 => ( ( member @ A @ A2 @ A5 )
% 3.10/1.60 => ( member @ B @ B2 @ ( image @ A @ B @ R4 @ A5 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_238_Image__empty2,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 3.10/1.60 ( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
% 3.10/1.60 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_239_Domain__Id__on,axiom,
% 3.10/1.60 ! [A: $tType,A5: set @ A] :
% 3.10/1.60 ( ( domain @ A @ A @ ( id_on @ A @ A5 ) )
% 3.10/1.60 = A5 ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_240_Image__empty1,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,X2: set @ B] :
% 3.10/1.60 ( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X2 )
% 3.10/1.60 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_241_Domain__empty,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType] :
% 3.10/1.60 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.10/1.60 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_242_univalent__finite_I2_J,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( univalent @ A @ B @ R )
% 3.10/1.60 => ( ( finite_finite2 @ A @ ( domain @ A @ B @ R ) )
% 3.10/1.60 = ( finite_finite2 @ ( product_prod @ A @ B ) @ R ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_243_Domain__O,axiom,
% 3.10/1.60 ! [A: $tType,C: $tType,B: $tType,A2: set @ A,X4: set @ ( product_prod @ A @ B ),Y3: set @ ( product_prod @ B @ C )] :
% 3.10/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( domain @ A @ B @ X4 ) )
% 3.10/1.60 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ X4 @ A2 ) @ ( domain @ B @ C @ Y3 ) )
% 3.10/1.60 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( domain @ A @ C @ ( relcomp @ A @ B @ C @ X4 @ Y3 ) ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_244_finite__Domain,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R4 )
% 3.10/1.60 => ( finite_finite2 @ A @ ( domain @ A @ B @ R4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_245_finite__Image,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A] :
% 3.10/1.60 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 3.10/1.60 => ( finite_finite2 @ B @ ( image @ A @ B @ R @ A5 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_246_Image__mono,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),R4: set @ ( product_prod @ A @ B ),A8: set @ A,A5: set @ A] :
% 3.10/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ R4 )
% 3.10/1.60 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ A5 )
% 3.10/1.60 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R5 @ A8 ) @ ( image @ A @ B @ R4 @ A5 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_247_Domain__mono,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S4 )
% 3.10/1.60 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R4 ) @ ( domain @ A @ B @ S4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_248_Domain_Oinducts,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,X4: A,R4: set @ ( product_prod @ A @ B ),P: A > $o] :
% 3.10/1.60 ( ( member @ A @ X4 @ ( domain @ A @ B @ R4 ) )
% 3.10/1.60 => ( ! [A4: A,B4: B] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R4 )
% 3.10/1.60 => ( P @ A4 ) )
% 3.10/1.60 => ( P @ X4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_249_Domain_ODomainI,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,A2: A,B2: B,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 3.10/1.60 => ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_250_Domain_Osimps,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 3.10/1.60 = ( ? [A7: A,B6: B] :
% 3.10/1.60 ( ( A2 = A7 )
% 3.10/1.60 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B6 ) @ R4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_251_Domain_Ocases,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 3.10/1.60 => ~ ! [B4: B] :
% 3.10/1.60 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R4 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_252_Domain__iff,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 3.10/1.60 = ( ? [Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y6 ) @ R4 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_253_DomainE,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,A2: A,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( member @ A @ A2 @ ( domain @ A @ B @ R4 ) )
% 3.10/1.60 => ~ ! [B4: B] :
% 3.10/1.60 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R4 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_254_rev__ImageI,axiom,
% 3.10/1.60 ! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,R4: set @ ( product_prod @ A @ B )] :
% 3.10/1.60 ( ( member @ A @ A2 @ A5 )
% 3.10/1.60 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R4 )
% 3.10/1.60 => ( member @ B @ B2 @ ( image @ A @ B @ R4 @ A5 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(fact_255_Image__iff,axiom,
% 3.10/1.60 ! [A: $tType,B: $tType,B2: A,R4: set @ ( product_prod @ B @ A ),A5: set @ B] :
% 3.10/1.60 ( ( member @ A @ B2 @ ( image @ B @ A @ R4 @ A5 ) )
% 3.10/1.60 = ( ? [X6: B] :
% 3.10/1.60 ( ( member @ B @ X6 @ A5 )
% 3.10/1.60 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X6 @ B2 ) @ R4 ) ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 3.10/1.60 ! [A9: $tType,A10: $tType] :
% 3.10/1.60 ( ( order_bot @ A10 )
% 3.10/1.60 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.10/1.60 ! [A9: $tType,A10: $tType] :
% 3.10/1.60 ( ( preorder @ A10 )
% 3.10/1.60 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 3.10/1.60 ! [A9: $tType,A10: $tType] :
% 3.10/1.60 ( ( ( finite_finite @ A9 )
% 3.10/1.60 & ( finite_finite @ A10 ) )
% 3.10/1.60 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.10/1.60 ! [A9: $tType,A10: $tType] :
% 3.10/1.60 ( ( order @ A10 )
% 3.10/1.60 => ( order @ ( A9 > A10 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_fun___Orderings_Oord,axiom,
% 3.10/1.60 ! [A9: $tType,A10: $tType] :
% 3.10/1.60 ( ( ord @ A10 )
% 3.10/1.60 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 3.10/1.60 order_bot @ nat ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 3.10/1.60 preorder @ nat ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.10/1.60 linorder @ nat ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 3.10/1.60 order @ nat ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 3.10/1.60 ord @ nat ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Set_Oset___Orderings_Oorder__bot_5,axiom,
% 3.10/1.60 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
% 3.10/1.60 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Set_Oset___Finite__Set_Ofinite_7,axiom,
% 3.10/1.60 ! [A9: $tType] :
% 3.10/1.60 ( ( finite_finite @ A9 )
% 3.10/1.60 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 3.10/1.60 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 3.10/1.60 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_HOL_Obool___Orderings_Oorder__bot_10,axiom,
% 3.10/1.60 order_bot @ $o ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
% 3.10/1.60 preorder @ $o ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_HOL_Obool___Orderings_Olinorder_12,axiom,
% 3.10/1.60 linorder @ $o ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_HOL_Obool___Finite__Set_Ofinite_13,axiom,
% 3.10/1.60 finite_finite @ $o ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 3.10/1.60 order @ $o ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 3.10/1.60 ord @ $o ).
% 3.10/1.60
% 3.10/1.60 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_16,axiom,
% 3.10/1.60 ! [A9: $tType,A10: $tType] :
% 3.10/1.60 ( ( ( finite_finite @ A9 )
% 3.10/1.60 & ( finite_finite @ A10 ) )
% 3.10/1.60 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.10/1.60
% 3.10/1.60 thf(conj_0,conjecture,
% 3.10/1.60 ( ( graph_homomorphism @ b @ nat @ nat @ ( product_fst @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) @ ( id_on @ nat @ ( labeled_vertices @ b @ nat @ ( product_fst @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) ) )
% 3.10/1.60 & ( ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) )
% 3.10/1.60 = ( restrict @ b @ nat @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) )
% 3.10/1.60 & ( finite_finite2 @ nat @ ( labeled_vertices @ b @ nat @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) )
% 3.10/1.60 & ( finite_finite2 @ ( product_prod @ b @ ( product_prod @ nat @ nat ) ) @ ( labeled_edges @ b @ nat @ ( product_snd @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( product_Pair @ ( labeled_graph @ b @ nat ) @ ( labeled_graph @ b @ nat ) @ ( translation @ b @ ( product_fst @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) @ ( translation @ b @ ( product_snd @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ ( product_Pair @ ( allegorical_term @ b ) @ ( allegorical_term @ b ) @ u @ ( allegorical_A_Int @ b @ u @ v ) ) ) ) ) ) ) ) ) ).
% 3.10/1.60 ------------------------
% 3.10/1.62 % (376)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/183Mi)
% 3.10/1.63 % (378)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/27Mi)
% 3.10/1.63 % (380)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/2Mi)
% 3.10/1.63 % (383)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/18Mi)
% 3.10/1.63 % (381)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/2Mi)
% 3.10/1.63 % (377)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/4Mi)
% 3.10/1.63 % (380)Instruction limit reached!
% 3.10/1.63 % (380)------------------------------
% 3.10/1.63 % (380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.63 % (380)Termination reason: Unknown
% 3.10/1.63 % (380)Termination phase: shuffling
% 3.10/1.63 % (381)Instruction limit reached!
% 3.10/1.63 % (381)------------------------------
% 3.10/1.63 % (381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.63 % (381)Termination reason: Unknown
% 3.10/1.63 % (381)Termination phase: shuffling
% 3.10/1.63
% 3.10/1.63 % (382)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/275Mi)
% 3.10/1.63 % (381)Memory used [KB]: 1535
% 3.10/1.63 % (381)Time elapsed: 0.003 s
% 3.10/1.63 % (381)Instructions burned: 3 (million)
% 3.10/1.63 % (381)------------------------------
% 3.10/1.63 % (381)------------------------------
% 3.10/1.63
% 3.10/1.63 % (380)Memory used [KB]: 1535
% 3.10/1.63 % (380)Time elapsed: 0.003 s
% 3.10/1.63 % (380)Instructions burned: 3 (million)
% 3.10/1.63 % (380)------------------------------
% 3.10/1.63 % (380)------------------------------
% 3.10/1.63 % (377)Instruction limit reached!
% 3.10/1.63 % (377)------------------------------
% 3.10/1.63 % (377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.63 % (377)Termination reason: Unknown
% 3.10/1.63 % (377)Termination phase: shuffling
% 3.10/1.63
% 3.10/1.63 % (377)Memory used [KB]: 1535
% 3.10/1.63 % (377)Time elapsed: 0.004 s
% 3.10/1.63 % (377)Instructions burned: 4 (million)
% 3.10/1.63 % (377)------------------------------
% 3.10/1.63 % (377)------------------------------
% 3.10/1.64 % (383)Instruction limit reached!
% 3.10/1.64 % (383)------------------------------
% 3.10/1.64 % (383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.64 % (383)Termination reason: Unknown
% 3.10/1.64 % (383)Termination phase: shuffling
% 3.10/1.64
% 3.10/1.64 % (383)Memory used [KB]: 1918
% 3.10/1.64 % (383)Time elapsed: 0.010 s
% 3.10/1.64 % (383)Instructions burned: 18 (million)
% 3.10/1.64 % (383)------------------------------
% 3.10/1.64 % (383)------------------------------
% 3.10/1.64 % (378)Instruction limit reached!
% 3.10/1.64 % (378)------------------------------
% 3.10/1.64 % (378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.64 % (378)Termination reason: Unknown
% 3.10/1.64 % (378)Termination phase: shuffling
% 3.10/1.64
% 3.10/1.64 % (378)Memory used [KB]: 2046
% 3.10/1.64 % (378)Time elapsed: 0.015 s
% 3.10/1.64 % (378)Instructions burned: 27 (million)
% 3.10/1.64 % (378)------------------------------
% 3.10/1.64 % (378)------------------------------
% 3.10/1.64 % (389)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/37Mi)
% 3.10/1.65 % (388)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/3Mi)
% 3.10/1.65 % (388)Instruction limit reached!
% 3.10/1.65 % (388)------------------------------
% 3.10/1.65 % (388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.65 % (388)Termination reason: Unknown
% 3.10/1.65 % (388)Termination phase: shuffling
% 3.10/1.65
% 3.10/1.65 % (388)Memory used [KB]: 1535
% 3.10/1.65 % (388)Time elapsed: 0.003 s
% 3.10/1.65 % (388)Instructions burned: 3 (million)
% 3.10/1.65 % (388)------------------------------
% 3.10/1.65 % (388)------------------------------
% 3.10/1.65 % (391)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/15Mi)
% 3.10/1.65 % (392)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/3Mi)
% 3.10/1.65 % (392)Instruction limit reached!
% 3.10/1.65 % (392)------------------------------
% 3.10/1.65 % (392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.65 % (392)Termination reason: Unknown
% 3.10/1.65 % (392)Termination phase: shuffling
% 3.10/1.65
% 3.10/1.65 % (392)Memory used [KB]: 1535
% 3.10/1.65 % (392)Time elapsed: 0.003 s
% 3.10/1.65 % (392)Instructions burned: 3 (million)
% 3.10/1.65 % (392)------------------------------
% 3.10/1.65 % (392)------------------------------
% 3.10/1.65 % (389)Instruction limit reached!
% 3.10/1.65 % (389)------------------------------
% 3.10/1.65 % (389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.65 % (389)Termination reason: Unknown
% 3.10/1.65 % (389)Termination phase: shuffling
% 3.10/1.65
% 3.10/1.65 % (389)Memory used [KB]: 2302
% 3.10/1.65 % (389)Time elapsed: 0.013 s
% 3.10/1.65 % (389)Instructions burned: 40 (million)
% 3.10/1.65 % (389)------------------------------
% 3.10/1.65 % (389)------------------------------
% 3.10/1.66 % (391)Instruction limit reached!
% 3.10/1.66 % (391)------------------------------
% 3.10/1.66 % (391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.66 % (391)Termination reason: Unknown
% 3.10/1.66 % (391)Termination phase: shuffling
% 3.10/1.66
% 3.10/1.66 % (391)Memory used [KB]: 1791
% 3.10/1.66 % (391)Time elapsed: 0.010 s
% 3.10/1.66 % (391)Instructions burned: 15 (million)
% 3.10/1.66 % (391)------------------------------
% 3.10/1.66 % (391)------------------------------
% 3.10/1.66 % (393)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/1041Mi)
% 3.10/1.66 % (396)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/16Mi)
% 3.10/1.66 % (397)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/3Mi)
% 3.10/1.67 % (397)Instruction limit reached!
% 3.10/1.67 % (397)------------------------------
% 3.10/1.67 % (397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.67 % (397)Termination reason: Unknown
% 3.10/1.67 % (397)Termination phase: shuffling
% 3.10/1.67
% 3.10/1.67 % (397)Memory used [KB]: 1663
% 3.10/1.67 % (397)Time elapsed: 0.003 s
% 3.10/1.67 % (397)Instructions burned: 5 (million)
% 3.10/1.67 % (397)------------------------------
% 3.10/1.67 % (397)------------------------------
% 3.10/1.67 % (394)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/7Mi)
% 3.10/1.67 % (396)Instruction limit reached!
% 3.10/1.67 % (396)------------------------------
% 3.10/1.67 % (396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.67 % (396)Termination reason: Unknown
% 3.10/1.67 % (396)Termination phase: shuffling
% 3.10/1.67
% 3.10/1.67 % (396)Memory used [KB]: 1918
% 3.10/1.67 % (396)Time elapsed: 0.007 s
% 3.10/1.67 % (396)Instructions burned: 18 (million)
% 3.10/1.67 % (396)------------------------------
% 3.10/1.67 % (396)------------------------------
% 3.10/1.67 % (394)Instruction limit reached!
% 3.10/1.67 % (394)------------------------------
% 3.10/1.67 % (394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.67 % (394)Termination reason: Unknown
% 3.10/1.67 % (394)Termination phase: shuffling
% 3.10/1.67
% 3.10/1.67 % (394)Memory used [KB]: 1663
% 3.10/1.67 % (394)Time elapsed: 0.005 s
% 3.10/1.67 % (394)Instructions burned: 7 (million)
% 3.10/1.67 % (394)------------------------------
% 3.10/1.67 % (394)------------------------------
% 3.10/1.68 % (399)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/3Mi)
% 3.10/1.68 % (400)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/7Mi)
% 3.10/1.68 % (399)Instruction limit reached!
% 3.10/1.68 % (399)------------------------------
% 3.10/1.68 % (399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.68 % (399)Termination reason: Unknown
% 3.10/1.68 % (399)Termination phase: shuffling
% 3.10/1.68
% 3.10/1.68 % (399)Memory used [KB]: 1535
% 3.10/1.68 % (399)Time elapsed: 0.004 s
% 3.10/1.68 % (399)Instructions burned: 4 (million)
% 3.10/1.68 % (399)------------------------------
% 3.10/1.68 % (399)------------------------------
% 3.10/1.68 % (400)Instruction limit reached!
% 3.10/1.68 % (400)------------------------------
% 3.10/1.68 % (400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.68 % (400)Termination reason: Unknown
% 3.10/1.68 % (400)Termination phase: shuffling
% 3.10/1.68
% 3.10/1.68 % (400)Memory used [KB]: 1663
% 3.10/1.68 % (400)Time elapsed: 0.004 s
% 3.10/1.68 % (400)Instructions burned: 8 (million)
% 3.10/1.68 % (400)------------------------------
% 3.10/1.68 % (400)------------------------------
% 3.10/1.68 % (402)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/3Mi)
% 3.10/1.68 % (402)Instruction limit reached!
% 3.10/1.68 % (402)------------------------------
% 3.10/1.68 % (402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.68 % (402)Termination reason: Unknown
% 3.10/1.68 % (402)Termination phase: shuffling
% 3.10/1.68
% 3.10/1.68 % (402)Memory used [KB]: 1535
% 3.10/1.68 % (402)Time elapsed: 0.002 s
% 3.10/1.68 % (402)Instructions burned: 3 (million)
% 3.10/1.68 % (402)------------------------------
% 3.10/1.68 % (402)------------------------------
% 3.10/1.69 % (403)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/4Mi)
% 3.10/1.69 % (407)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/6Mi)
% 3.10/1.69 % (403)Instruction limit reached!
% 3.10/1.69 % (403)------------------------------
% 3.10/1.69 % (403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.69 % (403)Termination reason: Unknown
% 3.10/1.69 % (403)Termination phase: shuffling
% 3.10/1.69
% 3.10/1.69 % (403)Memory used [KB]: 1663
% 3.10/1.69 % (403)Time elapsed: 0.004 s
% 3.10/1.69 % (403)Instructions burned: 5 (million)
% 3.10/1.69 % (403)------------------------------
% 3.10/1.69 % (403)------------------------------
% 3.10/1.69 % (407)Instruction limit reached!
% 3.10/1.69 % (407)------------------------------
% 3.10/1.69 % (407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.69 % (407)Termination reason: Unknown
% 3.10/1.69 % (407)Termination phase: shuffling
% 3.10/1.69
% 3.10/1.69 % (407)Memory used [KB]: 1663
% 3.10/1.69 % (407)Time elapsed: 0.004 s
% 3.10/1.69 % (407)Instructions burned: 8 (million)
% 3.10/1.69 % (407)------------------------------
% 3.10/1.69 % (407)------------------------------
% 3.10/1.69 % (382)Refutation not found, incomplete strategy
% 3.10/1.69 % (382)------------------------------
% 3.10/1.69 % (382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.69 % (382)Termination reason: Refutation not found, incomplete strategy
% 3.10/1.69
% 3.10/1.69
% 3.10/1.69 % (382)Memory used [KB]: 7419
% 3.10/1.69 % (382)Time elapsed: 0.065 s
% 3.10/1.69 % (382)Instructions burned: 122 (million)
% 3.10/1.69 % (382)------------------------------
% 3.10/1.69 % (382)------------------------------
% 3.10/1.69 % (406)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/710Mi)
% 3.10/1.70 % (404)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/18Mi)
% 3.10/1.70 % (376)Instruction limit reached!
% 3.10/1.70 % (376)------------------------------
% 3.10/1.70 % (376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.70 % (376)Termination reason: Unknown
% 3.10/1.70 % (376)Termination phase: Saturation
% 3.10/1.70
% 3.10/1.70 % (376)Memory used [KB]: 7419
% 3.10/1.70 % (376)Time elapsed: 0.076 s
% 3.10/1.70 % (376)Instructions burned: 183 (million)
% 3.10/1.70 % (376)------------------------------
% 3.10/1.70 % (376)------------------------------
% 3.10/1.70 % (409)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/902Mi)
% 3.10/1.70 % (404)Instruction limit reached!
% 3.10/1.70 % (404)------------------------------
% 3.10/1.70 % (404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.70 % (404)Termination reason: Unknown
% 3.10/1.70 % (404)Termination phase: shuffling
% 3.10/1.70
% 3.10/1.70 % (404)Memory used [KB]: 1791
% 3.10/1.70 % (404)Time elapsed: 0.008 s
% 3.10/1.70 % (404)Instructions burned: 18 (million)
% 3.10/1.70 % (404)------------------------------
% 3.10/1.70 % (404)------------------------------
% 3.10/1.70 % (410)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/21Mi)
% 3.10/1.70 % (411)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/5Mi)
% 3.10/1.71 % (411)Instruction limit reached!
% 3.10/1.71 % (411)------------------------------
% 3.10/1.71 % (411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.71 % (411)Termination reason: Unknown
% 3.10/1.71 % (411)Termination phase: shuffling
% 3.10/1.71
% 3.10/1.71 % (411)Memory used [KB]: 1663
% 3.10/1.71 % (411)Time elapsed: 0.003 s
% 3.10/1.71 % (411)Instructions burned: 5 (million)
% 3.10/1.71 % (411)------------------------------
% 3.10/1.71 % (411)------------------------------
% 3.10/1.71 % (413)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/6Mi)
% 3.10/1.71 % (413)Instruction limit reached!
% 3.10/1.71 % (413)------------------------------
% 3.10/1.71 % (413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.71 % (413)Termination reason: Unknown
% 3.10/1.71 % (413)Termination phase: shuffling
% 3.10/1.71
% 3.10/1.71 % (413)Memory used [KB]: 1663
% 3.10/1.71 % (413)Time elapsed: 0.003 s
% 3.10/1.71 % (413)Instructions burned: 6 (million)
% 3.10/1.71 % (413)------------------------------
% 3.10/1.71 % (413)------------------------------
% 3.10/1.71 % (410)Instruction limit reached!
% 3.10/1.71 % (410)------------------------------
% 3.10/1.71 % (410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.71 % (410)Termination reason: Unknown
% 3.10/1.71 % (410)Termination phase: shuffling
% 3.10/1.71
% 3.10/1.71 % (410)Memory used [KB]: 1918
% 3.10/1.71 % (410)Time elapsed: 0.009 s
% 3.10/1.71 % (410)Instructions burned: 23 (million)
% 3.10/1.71 % (410)------------------------------
% 3.10/1.71 % (410)------------------------------
% 3.10/1.71 % (414)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/377Mi)
% 3.10/1.71 % (415)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/779Mi)
% 3.10/1.72 % (416)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/19Mi)
% 3.10/1.72 % (417)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_32651 for (2999ds/879Mi)
% 3.10/1.72 % (416)Instruction limit reached!
% 3.10/1.72 % (416)------------------------------
% 3.10/1.72 % (416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.72 % (416)Termination reason: Unknown
% 3.10/1.72 % (416)Termination phase: shuffling
% 3.10/1.72
% 3.10/1.72 % (416)Memory used [KB]: 1918
% 3.10/1.72 % (416)Time elapsed: 0.007 s
% 3.10/1.72 % (416)Instructions burned: 20 (million)
% 3.10/1.72 % (416)------------------------------
% 3.10/1.72 % (416)------------------------------
% 3.10/1.73 % (418)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_32651 for (2998ds/17Mi)
% 3.10/1.74 % (418)Instruction limit reached!
% 3.10/1.74 % (418)------------------------------
% 3.10/1.74 % (418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.74 % (418)Termination reason: Unknown
% 3.10/1.74 % (418)Termination phase: shuffling
% 3.10/1.74
% 3.10/1.74 % (418)Memory used [KB]: 1791
% 3.10/1.74 % (418)Time elapsed: 0.007 s
% 3.10/1.74 % (418)Instructions burned: 18 (million)
% 3.10/1.74 % (418)------------------------------
% 3.10/1.74 % (418)------------------------------
% 3.10/1.75 % (419)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32651 for (2998ds/3Mi)
% 3.10/1.75 % (419)Instruction limit reached!
% 3.10/1.75 % (419)------------------------------
% 3.10/1.75 % (419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.75 % (419)Termination reason: Unknown
% 3.10/1.75 % (419)Termination phase: shuffling
% 3.10/1.75
% 3.10/1.75 % (419)Memory used [KB]: 1535
% 3.10/1.75 % (419)Time elapsed: 0.002 s
% 3.10/1.75 % (419)Instructions burned: 3 (million)
% 3.10/1.75 % (419)------------------------------
% 3.10/1.75 % (419)------------------------------
% 3.10/1.76 % (406)First to succeed.
% 3.10/1.76 % (420)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_32651 for (2998ds/30Mi)
% 3.10/1.76 % (409)Refutation not found, incomplete strategy
% 3.10/1.76 % (409)------------------------------
% 3.10/1.76 % (409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.76 % (409)Termination reason: Refutation not found, incomplete strategy
% 3.10/1.76
% 3.10/1.76
% 3.10/1.76 % (409)Memory used [KB]: 8059
% 3.10/1.76 % (409)Time elapsed: 0.060 s
% 3.10/1.76 % (409)Instructions burned: 194 (million)
% 3.10/1.76 % (409)------------------------------
% 3.10/1.76 % (409)------------------------------
% 3.10/1.76 % (406)Refutation found. Thanks to Tanya!
% 3.10/1.76 % SZS status Theorem for DTF2THF_32651
% 3.10/1.76 % SZS output start Proof for DTF2THF_32651
% 3.10/1.76 thf(type_def_5, type, set: ($tType) > $tType).
% 3.10/1.76 thf(type_def_7, type, allegorical_term: ($tType) > $tType).
% 3.10/1.76 thf(type_def_8, type, labeled_graph: ($tType * $tType) > $tType).
% 3.10/1.76 thf(type_def_9, type, product_prod: ($tType * $tType) > $tType).
% 3.10/1.76 thf(type_def_10, type, nat: $tType).
% 3.10/1.76 thf(type_def_11, type, itself: ($tType) > $tType).
% 3.10/1.76 thf(type_def_12, type, standard_Constant: ($tType) > $tType).
% 3.10/1.76 thf(type_def_13, type, a: $tType).
% 3.10/1.76 thf(type_def_14, type, b: $tType).
% 3.10/1.76 thf(func_def_0, type, standard_Constant: ($tType) > $tType).
% 3.10/1.76 thf(func_def_1, type, allegorical_term: ($tType) > $tType).
% 3.10/1.76 thf(func_def_2, type, labeled_graph: ($tType * $tType) > $tType).
% 3.10/1.76 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 3.10/1.76 thf(func_def_4, type, set: ($tType) > $tType).
% 3.10/1.76 thf(func_def_5, type, nat: $tType).
% 3.10/1.76 thf(func_def_6, type, itself: ($tType) > $tType).
% 3.10/1.76 thf(func_def_7, type, b: $tType).
% 3.10/1.76 thf(func_def_8, type, a: $tType).
% 3.10/1.76 thf(func_def_9, type, type: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_10, type, ord: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_11, type, order: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_12, type, finite_finite: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_15, type, order_bot: !>[X0: $tType]:($o)).
% 3.10/1.76 thf(func_def_16, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 3.10/1.76 thf(func_def_17, type, allegorical_A_Int: !>[X0: $tType]:(allegorical_term @ X0 > allegorical_term @ X0 > allegorical_term @ X0)).
% 3.10/1.76 thf(func_def_18, type, semantics: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > allegorical_term @ X0 > set @ product_prod @ X1 @ X1)).
% 3.10/1.76 thf(func_def_19, type, graph_homomorphism: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.10/1.76 thf(func_def_20, type, graph_union: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.10/1.76 thf(func_def_21, type, labeled_LG: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > labeled_graph @ X0 @ X1)).
% 3.10/1.76 thf(func_def_22, type, labele1974067554_graph: !>[X0: $tType, X1: $tType, X2: $tType]:((set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > X2) > labeled_graph @ X0 @ X1 > X2)).
% 3.10/1.76 thf(func_def_23, type, labeled_edges: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X0 @ product_prod @ X1 @ X1)).
% 3.10/1.76 thf(func_def_24, type, labeled_vertices: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ X1)).
% 3.10/1.76 thf(func_def_25, type, restrict: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.10/1.76 thf(func_def_26, type, univalent: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > $o)).
% 3.10/1.76 thf(func_def_27, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.10/1.76 thf(func_def_28, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.10/1.76 thf(func_def_29, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.10/1.76 thf(func_def_30, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.10/1.76 thf(func_def_31, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.10/1.76 thf(func_def_32, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.10/1.76 thf(func_def_33, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 3.10/1.76 thf(func_def_34, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 3.10/1.76 thf(func_def_35, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 3.10/1.76 thf(func_def_36, type, relcomp: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X0 @ X2)).
% 3.10/1.76 thf(func_def_37, type, inv_translation: set @ nat > $o).
% 3.10/1.76 thf(func_def_38, type, translation: !>[X0: $tType]:(allegorical_term @ X0 > labeled_graph @ X0 @ nat)).
% 3.10/1.76 thf(func_def_39, type, agree_on: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.10/1.76 thf(func_def_40, type, chain: !>[X0: $tType, X1: $tType]:((nat > labeled_graph @ X0 @ X1) > $o)).
% 3.10/1.76 thf(func_def_41, type, chain_sup: !>[X0: $tType, X1: $tType]:((nat > labeled_graph @ X0 @ X1) > labeled_graph @ X0 @ X1)).
% 3.10/1.76 thf(func_def_42, type, consequence_graph: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.10/1.76 thf(func_def_43, type, extensible: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.10/1.76 thf(func_def_44, type, fair_chain: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > (nat > labeled_graph @ X0 @ X2) > $o)).
% 3.10/1.76 thf(func_def_45, type, fin_maintained: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.10/1.76 thf(func_def_46, type, least: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(itself @ X0 > set @ product_prod @ labeled_graph @ X1 @ X2 @ labeled_graph @ X1 @ X2 > labeled_graph @ X1 @ X3 > labeled_graph @ X1 @ X3 > $o)).
% 3.10/1.76 thf(func_def_47, type, least_559130134_graph: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(itself @ X0 > set @ product_prod @ labeled_graph @ X1 @ X2 @ labeled_graph @ X1 @ X2 > labeled_graph @ X1 @ X3 > labeled_graph @ X1 @ X3 > $o)).
% 3.10/1.76 thf(func_def_48, type, maintained: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.10/1.76 thf(func_def_49, type, pushout_step: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(itself @ X0 > product_prod @ labeled_graph @ X1 @ X2 @ labeled_graph @ X1 @ X2 > labeled_graph @ X1 @ X3 > labeled_graph @ X1 @ X3 > $o)).
% 3.10/1.76 thf(func_def_50, type, set_of_graph_rules: !>[X0: $tType, X1: $tType]:(set @ product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > $o)).
% 3.10/1.76 thf(func_def_51, type, weak_universal: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(itself @ X0 > product_prod @ labeled_graph @ X1 @ X2 @ labeled_graph @ X1 @ X2 > labeled_graph @ X1 @ X3 > labeled_graph @ X1 @ X3 > set @ product_prod @ X2 @ X3 > set @ product_prod @ X2 @ X3 > $o)).
% 3.10/1.76 thf(func_def_52, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.10/1.76 thf(func_def_53, type, standa1138209853_rules: !>[X0: $tType]:(set @ X0 > set @ product_prod @ labeled_graph @ standard_Constant @ X0 @ nat @ labeled_graph @ standard_Constant @ X0 @ nat)).
% 3.10/1.76 thf(func_def_54, type, standa2002409347_rules: !>[X0: $tType]:(set @ standard_Constant @ X0 > set @ product_prod @ labeled_graph @ standard_Constant @ X0 @ nat @ labeled_graph @ standard_Constant @ X0 @ nat)).
% 3.10/1.76 thf(func_def_55, type, standa244753842y_rule: !>[X0: $tType]:(product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.10/1.76 thf(func_def_56, type, standa245363227y_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.10/1.76 thf(func_def_57, type, standa400804411p_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.10/1.76 thf(func_def_58, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.10/1.76 thf(func_def_59, type, c: set @ a).
% 3.10/1.76 thf(func_def_60, type, u: allegorical_term @ b).
% 3.10/1.76 thf(func_def_61, type, v: allegorical_term @ b).
% 3.10/1.76 thf(func_def_62, type, x: product_prod @ labeled_graph @ standard_Constant @ a @ nat @ labeled_graph @ standard_Constant @ a @ nat).
% 3.10/1.76 thf(func_def_80, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 3.10/1.76 thf(func_def_81, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 3.10/1.76 thf(func_def_82, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.10/1.76 thf(func_def_83, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.10/1.76 thf(func_def_84, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.10/1.76 thf(func_def_85, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.10/1.76 thf(func_def_86, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.10/1.76 thf(func_def_87, type, sK7: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0 > X1)).
% 3.10/1.76 thf(func_def_88, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.10/1.76 thf(func_def_89, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.10/1.76 thf(func_def_90, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.10/1.76 thf(func_def_91, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.10/1.76 thf(func_def_92, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.10/1.76 thf(func_def_93, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.10/1.76 thf(func_def_94, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.76 thf(func_def_95, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.76 thf(func_def_96, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.76 thf(func_def_97, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.76 thf(func_def_98, type, sK18: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.10/1.76 thf(func_def_99, type, sK19: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.10/1.76 thf(func_def_100, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.10/1.76 thf(func_def_101, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.10/1.76 thf(func_def_102, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.10/1.76 thf(func_def_103, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 3.10/1.76 thf(func_def_104, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 3.10/1.76 thf(func_def_105, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 3.10/1.76 thf(func_def_106, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 3.10/1.76 thf(func_def_107, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 3.10/1.76 thf(func_def_108, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 3.10/1.76 thf(func_def_109, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.76 thf(func_def_110, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.76 thf(func_def_111, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.76 thf(func_def_112, type, sK32: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > X0 > X1)).
% 3.10/1.76 thf(func_def_113, type, sK33: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.76 thf(func_def_114, type, sK34: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.77 thf(func_def_115, type, sK35: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.10/1.77 thf(func_def_116, type, sK36: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.10/1.77 thf(func_def_117, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 3.10/1.77 thf(func_def_118, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 3.10/1.77 thf(func_def_119, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 3.10/1.77 thf(func_def_120, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 3.10/1.77 thf(func_def_121, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 3.10/1.77 thf(func_def_122, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 3.10/1.77 thf(func_def_123, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 3.10/1.77 thf(func_def_124, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 3.10/1.77 thf(func_def_125, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.10/1.77 thf(func_def_126, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.10/1.77 thf(func_def_127, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.77 thf(func_def_128, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.77 thf(func_def_129, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.77 thf(func_def_130, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.77 thf(func_def_131, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.10/1.77 thf(func_def_132, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.77 thf(func_def_133, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.77 thf(func_def_134, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.77 thf(func_def_135, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.77 thf(func_def_136, type, sK56: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.77 thf(func_def_137, type, sK57: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.77 thf(func_def_138, type, sK58: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1)).
% 3.10/1.77 thf(func_def_139, type, sK59: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0)).
% 3.10/1.77 thf(func_def_140, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.77 thf(func_def_141, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.77 thf(func_def_142, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.77 thf(func_def_143, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.77 thf(func_def_144, type, sK64: (nat > $o) > nat).
% 3.10/1.77 thf(func_def_145, type, sK65: nat > (nat > $o) > nat).
% 3.10/1.77 thf(func_def_146, type, sK66: !>[X0: $tType]:(set @ X0 > product_prod @ X0 @ X0 > X0)).
% 3.10/1.77 thf(f1087,plain,(
% 3.10/1.77 $false),
% 3.10/1.77 inference(avatar_sat_refutation,[status(thm)],[f1024,f1066,f1073,f1077,f1086])).
% 3.10/1.77 thf(f1086,plain,(
% 3.10/1.77 spl67_4),
% 3.10/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1085])).
% 3.10/1.77 thf(f1085,plain,(
% 3.10/1.77 $false | spl67_4),
% 3.10/1.77 inference(subsumption_resolution,[status(thm)],[f1084,f953])).
% 3.10/1.77 thf(f953,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X1 : allegorical_term @ X0] : (((finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (translation @ X0 @ X1))) = $true)) )),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f543])).
% 3.10/1.77 thf(f543,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : allegorical_term @ X0] : ((finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (translation @ X0 @ X1))) = $true)),
% 3.10/1.77 inference(fool_elimination,[status(thm)],[f542])).
% 3.10/1.77 thf(f542,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : allegorical_term @ X0] : (finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (translation @ X0 @ X1)))),
% 3.10/1.77 inference(rectify,[status(thm)],[f9])).
% 3.10/1.77 thf(f9,axiom,(
% 3.10/1.77 ! [X0 : $tType,X7 : allegorical_term @ X0] : (finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (translation @ X0 @ X7)))),
% 3.10/1.77 file('/export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2THF_32651.p',fact_8_verts__in__translation__finite_I1_J)).
% 3.10/1.77 thf(f1084,plain,(
% 3.10/1.77 ((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) != $true) | spl67_4),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1023,f922])).
% 3.10/1.77 thf(f922,plain,(
% 3.10/1.77 ( ! [X1 : $tType,X0 : $tType,X2 : X0,X3 : X1] : (((product_snd @ X0 @ X1 @ (product_Pair @ X0 @ X1 @ X2 @ X3)) = X3)) )),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f767])).
% 3.10/1.77 thf(f767,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : $tType,X2 : X0,X3 : X1] : ((product_snd @ X0 @ X1 @ (product_Pair @ X0 @ X1 @ X2 @ X3)) = X3)),
% 3.10/1.77 inference(rectify,[status(thm)],[f34])).
% 3.10/1.77 thf(f34,axiom,(
% 3.10/1.77 ! [X39 : $tType,X0 : $tType,X18 : X39,X19 : X0] : ((product_snd @ X39 @ X0 @ (product_Pair @ X39 @ X0 @ X18 @ X19)) = X19)),
% 3.10/1.77 file('/export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2THF_32651.p',fact_33_snd__conv)).
% 3.10/1.77 thf(f1023,plain,(
% 3.10/1.77 ((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) != $true) | spl67_4),
% 3.10/1.77 inference(avatar_component_clause,[status(thm)],[f1021])).
% 3.10/1.77 thf(f1021,definition,(
% 3.10/1.77 spl67_4 <=> ((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) = $true)),
% 3.10/1.77 introduced(definition,[new_symbols(naming,[spl67_4])],[avatar_definition])).
% 3.10/1.77 thf(f1077,plain,(
% 3.10/1.77 spl67_3),
% 3.10/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1076])).
% 3.10/1.77 thf(f1076,plain,(
% 3.10/1.77 $false | spl67_3),
% 3.10/1.77 inference(subsumption_resolution,[status(thm)],[f1075,f1052])).
% 3.10/1.77 thf(f1052,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X2 : allegorical_term @ X0,X1 : allegorical_term @ X0] : (((translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)) = (restrict @ X0 @ nat @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))) )),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f949,f922])).
% 3.10/1.77 thf(f949,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X2 : allegorical_term @ X0,X1 : allegorical_term @ X0] : (((restrict @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))) = (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))))) )),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f735])).
% 3.10/1.77 thf(f735,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : allegorical_term @ X0,X2 : allegorical_term @ X0] : (($true = (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))))) & ((restrict @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))) = (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))) & ((graph_homomorphism @ X0 @ nat @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (id_on @ nat @ (labeled_vertices @ X0 @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))))) = $true) & ((finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))))) = $true))),
% 3.10/1.77 inference(fool_elimination,[status(thm)],[f734])).
% 3.10/1.77 thf(f734,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : allegorical_term @ X0,X2 : allegorical_term @ X0] : (((restrict @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))) = (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))) & (graph_homomorphism @ X0 @ nat @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (id_on @ nat @ (labeled_vertices @ X0 @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))))) & (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))))) & (finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))))))),
% 3.10/1.77 inference(rectify,[status(thm)],[f2])).
% 3.10/1.77 thf(f2,axiom,(
% 3.10/1.77 ! [X0 : $tType,X7 : allegorical_term @ X0,X8 : allegorical_term @ X0] : (((product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8)))) = (restrict @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8)))))) & (graph_homomorphism @ X0 @ nat @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8)))) @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8)))) @ (id_on @ nat @ (labeled_vertices @ X0 @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8))))))) & (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8)))))) & (finite_finite2 @ nat @ (labeled_vertices @ X0 @ nat @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X7) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X7 @ X8)))))))),
% 3.10/1.77 file('/export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2THF_32651.p',fact_1_graph__rule__translation)).
% 3.10/1.77 thf(f1075,plain,(
% 3.10/1.77 ((translation @ b @ (allegorical_A_Int @ b @ u @ v)) != (restrict @ b @ nat @ (translation @ b @ (allegorical_A_Int @ b @ u @ v)))) | spl67_3),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1074,f922])).
% 3.10/1.77 thf(f1074,plain,(
% 3.10/1.77 ((translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) != (restrict @ b @ nat @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) | spl67_3),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1019,f922])).
% 3.10/1.77 thf(f1019,plain,(
% 3.10/1.77 ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) != (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) | spl67_3),
% 3.10/1.77 inference(avatar_component_clause,[status(thm)],[f1017])).
% 3.10/1.77 thf(f1017,definition,(
% 3.10/1.77 spl67_3 <=> ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) = (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))),
% 3.10/1.77 introduced(definition,[new_symbols(naming,[spl67_3])],[avatar_definition])).
% 3.10/1.77 thf(f1073,plain,(
% 3.10/1.77 spl67_2),
% 3.10/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1072])).
% 3.10/1.77 thf(f1072,plain,(
% 3.10/1.77 $false | spl67_2),
% 3.10/1.77 inference(subsumption_resolution,[status(thm)],[f1071,f1063])).
% 3.10/1.77 thf(f1063,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X2 : allegorical_term @ X0,X1 : allegorical_term @ X0] : (($true = (graph_homomorphism @ X0 @ nat @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)) @ (id_on @ nat @ (labeled_vertices @ X0 @ nat @ (translation @ X0 @ X1)))))) )),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1062,f922])).
% 3.10/1.77 thf(f1062,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X2 : allegorical_term @ X0,X1 : allegorical_term @ X0] : (((graph_homomorphism @ X0 @ nat @ nat @ (translation @ X0 @ X1) @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (id_on @ nat @ (labeled_vertices @ X0 @ nat @ (translation @ X0 @ X1)))) = $true)) )),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f948,f1005])).
% 3.10/1.77 thf(f1005,plain,(
% 3.10/1.77 ( ! [X1 : $tType,X0 : $tType,X3 : X1,X4 : X0] : (((product_fst @ X1 @ X0 @ (product_Pair @ X1 @ X0 @ X3 @ X4)) = X3)) )),
% 3.10/1.77 inference(equality_resolution,[status(thm)],[f980])).
% 3.10/1.77 thf(f980,plain,(
% 3.10/1.77 ( ! [X1 : $tType,X0 : $tType,X2 : product_prod @ X1 @ X0,X3 : X1,X4 : X0] : (((product_Pair @ X1 @ X0 @ X3 @ X4) != X2) | ((product_fst @ X1 @ X0 @ X2) = X3)) )),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f806])).
% 3.10/1.77 thf(f806,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : $tType,X2 : product_prod @ X1 @ X0,X3 : X1,X4 : X0] : (((product_Pair @ X1 @ X0 @ X3 @ X4) != X2) | ((product_fst @ X1 @ X0 @ X2) = X3))),
% 3.10/1.77 inference(ennf_transformation,[status(thm)],[f763])).
% 3.10/1.77 thf(f763,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : $tType,X2 : product_prod @ X1 @ X0,X3 : X1,X4 : X0] : (((product_Pair @ X1 @ X0 @ X3 @ X4) = X2) => ((product_fst @ X1 @ X0 @ X2) = X3))),
% 3.10/1.77 inference(rectify,[status(thm)],[f83])).
% 3.10/1.77 thf(f83,axiom,(
% 3.10/1.77 ! [X2 : $tType,X0 : $tType,X22 : product_prod @ X0 @ X2,X26 : X0,X75 : X2] : (((product_Pair @ X0 @ X2 @ X26 @ X75) = X22) => ((product_fst @ X0 @ X2 @ X22) = X26))),
% 3.10/1.77 file('/export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2THF_32651.p',fact_82_fstI)).
% 3.10/1.77 thf(f948,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X2 : allegorical_term @ X0,X1 : allegorical_term @ X0] : (((graph_homomorphism @ X0 @ nat @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (product_snd @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2)))) @ (id_on @ nat @ (labeled_vertices @ X0 @ nat @ (product_fst @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (product_Pair @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat @ (translation @ X0 @ X1) @ (translation @ X0 @ (allegorical_A_Int @ X0 @ X1 @ X2))))))) = $true)) )),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f735])).
% 3.10/1.77 thf(f1071,plain,(
% 3.10/1.77 ((graph_homomorphism @ b @ nat @ nat @ (translation @ b @ u) @ (translation @ b @ (allegorical_A_Int @ b @ u @ v)) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (translation @ b @ u)))) != $true) | spl67_2),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1070,f922])).
% 3.10/1.77 thf(f1070,plain,(
% 3.10/1.77 ($true != (graph_homomorphism @ b @ nat @ nat @ (translation @ b @ u) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (translation @ b @ u))))) | spl67_2),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1069,f922])).
% 3.10/1.77 thf(f1069,plain,(
% 3.10/1.77 ($true != (graph_homomorphism @ b @ nat @ nat @ (translation @ b @ u) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ u) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (translation @ b @ u))))) | spl67_2),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1068,f1005])).
% 3.10/1.77 thf(f1068,plain,(
% 3.10/1.77 ($true != (graph_homomorphism @ b @ nat @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) | spl67_2),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1015,f1005])).
% 3.10/1.77 thf(f1015,plain,(
% 3.10/1.77 ((graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) != $true) | spl67_2),
% 3.10/1.77 inference(avatar_component_clause,[status(thm)],[f1013])).
% 3.10/1.77 thf(f1013,definition,(
% 3.10/1.77 spl67_2 <=> ((graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) = $true)),
% 3.10/1.77 introduced(definition,[new_symbols(naming,[spl67_2])],[avatar_definition])).
% 3.10/1.77 thf(f1066,plain,(
% 3.10/1.77 spl67_1),
% 3.10/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1065])).
% 3.10/1.77 thf(f1065,plain,(
% 3.10/1.77 $false | spl67_1),
% 3.10/1.77 inference(subsumption_resolution,[status(thm)],[f1064,f895])).
% 3.10/1.77 thf(f895,plain,(
% 3.10/1.77 ( ! [X0 : $tType,X1 : allegorical_term @ X0] : (($true = (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (translation @ X0 @ X1))))) )),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f713])).
% 3.10/1.77 thf(f713,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : allegorical_term @ X0] : ($true = (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (translation @ X0 @ X1))))),
% 3.10/1.77 inference(fool_elimination,[status(thm)],[f712])).
% 3.10/1.77 thf(f712,plain,(
% 3.10/1.77 ! [X0 : $tType,X1 : allegorical_term @ X0] : (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (translation @ X0 @ X1)))),
% 3.10/1.77 inference(rectify,[status(thm)],[f6])).
% 3.10/1.77 thf(f6,axiom,(
% 3.10/1.77 ! [X0 : $tType,X7 : allegorical_term @ X0] : (finite_finite2 @ product_prod @ X0 @ product_prod @ nat @ nat @ (labeled_edges @ X0 @ nat @ (translation @ X0 @ X7)))),
% 3.10/1.77 file('/export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2THF_32651.p',fact_5_verts__in__translation__finite_I2_J)).
% 3.10/1.77 thf(f1064,plain,(
% 3.10/1.77 ($true != (finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))) | spl67_1),
% 3.10/1.77 inference(forward_demodulation,[status(thm)],[f1011,f922])).
% 3.10/1.77 thf(f1011,plain,(
% 3.10/1.77 ((finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) != $true) | spl67_1),
% 3.10/1.77 inference(avatar_component_clause,[status(thm)],[f1009])).
% 3.10/1.77 thf(f1009,definition,(
% 3.10/1.77 spl67_1 <=> ((finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) = $true)),
% 3.10/1.77 introduced(definition,[new_symbols(naming,[spl67_1])],[avatar_definition])).
% 3.10/1.77 thf(f1024,plain,(
% 3.10/1.77 ~spl67_1 | ~spl67_2 | ~spl67_3 | ~spl67_4),
% 3.10/1.77 inference(avatar_split_clause,[status(thm)],[f955,f1021,f1017,f1013,f1009])).
% 3.10/1.77 thf(f955,plain,(
% 3.10/1.77 ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) != (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) | ((graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) != $true) | ((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) != $true) | ((finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) != $true)),
% 3.10/1.77 inference(cnf_transformation,[status(thm)],[f791])).
% 3.10/1.77 thf(f791,plain,(
% 3.10/1.77 ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) != (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) | ((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) != $true) | ((finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) != $true) | ((graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) != $true)),
% 3.10/1.77 inference(ennf_transformation,[status(thm)],[f495])).
% 3.10/1.77 thf(f495,plain,(
% 3.10/1.77 ~(((graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) = $true) & ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) = (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & ((finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) = $true) & ((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) = $true))),
% 3.10/1.77 inference(fool_elimination,[status(thm)],[f494])).
% 3.10/1.77 thf(f494,plain,(
% 3.10/1.77 ~((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & (finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & (graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) & ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) = (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))))),
% 3.10/1.77 inference(rectify,[status(thm)],[f280])).
% 3.10/1.77 thf(f280,negated_conjecture,(
% 3.10/1.77 ~((finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & (finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & (graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) & ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) = (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))))),
% 3.10/1.77 inference(negated_conjecture,[status(cth)],[f279])).
% 3.10/1.77 thf(f279,conjecture,(
% 3.10/1.77 (finite_finite2 @ nat @ (labeled_vertices @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & (finite_finite2 @ product_prod @ b @ product_prod @ nat @ nat @ (labeled_edges @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))))) & (graph_homomorphism @ b @ nat @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) @ (id_on @ nat @ (labeled_vertices @ b @ nat @ (product_fst @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))) & ((product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))))) = (restrict @ b @ nat @ (product_snd @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (product_Pair @ labeled_graph @ b @ nat @ labeled_graph @ b @ nat @ (translation @ b @ (product_fst @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v)))) @ (translation @ b @ (product_snd @ allegorical_term @ b @ allegorical_term @ b @ (product_Pair @ allegorical_term @ b @ allegorical_term @ b @ u @ (allegorical_A_Int @ b @ u @ v))))))))),
% 3.10/1.77 file('/export/starexec/sandbox2/tmp/tmp.pyPejbcydM/DTF2THF_32651.p',conj_0)).
% 3.10/1.77 % SZS output end Proof for DTF2THF_32651
% 3.10/1.77 % (406)------------------------------
% 3.10/1.77 % (406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.77 % (406)Termination reason: Refutation
% 3.10/1.77
% 3.10/1.77 % (406)Memory used [KB]: 7803
% 3.10/1.77 % (406)Time elapsed: 0.071 s
% 3.10/1.77 % (406)Instructions burned: 182 (million)
% 3.10/1.77 % (406)------------------------------
% 3.10/1.77 % (406)------------------------------
% 3.10/1.77 % (368)Success in time 0.152 s
%------------------------------------------------------------------------------