%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP180^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n028.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 08:56:12 AM UTC 2026
% Result : Theorem 3.08s 1.74s
% Output : Refutation 3.08s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP180^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n028.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:29:12 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : ITP180^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.47 % Domain : Interactive Theorem Proving
% 0.30/0.47 % Problem : Sledgehammer StandardRules problem prob_651__5393624_1
% 0.30/0.47 % Version : Especial.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Des21]
% 0.30/0.47 % Names : StandardRules/prob_651__5393624_1 [Des21]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.33 v8.1.0, 0.75 v7.5.0
% 0.30/0.47 % Syntax : Number of formulae : 343 ( 144 unt; 64 typ; 0 def)
% 0.30/0.47 % Number of atoms : 623 ( 280 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 6 ( 2 avg)
% 0.30/0.47 % Number of connectives : 5623 ( 63 ~; 11 |; 40 &;5289 @)
% 0.30/0.47 % ( 0 <=>; 220 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 28 ( 8 avg)
% 0.30/0.47 % Number of types : 4 ( 3 usr)
% 0.30/0.47 % Number of type conns : 173 ( 173 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 62 ( 61 usr; 6 con; 0-8 aty)
% 0.30/0.47 % Number of variables : 1204 ( 46 ^;1064 !; 12 ?;1204 :)
% 0.30/0.47 % ( 82 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.47 % SPC : TH1_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:22.396
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % Could-be-implicit typings (9)
% 0.30/0.47 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 0.30/0.47 standard_Constant: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 0.30/0.47 allegorical_term: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 0.30/0.47 labeled_graph: $tType > $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.47 product_prod: $tType > $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Set_Oset,type,
% 0.30/0.47 set: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Nat_Onat,type,
% 0.30/0.47 nat: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_itself,type,
% 0.30/0.47 itself: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_tf_V_H,type,
% 0.30/0.47 v: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_tf_V,type,
% 0.30/0.47 v2: $tType ).
% 0.30/0.47
% 0.30/0.47 % Explicit typings (55)
% 0.30/0.47 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.30/0.47 bounded_lattice:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_HOL_Otype,type,
% 0.30/0.47 type:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.47 bot:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Lattices_Olattice,type,
% 0.30/0.47 lattice:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.47 preorder:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.30/0.47 semilattice_sup:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.30/0.47 bounde1808546759up_bot:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Def_OGr,type,
% 0.30/0.47 bNF_Gr:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Fun_Ocomp,type,
% 0.30/0.47 comp:
% 0.30/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Bot,type,
% 0.30/0.47 standard_S_Bot:
% 0.30/0.47 !>[V: $tType] : ( standard_Constant @ V ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Const,type,
% 0.30/0.47 standard_S_Const:
% 0.30/0.47 !>[V: $tType] : ( V > ( standard_Constant @ V ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Top,type,
% 0.30/0.47 standard_S_Top:
% 0.30/0.47 !>[V: $tType] : ( standard_Constant @ V ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 0.30/0.47 getRel:
% 0.30/0.47 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 0.30/0.47 semantics:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Oedge__preserving,type,
% 0.30/0.47 edge_preserving:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) > ( set @ ( product_prod @ C @ ( product_prod @ B @ B ) ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 0.30/0.47 graph_homomorphism:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 0.30/0.47 graph_union:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 0.30/0.47 labeled_LG:
% 0.30/0.47 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 0.30/0.47 labele1974067554_graph:
% 0.30/0.47 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 0.30/0.47 labeled_edges:
% 0.30/0.47 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 0.30/0.47 labeled_vertices:
% 0.30/0.47 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Omap__graph,type,
% 0.30/0.47 map_graph:
% 0.30/0.47 !>[C: $tType,B: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ B ) ) > ( labeled_graph @ A @ C ) > ( labeled_graph @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_LabeledGraphs_Orestrict,type,
% 0.30/0.47 restrict:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.30/0.47 sup_sup:
% 0.30/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.47 bot_bot:
% 0.30/0.47 !>[A: $tType] : A ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.47 ord_less_eq:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.47 product_Pair:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.47 product_rec_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Relation_OField,type,
% 0.30/0.47 field:
% 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Relation_OId__on,type,
% 0.30/0.47 id_on:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Relation_Ototal__on,type,
% 0.30/0.47 total_on:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_RulesAndChains_Oagree__on,type,
% 0.30/0.47 agree_on:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 0.30/0.47 consequence_graph:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_RulesAndChains_Oextensible,type,
% 0.30/0.47 extensible:
% 0.30/0.47 !>[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 ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_RulesAndChains_Oleast,type,
% 0.30/0.47 least:
% 0.30/0.47 !>[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 ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 0.30/0.47 least_559130134_graph:
% 0.30/0.47 !>[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 ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_RulesAndChains_Omaintained,type,
% 0.30/0.47 maintained:
% 0.30/0.47 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_OCollect,type,
% 0.30/0.47 collect:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Oinsert,type,
% 0.30/0.47 insert:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.30/0.47 is_singleton:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.30/0.47 the_elem:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules,type,
% 0.30/0.47 standa1138209853_rules:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
% 0.30/0.47 standa2002409347_rules:
% 0.30/0.47 !>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 0.30/0.47 standa244753842y_rule:
% 0.30/0.47 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ostandard__rules,type,
% 0.30/0.47 standa438229444_rules:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule,type,
% 0.30/0.47 standa400804411p_rule:
% 0.30/0.47 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_member,type,
% 0.30/0.47 member:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_C,type,
% 0.30/0.47 c: set @ v2 ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_G_H,type,
% 0.30/0.47 g: labeled_graph @ ( standard_Constant @ v2 ) @ v ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_L,type,
% 0.30/0.47 l: set @ ( standard_Constant @ v2 ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_h____,type,
% 0.30/0.47 h: v > v ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_m____,type,
% 0.30/0.47 m: v2 > v ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_x____,type,
% 0.30/0.47 x: v ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_xa____,type,
% 0.30/0.47 xa: v2 ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_y____,type,
% 0.30/0.47 y: v2 ).
% 0.30/0.47
% 0.30/0.47 % Relevant facts (256)
% 0.30/0.47 thf(fact_0_x,axiom,
% 0.30/0.47 member @ v @ x @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) ).
% 0.30/0.47
% 0.30/0.47 % x
% 0.30/0.47 thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.47 ~ ! [X2: v] :
% 0.30/0.47 ~ ( member @ v @ X2 @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>thesis. (\<And>x. x \<in> vertices G' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.47 thf(fact_2__C1_C_I3_J,axiom,
% 0.30/0.47 ( ( m @ xa )
% 0.30/0.47 = ( m @ y ) ) ).
% 0.30/0.47
% 0.30/0.47 % "1"(3)
% 0.30/0.47 thf(fact_3__C1_C_I1_J,axiom,
% 0.30/0.47 member @ v2 @ xa @ c ).
% 0.30/0.47
% 0.30/0.47 % "1"(1)
% 0.30/0.47 thf(fact_4__C1_C_I2_J,axiom,
% 0.30/0.47 member @ v2 @ y @ c ).
% 0.30/0.47
% 0.30/0.47 % "1"(2)
% 0.30/0.47 thf(fact_5__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 0.30/0.47 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ xa ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>(m x, m x) \<in> getRel (S_Const x) (map_graph_fn G' h)\<close>
% 0.30/0.47 thf(fact_6__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ay_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 0.30/0.47 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ y ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>(m x, m x) \<in> getRel (S_Const y) (map_graph_fn G' h)\<close>
% 0.30/0.47 thf(fact_7_gr_I1_J,axiom,
% 0.30/0.47 ( g
% 0.30/0.47 = ( restrict @ ( standard_Constant @ v2 ) @ v @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % gr(1)
% 0.30/0.47 thf(fact_8_h_I3_J,axiom,
% 0.30/0.47 graph_homomorphism @ ( standard_Constant @ v2 ) @ v @ v @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) @ g @ ( id_on @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % h(3)
% 0.30/0.47 thf(fact_9_Standard__Constant_Oinject,axiom,
% 0.30/0.47 ! [V: $tType,X4: V,Y4: V] :
% 0.30/0.47 ( ( ( standard_S_Const @ V @ X4 )
% 0.30/0.47 = ( standard_S_Const @ V @ Y4 ) )
% 0.30/0.47 = ( X4 = Y4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Standard_Constant.inject
% 0.30/0.47 thf(fact_10_prod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.30/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.30/0.47 = ( ( X1 = Y1 )
% 0.30/0.47 & ( X22 = Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.inject
% 0.30/0.47 thf(fact_11_old_Oprod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/0.47 = ( ( A2 = A3 )
% 0.30/0.47 & ( B2 = B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inject
% 0.30/0.47 thf(fact_12_m,axiom,
% 0.30/0.47 ! [X3: v2] :
% 0.30/0.47 ( ( member @ v2 @ X3 @ c )
% 0.30/0.47 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ X3 ) @ ( m @ X3 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X3 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % m
% 0.30/0.47 thf(fact_13_top__nonempty,axiom,
% 0.30/0.47 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ x @ x ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Top @ v2 ) @ g ) ).
% 0.30/0.47
% 0.30/0.47 % top_nonempty
% 0.30/0.47 thf(fact_14_cf,axiom,
% 0.30/0.47 ( ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Bot @ v2 ) @ g )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ v @ v ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % cf
% 0.30/0.47 thf(fact_15__092_060open_062_092_060And_062c_O_Ac_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_092_060exists_062v_O_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 0.30/0.47 ! [C2: v2] :
% 0.30/0.47 ( ( member @ v2 @ C2 @ c )
% 0.30/0.47 => ? [V2: v] : ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ V2 @ V2 ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ C2 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>c. c \<in> C \<Longrightarrow> \<exists>v. (v, v) \<in> getRel (S_Const c) (map_graph_fn G' h)\<close>
% 0.30/0.47 thf(fact_16__092_060open_062_092_060forall_062c_O_A_092_060exists_062v_O_Ac_A_092_060in_062_AC_A_092_060longrightarrow_062_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 0.30/0.47 ! [C3: v2] :
% 0.30/0.47 ? [V2: v] :
% 0.30/0.47 ( ( member @ v2 @ C3 @ c )
% 0.30/0.47 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ V2 @ V2 ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ C3 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<forall>c. \<exists>v. c \<in> C \<longrightarrow> (v, v) \<in> getRel (S_Const c) (map_graph_fn G' h)\<close>
% 0.30/0.47 thf(fact_17__092_060open_062_092_060exists_062f_O_A_092_060forall_062x_O_Ax_A_092_060in_062_AC_A_092_060longrightarrow_062_A_If_Ax_M_Af_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 0.30/0.47 ? [F: v2 > v] :
% 0.30/0.47 ! [X5: v2] :
% 0.30/0.47 ( ( member @ v2 @ X5 @ c )
% 0.30/0.47 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( F @ X5 ) @ ( F @ X5 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X5 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<exists>f. \<forall>x. x \<in> C \<longrightarrow> (f x, f x) \<in> getRel (S_Const x) (map_graph_fn G' h)\<close>
% 0.30/0.47 thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.47 ~ ! [M: v2 > v] :
% 0.30/0.47 ~ ! [X5: v2] :
% 0.30/0.47 ( ( member @ v2 @ X5 @ c )
% 0.30/0.47 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( M @ X5 ) @ ( M @ X5 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X5 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>thesis. (\<And>m. (\<And>x. x \<in> C \<Longrightarrow> (m x, m x) \<in> getRel (S_Const x) (map_graph_fn G' h)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.47 thf(fact_19_getRel__homR,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,U: C,F2: set @ ( product_prod @ A @ C ),V3: C] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Y @ U ) @ F2 )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Z @ V3 ) @ F2 )
% 0.30/0.47 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ U @ V3 ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ F2 @ G ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % getRel_homR
% 0.30/0.47 thf(fact_20_surj__pair,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.30/0.47 ? [X2: A,Y3: B] :
% 0.30/0.47 ( P
% 0.30/0.47 = ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_pair
% 0.30/0.47 thf(fact_21_h_I1_J,axiom,
% 0.30/0.47 ( ( comp @ v @ v @ v @ h @ h )
% 0.30/0.47 = h ) ).
% 0.30/0.47
% 0.30/0.47 % h(1)
% 0.30/0.47 thf(fact_22_getRel__map__fn,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,A22: A,G: labeled_graph @ B @ A,B22: A,L2: B,F2: A > C,A2: C,B2: C] :
% 0.30/0.47 ( ( member @ A @ A22 @ ( labeled_vertices @ B @ A @ G ) )
% 0.30/0.47 => ( ( member @ A @ B22 @ ( labeled_vertices @ B @ A @ G ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B22 ) @ ( getRel @ B @ A @ L2 @ G ) )
% 0.30/0.47 => ( ( ( F2 @ A22 )
% 0.30/0.47 = A2 )
% 0.30/0.47 => ( ( ( F2 @ B22 )
% 0.30/0.47 = B2 )
% 0.30/0.47 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A2 @ B2 ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % getRel_map_fn
% 0.30/0.47 thf(fact_23_getRel__hom,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,F2: A > C] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( labeled_vertices @ B @ A @ G ) )
% 0.30/0.47 => ( ( member @ A @ Z @ ( labeled_vertices @ B @ A @ G ) )
% 0.30/0.47 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % getRel_hom
% 0.30/0.47 thf(fact_24_Standard__Constant_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [V: $tType] :
% 0.30/0.47 ( ( standard_S_Top @ V )
% 0.30/0.47 != ( standard_S_Bot @ V ) ) ).
% 0.30/0.47
% 0.30/0.47 % Standard_Constant.distinct(1)
% 0.30/0.47 thf(fact_25_Standard__Constant_Odistinct_I5_J,axiom,
% 0.30/0.47 ! [V: $tType,X4: V] :
% 0.30/0.47 ( ( standard_S_Top @ V )
% 0.30/0.47 != ( standard_S_Const @ V @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Standard_Constant.distinct(5)
% 0.30/0.47 thf(fact_26_Standard__Constant_Odistinct_I9_J,axiom,
% 0.30/0.47 ! [V: $tType,X4: V] :
% 0.30/0.47 ( ( standard_S_Bot @ V )
% 0.30/0.47 != ( standard_S_Const @ V @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Standard_Constant.distinct(9)
% 0.30/0.47 thf(fact_27_getRel__dom_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 0.30/0.47 ( ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G ) )
% 0.30/0.47 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % getRel_dom(1)
% 0.30/0.47 thf(fact_28_getRel__dom_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 0.30/0.47 ( ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G ) )
% 0.30/0.47 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % getRel_dom(2)
% 0.30/0.47 thf(fact_29_getRel__subgraph,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,G2: labeled_graph @ B @ A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
% 0.30/0.47 => ( ( graph_homomorphism @ B @ A @ A @ G @ G2 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G ) ) )
% 0.30/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % getRel_subgraph
% 0.30/0.47 thf(fact_30_old_Oprod_Oinducts,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.47 => ( P2 @ Prod ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inducts
% 0.30/0.47 thf(fact_31_old_Oprod_Oexhaust,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.30/0.47 ~ ! [A4: A,B4: B] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.exhaust
% 0.30/0.47 thf(fact_32_prod__induct7,axiom,
% 0.30/0.47 ! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.30/0.47 ( ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F @ G4 ) ) ) ) ) ) )
% 0.30/0.47 => ( P2 @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct7
% 0.30/0.47 thf(fact_33_prod__induct6,axiom,
% 0.30/0.47 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.30/0.47 ( ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) )
% 0.30/0.47 => ( P2 @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct6
% 0.30/0.47 thf(fact_34_prod__induct5,axiom,
% 0.30/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.47 ( ! [A4: A,B4: B,C4: C,D2: D,E2: E] : ( P2 @ ( 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 ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.30/0.47 => ( P2 @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct5
% 0.30/0.47 thf(fact_35_prod__induct4,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.47 ( ! [A4: A,B4: B,C4: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
% 0.30/0.47 => ( P2 @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct4
% 0.30/0.47 thf(fact_36_prod__induct3,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ! [A4: A,B4: B,C4: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
% 0.30/0.47 => ( P2 @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct3
% 0.30/0.47 thf(fact_37_prod__cases7,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.30/0.47 ~ ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3,G4: G3] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F @ G4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases7
% 0.30/0.47 thf(fact_38_prod__cases6,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.30/0.47 ~ ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases6
% 0.30/0.47 thf(fact_39_prod__cases5,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.47 ~ ! [A4: A,B4: B,C4: C,D2: D,E2: E] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( 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 ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases5
% 0.30/0.47 thf(fact_40_prod__cases4,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.47 ~ ! [A4: A,B4: B,C4: C,D2: D] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases4
% 0.30/0.47 thf(fact_41_prod__cases3,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ~ ! [A4: A,B4: B,C4: C] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases3
% 0.30/0.47 thf(fact_42_Pair__inject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/0.47 => ~ ( ( A2 = A3 )
% 0.30/0.47 => ( B2 != B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_inject
% 0.30/0.47 thf(fact_43_prod__cases,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.47 => ( P2 @ P ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases
% 0.30/0.47 thf(fact_44_mg,axiom,
% 0.30/0.47 ! [B5: $tType,R: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ B5 ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ B5 )] :
% 0.30/0.47 ( ( maintained @ ( standard_Constant @ v2 ) @ B5 @ v @ R @ g )
% 0.30/0.47 => ( maintained @ ( standard_Constant @ v2 ) @ B5 @ v @ R @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mg
% 0.30/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,P2: A > $o] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.30/0.47 = ( P2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X7: A] : ( member @ A @ X7 @ A5 ) )
% 0.30/0.47 = A5 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_47_Collect__cong,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( P2 @ X2 )
% 0.30/0.47 = ( Q @ X2 ) )
% 0.30/0.47 => ( ( collect @ A @ P2 )
% 0.30/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_cong
% 0.30/0.47 thf(fact_48_ext,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F2: A > B,G5: A > B] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( F2 @ X2 )
% 0.30/0.47 = ( G5 @ X2 ) )
% 0.30/0.47 => ( F2 = G5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % ext
% 0.30/0.47 thf(fact_49_graph__homo,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,F2: B > C] :
% 0.30/0.47 ( ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) )
% 0.30/0.47 => ( graph_homomorphism @ A @ B @ C @ G @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_homo
% 0.30/0.47 thf(fact_50_map__graph__fn__id_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X8: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( map_graph @ B @ B @ A @ ( id_on @ B @ ( labeled_vertices @ A @ B @ X8 ) ) @ X8 )
% 0.30/0.47 = ( restrict @ A @ B @ X8 ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_fn_id(2)
% 0.30/0.47 thf(fact_51_map__graph__fn__graphI,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: C > B] :
% 0.30/0.47 ( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F2 ) @ G )
% 0.30/0.47 = ( restrict @ A @ B @ ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F2 ) @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_fn_graphI
% 0.30/0.47 thf(fact_52_map__graph__preserves__subgraph,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.30/0.47 => ( graph_homomorphism @ A @ C @ C @ ( map_graph @ B @ C @ A @ F2 @ A5 ) @ ( map_graph @ B @ C @ A @ F2 @ B6 ) @ ( id_on @ C @ ( labeled_vertices @ A @ C @ ( map_graph @ B @ C @ A @ F2 @ A5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_preserves_subgraph
% 0.30/0.47 thf(fact_53_subgraph__refl,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 0.30/0.47 = ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_refl
% 0.30/0.47 thf(fact_54_subgraph__restrict,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 0.30/0.47 = ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_restrict
% 0.30/0.47 thf(fact_55_graph__homomorphism__Id,axiom,
% 0.30/0.47 ! [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 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_homomorphism_Id
% 0.30/0.47 thf(fact_56_Id__on__empty,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Id_on_empty
% 0.30/0.47 thf(fact_57_Gr__empty,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F2: A > B] :
% 0.30/0.47 ( ( bNF_Gr @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F2 )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Gr_empty
% 0.30/0.47 thf(fact_58_restrict__idemp,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X6: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X6 ) )
% 0.30/0.47 = ( restrict @ A @ B @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % restrict_idemp
% 0.30/0.47 thf(fact_59_in__Gr,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X6: A,Y: B,A5: set @ A,F2: A > B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
% 0.30/0.47 = ( ( member @ A @ X6 @ A5 )
% 0.30/0.47 & ( ( F2 @ X6 )
% 0.30/0.47 = Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_Gr
% 0.30/0.47 thf(fact_60_Gr__not__in,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X6: A,F4: set @ A,F2: A > B,Y: B] :
% 0.30/0.47 ( ( ~ ( member @ A @ X6 @ F4 )
% 0.30/0.47 | ( ( F2 @ X6 )
% 0.30/0.47 != Y ) )
% 0.30/0.47 => ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y ) @ ( bNF_Gr @ A @ B @ F4 @ F2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Gr_not_in
% 0.30/0.47 thf(fact_61_Id__onI,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ A5 )
% 0.30/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Id_onI
% 0.30/0.47 thf(fact_62_vertices__restrict,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
% 0.30/0.47 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
% 0.30/0.47 = ( labeled_vertices @ B @ A @ G ) ) ).
% 0.30/0.47
% 0.30/0.47 % vertices_restrict
% 0.30/0.47 thf(fact_63_map__graph__preserves__restricted,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) )
% 0.30/0.47 => ( ( map_graph @ B @ C @ A @ F2 @ G )
% 0.30/0.47 = ( restrict @ A @ C @ ( map_graph @ B @ C @ A @ F2 @ G ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_preserves_restricted
% 0.30/0.47 thf(fact_64_map__graph__fn__comp,axiom,
% 0.30/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: D > B,G5: C > D] :
% 0.30/0.47 ( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ ( comp @ D @ B @ C @ F2 @ G5 ) ) @ G )
% 0.30/0.47 = ( map_graph @ D @ B @ A @ ( bNF_Gr @ D @ B @ ( labeled_vertices @ A @ D @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ G ) @ G5 ) @ G ) ) @ F2 ) @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ G ) @ G5 ) @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_fn_comp
% 0.30/0.47 thf(fact_65_assms_I1_J,axiom,
% 0.30/0.47 ! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
% 0.30/0.47 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( standa438229444_rules @ v2 @ c @ l ) )
% 0.30/0.47 => ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % assms(1)
% 0.30/0.47 thf(fact_66_ne,axiom,
% 0.30/0.47 maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ g ).
% 0.30/0.47
% 0.30/0.47 % ne
% 0.30/0.47 thf(fact_67__092_060open_062maintained_Anonempty__rule_AG_H_A_061_A_Ivertices_AG_H_A_092_060noteq_062_A_123_125_J_092_060close_062,axiom,
% 0.30/0.47 ( ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ g )
% 0.30/0.47 = ( ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g )
% 0.30/0.47 != ( bot_bot @ ( set @ v ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>maintained nonempty_rule G' = (vertices G' \<noteq> {})\<close>
% 0.30/0.47 thf(fact_68_maintained__preserved__by__isomorphism,axiom,
% 0.30/0.47 ! [C: $tType,D: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F2: C > A,G5: A > C,R: product_prod @ ( labeled_graph @ B @ D ) @ ( labeled_graph @ B @ D )] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ ( labeled_vertices @ B @ A @ G ) )
% 0.30/0.47 => ( ( comp @ C @ A @ A @ F2 @ G5 @ X2 )
% 0.30/0.47 = X2 ) )
% 0.30/0.47 => ( ( G
% 0.30/0.47 = ( restrict @ B @ A @ G ) )
% 0.30/0.47 => ( ( maintained @ B @ D @ C @ R @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G5 ) @ G ) )
% 0.30/0.47 => ( maintained @ B @ D @ A @ R @ G ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % maintained_preserved_by_isomorphism
% 0.30/0.47 thf(fact_69_idemp__embedding__maintained__preserved,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F2: B > B,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) ) ) )
% 0.30/0.47 => ( ! [X2: B] :
% 0.30/0.47 ( ( member @ B @ X2 @ ( labeled_vertices @ A @ B @ G ) )
% 0.30/0.47 => ( ( comp @ B @ B @ B @ F2 @ F2 @ X2 )
% 0.30/0.47 = ( F2 @ X2 ) ) )
% 0.30/0.47 => ( ( maintained @ A @ C @ B @ R @ G )
% 0.30/0.47 => ( maintained @ A @ C @ B @ R @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % idemp_embedding_maintained_preserved
% 0.30/0.47 thf(fact_70_Id__onE,axiom,
% 0.30/0.47 ! [A: $tType,C5: product_prod @ A @ A,A5: set @ A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ C5 @ ( id_on @ A @ A5 ) )
% 0.30/0.47 => ~ ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A5 )
% 0.30/0.47 => ( C5
% 0.30/0.47 != ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Id_onE
% 0.30/0.47 thf(fact_71_Id__on__eqI,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.30/0.47 ( ( A2 = B2 )
% 0.30/0.47 => ( ( member @ A @ A2 @ A5 )
% 0.30/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Id_on_eqI
% 0.30/0.47 thf(fact_72_Id__on__iff,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,Y: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Y ) @ ( id_on @ A @ A5 ) )
% 0.30/0.47 = ( ( X6 = Y )
% 0.30/0.47 & ( member @ A @ X6 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Id_on_iff
% 0.30/0.47 thf(fact_73_subgraph__preserves__hom,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,X8: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.30/0.47 => ( ( graph_homomorphism @ A @ C @ B @ X8 @ A5 @ H )
% 0.30/0.47 => ( graph_homomorphism @ A @ C @ B @ X8 @ B6 @ H ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_preserves_hom
% 0.30/0.47 thf(fact_74_subgraph__trans,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.30/0.47 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 0.30/0.47 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_trans
% 0.30/0.47 thf(fact_75_map__graph__fn__eqI,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F2: A > C,G5: A > C] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ ( labeled_vertices @ B @ A @ G ) )
% 0.30/0.47 => ( ( F2 @ X2 )
% 0.30/0.47 = ( G5 @ X2 ) ) )
% 0.30/0.47 => ( ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G )
% 0.30/0.47 = ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G5 ) @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_fn_eqI
% 0.30/0.47 thf(fact_76_tr,axiom,
% 0.30/0.47 maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa400804411p_rule @ ( standard_Constant @ v2 ) @ ( standard_S_Top @ v2 ) ) @ g ).
% 0.30/0.47
% 0.30/0.47 % tr
% 0.30/0.47 thf(fact_77_h_I4_J,axiom,
% 0.30/0.47 ! [L2: standard_Constant @ v2,X6: v,Y: v] :
% 0.30/0.47 ( ( member @ ( product_prod @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) ) @ ( product_Pair @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) @ L2 @ ( product_Pair @ v @ v @ X6 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ v2 ) @ v @ g ) )
% 0.30/0.47 = ( member @ ( product_prod @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) ) @ ( product_Pair @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) @ L2 @ ( product_Pair @ v @ v @ ( h @ X6 ) @ ( h @ Y ) ) ) @ ( labeled_edges @ ( standard_Constant @ v2 ) @ v @ g ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % h(4)
% 0.30/0.47 thf(fact_78_old_Oprod_Orec,axiom,
% 0.30/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.30/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.rec
% 0.30/0.47 thf(fact_79_maintained__refl,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G ) ).
% 0.30/0.47
% 0.30/0.47 % maintained_refl
% 0.30/0.47 thf(fact_80_comp__apply,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.47 ( ( comp @ B @ A @ C )
% 0.30/0.47 = ( ^ [F5: B > A,G6: C > B,X7: C] : ( F5 @ ( G6 @ X7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % comp_apply
% 0.30/0.47 thf(fact_81_empty__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ P2 ) )
% 0.30/0.47 = ( ! [X7: A] :
% 0.30/0.47 ~ ( P2 @ X7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_Collect_eq
% 0.30/0.47 thf(fact_82_Collect__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( collect @ A @ P2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ! [X7: A] :
% 0.30/0.47 ~ ( P2 @ X7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_empty_eq
% 0.30/0.47 thf(fact_83_empty__iff,axiom,
% 0.30/0.47 ! [A: $tType,C5: A] :
% 0.30/0.47 ~ ( member @ A @ C5 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_iff
% 0.30/0.47 thf(fact_84_all__not__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( ! [X7: A] :
% 0.30/0.47 ~ ( member @ A @ X7 @ A5 ) )
% 0.30/0.47 = ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % all_not_in_conv
% 0.30/0.47 thf(fact_85_nonempty__rule,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) )
% 0.30/0.47 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
% 0.30/0.47 = ( ( labeled_vertices @ A @ B @ G )
% 0.30/0.47 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nonempty_rule
% 0.30/0.47 thf(fact_86__092_060open_062maintainedA_A_Iidentity__rules_AL_J_AG_H_092_060close_062,axiom,
% 0.30/0.47 ! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
% 0.30/0.47 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( standa2002409347_rules @ v2 @ l ) )
% 0.30/0.47 => ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>maintainedA (identity_rules L) G'\<close>
% 0.30/0.47 thf(fact_87_labeled__graph_Oexpand,axiom,
% 0.30/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 0.30/0.47 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 0.30/0.47 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 0.30/0.47 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 0.30/0.47 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 0.30/0.47 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.expand
% 0.30/0.47 thf(fact_88_emptyE,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] :
% 0.30/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % emptyE
% 0.30/0.47 thf(fact_89_equals0D,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,A2: A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 => ~ ( member @ A @ A2 @ A5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0D
% 0.30/0.47 thf(fact_90_equals0I,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ! [Y3: A] :
% 0.30/0.47 ~ ( member @ A @ Y3 @ A5 )
% 0.30/0.47 => ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0I
% 0.30/0.47 thf(fact_91_ex__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( ? [X7: A] : ( member @ A @ X7 @ A5 ) )
% 0.30/0.47 = ( A5
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_in_conv
% 0.30/0.47 thf(fact_92_comp__def,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.47 ( ( comp @ B @ C @ A )
% 0.30/0.47 = ( ^ [F5: B > C,G6: A > B,X7: A] : ( F5 @ ( G6 @ X7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % comp_def
% 0.30/0.47 thf(fact_93_comp__assoc,axiom,
% 0.30/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G5: C > D,H: A > C] :
% 0.30/0.47 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G5 ) @ H )
% 0.30/0.47 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G5 @ H ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % comp_assoc
% 0.30/0.47 thf(fact_94_comp__eq__dest,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C5: D > B,D3: A > D,V3: A] :
% 0.30/0.47 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.30/0.47 = ( comp @ D @ B @ A @ C5 @ D3 ) )
% 0.30/0.47 => ( ( A2 @ ( B2 @ V3 ) )
% 0.30/0.47 = ( C5 @ ( D3 @ V3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % comp_eq_dest
% 0.30/0.47 thf(fact_95_comp__eq__elim,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C5: D > B,D3: A > D] :
% 0.30/0.47 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.30/0.47 = ( comp @ D @ B @ A @ C5 @ D3 ) )
% 0.30/0.47 => ! [V4: A] :
% 0.30/0.47 ( ( A2 @ ( B2 @ V4 ) )
% 0.30/0.47 = ( C5 @ ( D3 @ V4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % comp_eq_elim
% 0.30/0.47 thf(fact_96_comp__eq__dest__lhs,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C5: A > B,V3: A] :
% 0.30/0.47 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.30/0.47 = C5 )
% 0.30/0.47 => ( ( A2 @ ( B2 @ V3 ) )
% 0.30/0.47 = ( C5 @ V3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % comp_eq_dest_lhs
% 0.30/0.47 thf(fact_97_edge__preserving__on__graphI,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,X8: labeled_graph @ A @ B,F2: B > C,Y5: set @ ( product_prod @ A @ ( product_prod @ C @ C ) )] :
% 0.30/0.47 ( ! [L3: A,X2: B,Y3: B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L3 @ ( product_Pair @ B @ B @ X2 @ Y3 ) ) @ ( labeled_edges @ A @ B @ X8 ) )
% 0.30/0.47 => ( ( member @ B @ X2 @ ( labeled_vertices @ A @ B @ X8 ) )
% 0.30/0.47 => ( ( member @ B @ Y3 @ ( labeled_vertices @ A @ B @ X8 ) )
% 0.30/0.47 => ( member @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L3 @ ( product_Pair @ C @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
% 0.30/0.47 => ( edge_preserving @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ X8 ) @ F2 ) @ ( labeled_edges @ A @ B @ X8 ) @ Y5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % edge_preserving_on_graphI
% 0.30/0.47 thf(fact_98_bot__apply,axiom,
% 0.30/0.47 ! [C: $tType,D: $tType] :
% 0.30/0.47 ( ( bot @ C )
% 0.30/0.47 => ( ( bot_bot @ ( D > C ) )
% 0.30/0.47 = ( ^ [X7: D] : ( bot_bot @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_apply
% 0.30/0.47 thf(fact_99_least__consequence__graphI,axiom,
% 0.30/0.47 ! [X: $tType,A: $tType,C: $tType,L: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G: labeled_graph @ L @ C,S: labeled_graph @ L @ C,T2: itself @ X] :
% 0.30/0.47 ( ( consequence_graph @ L @ A @ C @ Rs @ G )
% 0.30/0.47 => ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 0.30/0.47 => ( ! [C6: labeled_graph @ L @ X] :
% 0.30/0.47 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 0.30/0.47 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C6 ) )
% 0.30/0.47 => ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % least_consequence_graphI
% 0.30/0.47 thf(fact_100_GrD2,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X6: A,Fx: B,A5: set @ A,F2: A > B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
% 0.30/0.47 => ( ( F2 @ X6 )
% 0.30/0.47 = Fx ) ) ).
% 0.30/0.47
% 0.30/0.47 % GrD2
% 0.30/0.47 thf(fact_101_GrD1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X6: A,Fx: B,A5: set @ A,F2: A > B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
% 0.30/0.47 => ( member @ A @ X6 @ A5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % GrD1
% 0.30/0.47 thf(fact_102_leastI,axiom,
% 0.30/0.47 ! [X: $tType,A: $tType,C: $tType,L: $tType,S: labeled_graph @ L @ C,G: labeled_graph @ L @ C,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T2: itself @ X] :
% 0.30/0.47 ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 0.30/0.47 => ( ! [C6: labeled_graph @ L @ X] :
% 0.30/0.47 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 0.30/0.47 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C6 ) )
% 0.30/0.47 => ( least @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % leastI
% 0.30/0.47 thf(fact_103_edge__preserving__Id,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: set @ A,X6: set @ ( product_prod @ B @ ( product_prod @ A @ A ) )] : ( edge_preserving @ A @ A @ B @ ( id_on @ A @ Y ) @ X6 @ X6 ) ).
% 0.30/0.47
% 0.30/0.47 % edge_preserving_Id
% 0.30/0.47 thf(fact_104_map__graph__edge__preserving,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,F2: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] : ( edge_preserving @ A @ B @ C @ F2 @ ( labeled_edges @ C @ A @ G ) @ ( labeled_edges @ C @ B @ ( map_graph @ A @ B @ C @ F2 @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_edge_preserving
% 0.30/0.47 thf(fact_105_least__consequence__graph__def,axiom,
% 0.30/0.47 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.30/0.47 ( ( least_559130134_graph @ X @ L @ V @ C )
% 0.30/0.47 = ( ^ [T3: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 0.30/0.47 ( ( consequence_graph @ L @ V @ C @ Rs2 @ G7 )
% 0.30/0.47 & ( least @ X @ L @ V @ C @ T3 @ Rs2 @ S2 @ G7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % least_consequence_graph_def
% 0.30/0.47 thf(fact_106_bot__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.30/0.47 = ( ^ [X7: A] : ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_empty_eq
% 0.30/0.47 thf(fact_107_bot__set__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_set_def
% 0.30/0.47 thf(fact_108_consequence__graphD_I3_J,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 0.30/0.47 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 0.30/0.47 => ( G
% 0.30/0.47 = ( restrict @ A @ C @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % consequence_graphD(3)
% 0.30/0.47 thf(fact_109_consequence__graphD_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R2: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.30/0.47 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 0.30/0.47 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
% 0.30/0.47 => ( maintained @ A @ B @ C @ R2 @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % consequence_graphD(1)
% 0.30/0.47 thf(fact_110_edge__preserving__atomic,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E22: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),V1: A,V12: B,V22: A,V23: B,K: C] :
% 0.30/0.47 ( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E22 )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V1 @ V12 ) @ H1 )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V22 @ V23 ) @ H1 )
% 0.30/0.47 => ( ( member @ ( product_prod @ C @ ( product_prod @ A @ A ) ) @ ( product_Pair @ C @ ( product_prod @ A @ A ) @ K @ ( product_Pair @ A @ A @ V1 @ V22 ) ) @ E1 )
% 0.30/0.47 => ( member @ ( product_prod @ C @ ( product_prod @ B @ B ) ) @ ( product_Pair @ C @ ( product_prod @ B @ B ) @ K @ ( product_Pair @ B @ B @ V12 @ V23 ) ) @ E22 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % edge_preserving_atomic
% 0.30/0.47 thf(fact_111_bot__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( bot @ B )
% 0.30/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.47 = ( ^ [X7: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_fun_def
% 0.30/0.47 thf(fact_112_least__def,axiom,
% 0.30/0.47 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.30/0.47 ( ( least @ X @ L @ V @ C )
% 0.30/0.47 = ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 0.30/0.47 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G7 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 0.30/0.47 & ! [C7: labeled_graph @ L @ X] :
% 0.30/0.47 ( ( consequence_graph @ L @ V @ X @ Rs2 @ C7 )
% 0.30/0.47 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G7 ) @ C7 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % least_def
% 0.30/0.47 thf(fact_113_mnt,axiom,
% 0.30/0.47 ! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
% 0.30/0.47 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) @ ( standa1138209853_rules @ v2 @ c ) @ ( standa2002409347_rules @ v2 @ l ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ ( standa400804411p_rule @ ( standard_Constant @ v2 ) @ ( standard_S_Top @ v2 ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) ) ) ) ) )
% 0.30/0.47 => ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% 0.30/0.47
% 0.30/0.47 % mnt
% 0.30/0.47 thf(fact_114_map__graph__in,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A,F2: B > C] :
% 0.30/0.47 ( ( G
% 0.30/0.47 = ( restrict @ A @ B @ G ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ G @ E3 ) )
% 0.30/0.47 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) @ ( semantics @ A @ C @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ E3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_graph_in
% 0.30/0.47 thf(fact_115_maintainedI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B6: labeled_graph @ C @ A] :
% 0.30/0.47 ( ! [F: set @ ( product_prod @ A @ B )] :
% 0.30/0.47 ( ( graph_homomorphism @ C @ A @ B @ A5 @ G @ F )
% 0.30/0.47 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B6 ) @ G @ F ) )
% 0.30/0.47 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B6 ) @ G ) ) ).
% 0.30/0.47
% 0.30/0.47 % maintainedI
% 0.30/0.47 thf(fact_116_graph__homomorphism__empty,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F2 )
% 0.30/0.47 = ( ( F2
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 0.30/0.47 & ( G
% 0.30/0.47 = ( restrict @ A @ C @ G ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_homomorphism_empty
% 0.30/0.47 thf(fact_117_subgraph__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.30/0.47 = ( ( G_1
% 0.30/0.47 = ( restrict @ A @ B @ G_1 ) )
% 0.30/0.47 & ( G_2
% 0.30/0.47 = ( restrict @ A @ B @ G_2 ) )
% 0.30/0.47 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.30/0.47 = G_2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_def
% 0.30/0.47 thf(fact_118_insert__absorb2,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,A5: set @ A] :
% 0.30/0.47 ( ( insert @ A @ X6 @ ( insert @ A @ X6 @ A5 ) )
% 0.30/0.47 = ( insert @ A @ X6 @ A5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_absorb2
% 0.30/0.47 thf(fact_119_insert__iff,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 0.30/0.47 = ( ( A2 = B2 )
% 0.30/0.47 | ( member @ A @ A2 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_iff
% 0.30/0.47 thf(fact_120_insertCI,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B6: set @ A,B2: A] :
% 0.30/0.47 ( ( ~ ( member @ A @ A2 @ B6 )
% 0.30/0.47 => ( A2 = B2 ) )
% 0.30/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertCI
% 0.30/0.47 thf(fact_121_Un__iff,axiom,
% 0.30/0.47 ! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 0.30/0.47 = ( ( member @ A @ C5 @ A5 )
% 0.30/0.47 | ( member @ A @ C5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_iff
% 0.30/0.47 thf(fact_122_UnCI,axiom,
% 0.30/0.47 ! [A: $tType,C5: A,B6: set @ A,A5: set @ A] :
% 0.30/0.47 ( ( ~ ( member @ A @ C5 @ B6 )
% 0.30/0.47 => ( member @ A @ C5 @ A5 ) )
% 0.30/0.47 => ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnCI
% 0.30/0.47 thf(fact_123_labeled__graph_Oinject,axiom,
% 0.30/0.47 ! [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] :
% 0.30/0.47 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 0.30/0.47 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 0.30/0.47 = ( ( X1 = Y1 )
% 0.30/0.47 & ( X22 = Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.inject
% 0.30/0.47 thf(fact_124_graph__union__idemp_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_union @ A @ B @ A5 @ A5 )
% 0.30/0.47 = A5 ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_idemp(1)
% 0.30/0.47 thf(fact_125_graph__union__idemp_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B6 ) )
% 0.30/0.47 = ( graph_union @ A @ B @ A5 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_idemp(2)
% 0.30/0.47 thf(fact_126_graph__union__idemp_I3_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B6 @ A5 ) )
% 0.30/0.47 = ( graph_union @ A @ B @ B6 @ A5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_idemp(3)
% 0.30/0.47 thf(fact_127_singletonI,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singletonI
% 0.30/0.47 thf(fact_128_Un__empty,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_empty
% 0.30/0.47 thf(fact_129_Un__insert__right,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,A2: A,B6: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) )
% 0.30/0.47 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_insert_right
% 0.30/0.47 thf(fact_130_Un__insert__left,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B6: set @ A,C8: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B6 ) @ C8 )
% 0.30/0.47 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_insert_left
% 0.30/0.47 thf(fact_131_graph__union__preserves__restrict,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( G_1
% 0.30/0.47 = ( restrict @ A @ B @ G_1 ) )
% 0.30/0.47 => ( ( G_2
% 0.30/0.47 = ( restrict @ A @ B @ G_2 ) )
% 0.30/0.47 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.30/0.47 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_preserves_restrict
% 0.30/0.47 thf(fact_132_graph__union__vertices,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,G_1: labeled_graph @ B @ A,G_2: labeled_graph @ B @ A] :
% 0.30/0.47 ( ( labeled_vertices @ B @ A @ ( graph_union @ B @ A @ G_1 @ G_2 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ ( labeled_vertices @ B @ A @ G_1 ) @ ( labeled_vertices @ B @ A @ G_2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_vertices
% 0.30/0.47 thf(fact_133_labeled__graph_Ocollapse,axiom,
% 0.30/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.30/0.47 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 0.30/0.47 = Labeled_graph ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.collapse
% 0.30/0.47 thf(fact_134_extensible__refl,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ C @ R2 @ G @ F2 )
% 0.30/0.47 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G @ F2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % extensible_refl
% 0.30/0.47 thf(fact_135_graph__empty__e,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,V3: set @ B] :
% 0.30/0.47 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
% 0.30/0.47 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_empty_e
% 0.30/0.47 thf(fact_136_constant__rules__empty,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( standa1138209853_rules @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % constant_rules_empty
% 0.30/0.47 thf(fact_137_Gr__insert,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X6: A,F4: set @ A,F2: A > B] :
% 0.30/0.47 ( ( bNF_Gr @ A @ B @ ( insert @ A @ X6 @ F4 ) @ F2 )
% 0.30/0.47 = ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ ( F2 @ X6 ) ) @ ( bNF_Gr @ A @ B @ F4 @ F2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Gr_insert
% 0.30/0.47 thf(fact_138_extensible__refl__concr,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V3: set @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ G @ F2 )
% 0.30/0.47 => ( ( 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 @ F2 )
% 0.30/0.47 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V3 ) @ G @ F2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % extensible_refl_concr
% 0.30/0.47 thf(fact_139_graph__single,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,C5: B] :
% 0.30/0.47 ( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C5 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C5 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.30/0.47 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C5 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C5 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_single
% 0.30/0.47 thf(fact_140_insert__is__Un,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( insert @ A )
% 0.30/0.47 = ( ^ [A6: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_is_Un
% 0.30/0.47 thf(fact_141_Un__singleton__iff,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,X6: A] :
% 0.30/0.47 ( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( ( ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.47 | ( ( A5
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 | ( ( A5
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_singleton_iff
% 0.30/0.47 thf(fact_142_singleton__Un__iff,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 0.30/0.47 = ( ( ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.47 | ( ( A5
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 | ( ( A5
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_Un_iff
% 0.30/0.47 thf(fact_143_graph__union__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( graph_union @ A @ B )
% 0.30/0.47 = ( ^ [G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B] : ( labeled_LG @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_12 ) @ ( labeled_edges @ A @ B @ G_22 ) ) @ ( sup_sup @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_12 ) @ ( labeled_vertices @ A @ B @ G_22 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_def
% 0.30/0.47 thf(fact_144_Un__empty__left,axiom,
% 0.30/0.47 ! [A: $tType,B6: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 0.30/0.47 = B6 ) ).
% 0.30/0.47
% 0.30/0.47 % Un_empty_left
% 0.30/0.47 thf(fact_145_Un__empty__right,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = A5 ) ).
% 0.30/0.47
% 0.30/0.47 % Un_empty_right
% 0.30/0.47 thf(fact_146_singletonD,axiom,
% 0.30/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.30/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 => ( B2 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % singletonD
% 0.30/0.47 thf(fact_147_singleton__iff,axiom,
% 0.30/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.30/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( B2 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_iff
% 0.30/0.47 thf(fact_148_doubleton__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B2: A,C5: A,D3: A] :
% 0.30/0.47 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( insert @ A @ C5 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.47 = ( ( ( A2 = C5 )
% 0.30/0.47 & ( B2 = D3 ) )
% 0.30/0.47 | ( ( A2 = D3 )
% 0.30/0.47 & ( B2 = C5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % doubleton_eq_iff
% 0.30/0.47 thf(fact_149_insert__not__empty,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.30/0.47 ( ( insert @ A @ A2 @ A5 )
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_not_empty
% 0.30/0.47 thf(fact_150_singleton__inject,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B2: A] :
% 0.30/0.47 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 => ( A2 = B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_inject
% 0.30/0.47 thf(fact_151_labeled__graph_Oexhaust,axiom,
% 0.30/0.47 ! [L: $tType,V: $tType,Y: labeled_graph @ L @ V] :
% 0.30/0.47 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.exhaust
% 0.30/0.47 thf(fact_152_labeled__graph_Oinduct,axiom,
% 0.30/0.47 ! [V: $tType,L: $tType,P2: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 0.30/0.47 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P2 @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 0.30/0.47 => ( P2 @ Labeled_graph ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.induct
% 0.30/0.47 thf(fact_153_mk__disjoint__insert,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ A5 )
% 0.30/0.47 => ? [B7: set @ A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( insert @ A @ A2 @ B7 ) )
% 0.30/0.47 & ~ ( member @ A @ A2 @ B7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mk_disjoint_insert
% 0.30/0.47 thf(fact_154_Un__left__commute,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A5 @ C8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_left_commute
% 0.30/0.47 thf(fact_155_insert__commute,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,Y: A,A5: set @ A] :
% 0.30/0.47 ( ( insert @ A @ X6 @ ( insert @ A @ Y @ A5 ) )
% 0.30/0.47 = ( insert @ A @ Y @ ( insert @ A @ X6 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_commute
% 0.30/0.47 thf(fact_156_Un__left__absorb,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_left_absorb
% 0.30/0.47 thf(fact_157_insert__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A5: set @ A,B2: A,B6: set @ A] :
% 0.30/0.47 ( ~ ( member @ A @ A2 @ A5 )
% 0.30/0.47 => ( ~ ( member @ A @ B2 @ B6 )
% 0.30/0.47 => ( ( ( insert @ A @ A2 @ A5 )
% 0.30/0.47 = ( insert @ A @ B2 @ B6 ) )
% 0.30/0.47 = ( ( ( A2 = B2 )
% 0.30/0.47 => ( A5 = B6 ) )
% 0.30/0.47 & ( ( A2 != B2 )
% 0.30/0.47 => ? [C7: set @ A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( insert @ A @ B2 @ C7 ) )
% 0.30/0.47 & ~ ( member @ A @ B2 @ C7 )
% 0.30/0.47 & ( B6
% 0.30/0.47 = ( insert @ A @ A2 @ C7 ) )
% 0.30/0.47 & ~ ( member @ A @ A2 @ C7 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_eq_iff
% 0.30/0.47 thf(fact_158_insert__absorb,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ A5 )
% 0.30/0.47 => ( ( insert @ A @ A2 @ A5 )
% 0.30/0.47 = A5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_absorb
% 0.30/0.47 thf(fact_159_insert__ident,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ~ ( member @ A @ X6 @ A5 )
% 0.30/0.47 => ( ~ ( member @ A @ X6 @ B6 )
% 0.30/0.47 => ( ( ( insert @ A @ X6 @ A5 )
% 0.30/0.47 = ( insert @ A @ X6 @ B6 ) )
% 0.30/0.47 = ( A5 = B6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_ident
% 0.30/0.47 thf(fact_160_Set_Oset__insert,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ X6 @ A5 )
% 0.30/0.47 => ~ ! [B7: set @ A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( insert @ A @ X6 @ B7 ) )
% 0.30/0.47 => ( member @ A @ X6 @ B7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Set.set_insert
% 0.30/0.47 thf(fact_161_Un__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B8: set @ A] : ( sup_sup @ ( set @ A ) @ B8 @ A7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_commute
% 0.30/0.47 thf(fact_162_Un__absorb,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
% 0.30/0.47 = A5 ) ).
% 0.30/0.47
% 0.30/0.47 % Un_absorb
% 0.30/0.47 thf(fact_163_insertI2,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B6: set @ A,B2: A] :
% 0.30/0.47 ( ( member @ A @ A2 @ B6 )
% 0.30/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertI2
% 0.30/0.47 thf(fact_164_insertI1,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B6: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertI1
% 0.30/0.47 thf(fact_165_Un__assoc,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) @ C8 )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_assoc
% 0.30/0.47 thf(fact_166_insertE,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 0.30/0.47 => ( ( A2 != B2 )
% 0.30/0.47 => ( member @ A @ A2 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertE
% 0.30/0.47 thf(fact_167_ball__Un,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,P2: A > $o] :
% 0.30/0.47 ( ( ! [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 0.30/0.47 => ( P2 @ X7 ) ) )
% 0.30/0.47 = ( ! [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ A5 )
% 0.30/0.47 => ( P2 @ X7 ) )
% 0.30/0.47 & ! [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ B6 )
% 0.30/0.47 => ( P2 @ X7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_Un
% 0.30/0.47 thf(fact_168_bex__Un,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,P2: A > $o] :
% 0.30/0.47 ( ( ? [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 0.30/0.47 & ( P2 @ X7 ) ) )
% 0.30/0.47 = ( ? [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ A5 )
% 0.30/0.47 & ( P2 @ X7 ) )
% 0.30/0.47 | ? [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ B6 )
% 0.30/0.47 & ( P2 @ X7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_Un
% 0.30/0.47 thf(fact_169_UnI2,axiom,
% 0.30/0.47 ! [A: $tType,C5: A,B6: set @ A,A5: set @ A] :
% 0.30/0.47 ( ( member @ A @ C5 @ B6 )
% 0.30/0.47 => ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnI2
% 0.30/0.47 thf(fact_170_UnI1,axiom,
% 0.30/0.47 ! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( member @ A @ C5 @ A5 )
% 0.30/0.47 => ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnI1
% 0.30/0.47 thf(fact_171_UnE,axiom,
% 0.30/0.47 ! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 0.30/0.47 => ( ~ ( member @ A @ C5 @ A5 )
% 0.30/0.47 => ( member @ A @ C5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnE
% 0.30/0.47 thf(fact_172_labeled__graph_Osel_I2_J,axiom,
% 0.30/0.47 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.30/0.47 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.30/0.47 = X22 ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.sel(2)
% 0.30/0.47 thf(fact_173_labeled__graph_Osel_I1_J,axiom,
% 0.30/0.47 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.30/0.47 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.30/0.47 = X1 ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.sel(1)
% 0.30/0.47 thf(fact_174_graph__homomorphism__semantics,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B6 @ F2 )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F2 )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F2 )
% 0.30/0.47 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B6 @ E3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_homomorphism_semantics
% 0.30/0.47 thf(fact_175_graph__homomorphism__nonempty,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B6 @ F2 )
% 0.30/0.47 => ( ( ( semantics @ A @ B @ A5 @ E3 )
% 0.30/0.47 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.30/0.47 => ( ( semantics @ A @ C @ B6 @ E3 )
% 0.30/0.47 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_homomorphism_nonempty
% 0.30/0.47 thf(fact_176_labeled__graph_Oexhaust__sel,axiom,
% 0.30/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.30/0.47 ( Labeled_graph
% 0.30/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.exhaust_sel
% 0.30/0.47 thf(fact_177_standard__rules__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( standa438229444_rules @ A )
% 0.30/0.47 = ( ^ [C7: set @ A,L4: set @ ( standard_Constant @ A )] : ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) @ ( standa1138209853_rules @ A @ C7 ) @ ( standa2002409347_rules @ A @ L4 ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa400804411p_rule @ ( standard_Constant @ A ) @ ( standard_S_Top @ A ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa244753842y_rule @ ( standard_Constant @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % standard_rules_def
% 0.30/0.47 thf(fact_178_semantics__in__vertices_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( restrict @ A @ B @ A5 ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.30/0.47 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semantics_in_vertices(2)
% 0.30/0.47 thf(fact_179_semantics__in__vertices_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( restrict @ A @ B @ A5 ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.30/0.47 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semantics_in_vertices(1)
% 0.30/0.47 thf(fact_180_subgraph__semantics,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.30/0.47 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B6 @ E3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_semantics
% 0.30/0.47 thf(fact_181_maintainedD,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B6 ) @ G )
% 0.30/0.47 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F2 )
% 0.30/0.47 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B6 ) @ G @ F2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % maintainedD
% 0.30/0.47 thf(fact_182_sup__bot_Oright__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.right_neutral
% 0.30/0.47 thf(fact_183_sup__bot_Oneutr__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ( bot_bot @ A )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ B2 ) )
% 0.30/0.47 = ( ( A2
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 & ( B2
% 0.30/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.neutr_eq_iff
% 0.30/0.47 thf(fact_184_sup__bot_Oleft__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.left_neutral
% 0.30/0.47 thf(fact_185_sup__bot_Oeq__neutr__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ( sup_sup @ A @ A2 @ B2 )
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 = ( ( A2
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 & ( B2
% 0.30/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.eq_neutr_iff
% 0.30/0.47 thf(fact_186_sup_Oright__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.right_idem
% 0.30/0.47 thf(fact_187_sup__left__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,Y: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) )
% 0.30/0.47 = ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_left_idem
% 0.30/0.47 thf(fact_188_sup_Oleft__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.left_idem
% 0.30/0.47 thf(fact_189_sup__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X6 @ X6 )
% 0.30/0.47 = X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_idem
% 0.30/0.47 thf(fact_190_sup_Oidem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ A2 @ A2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.idem
% 0.30/0.47 thf(fact_191_sup__apply,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ B )
% 0.30/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.47 = ( ^ [F5: A > B,G6: A > B,X7: A] : ( sup_sup @ B @ ( F5 @ X7 ) @ ( G6 @ X7 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_apply
% 0.30/0.47 thf(fact_192_sup__bot__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [X6: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X6 )
% 0.30/0.47 = X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot_left
% 0.30/0.47 thf(fact_193_sup__bot__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [X6: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X6 @ ( bot_bot @ A ) )
% 0.30/0.47 = X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot_right
% 0.30/0.47 thf(fact_194_bot__eq__sup__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [X6: A,Y: A] :
% 0.30/0.47 ( ( ( bot_bot @ A )
% 0.30/0.47 = ( sup_sup @ A @ X6 @ Y ) )
% 0.30/0.47 = ( ( X6
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 & ( Y
% 0.30/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_eq_sup_iff
% 0.30/0.47 thf(fact_195_sup__eq__bot__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A )
% 0.30/0.47 => ! [X6: A,Y: A] :
% 0.30/0.47 ( ( ( sup_sup @ A @ X6 @ Y )
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 = ( ( X6
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 & ( Y
% 0.30/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_eq_bot_iff
% 0.30/0.47 thf(fact_196_edge__preserving__unionI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,F2: set @ ( product_prod @ A @ B ),A5: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),G: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),B6: set @ ( product_prod @ C @ ( product_prod @ A @ A ) )] :
% 0.30/0.47 ( ( edge_preserving @ A @ B @ C @ F2 @ A5 @ G )
% 0.30/0.47 => ( ( edge_preserving @ A @ B @ C @ F2 @ B6 @ G )
% 0.30/0.47 => ( edge_preserving @ A @ B @ C @ F2 @ ( sup_sup @ ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) @ A5 @ B6 ) @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % edge_preserving_unionI
% 0.30/0.47 thf(fact_197_graph__union__edges,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( labeled_edges @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_union_edges
% 0.30/0.47 thf(fact_198_sup__left__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) )
% 0.30/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X6 @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_left_commute
% 0.30/0.47 thf(fact_199_sup_Oleft__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C5: A] :
% 0.30/0.47 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C5 ) )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.left_commute
% 0.30/0.47 thf(fact_200_sup__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ( ( sup_sup @ A )
% 0.30/0.47 = ( ^ [X7: A,Y6: A] : ( sup_sup @ A @ Y6 @ X7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_commute
% 0.30/0.47 thf(fact_201_sup_Ocommute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ( ( sup_sup @ A )
% 0.30/0.47 = ( ^ [A6: A,B9: A] : ( sup_sup @ A @ B9 @ A6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.commute
% 0.30/0.47 thf(fact_202_sup__assoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
% 0.30/0.47 = ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_assoc
% 0.30/0.47 thf(fact_203_sup_Oassoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C5: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C5 )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.assoc
% 0.30/0.47 thf(fact_204_boolean__algebra__cancel_Osup2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [B6: A,K: A,B2: A,A2: A] :
% 0.30/0.47 ( ( B6
% 0.30/0.47 = ( sup_sup @ A @ K @ B2 ) )
% 0.30/0.47 => ( ( sup_sup @ A @ A2 @ B6 )
% 0.30/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % boolean_algebra_cancel.sup2
% 0.30/0.47 thf(fact_205_boolean__algebra__cancel_Osup1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A5: A,K: A,A2: A,B2: A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 = ( sup_sup @ A @ K @ A2 ) )
% 0.30/0.47 => ( ( sup_sup @ A @ A5 @ B2 )
% 0.30/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % boolean_algebra_cancel.sup1
% 0.30/0.47 thf(fact_206_sup__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ B )
% 0.30/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.47 = ( ^ [F5: A > B,G6: A > B,X7: A] : ( sup_sup @ B @ ( F5 @ X7 ) @ ( G6 @ X7 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_fun_def
% 0.30/0.47 thf(fact_207_inf__sup__aci_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A )
% 0.30/0.47 => ( ( sup_sup @ A )
% 0.30/0.47 = ( ^ [X7: A,Y6: A] : ( sup_sup @ A @ Y6 @ X7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(5)
% 0.30/0.47 thf(fact_208_inf__sup__aci_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A )
% 0.30/0.47 => ! [X6: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
% 0.30/0.47 = ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(6)
% 0.30/0.47 thf(fact_209_inf__sup__aci_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A )
% 0.30/0.47 => ! [X6: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) )
% 0.30/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X6 @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(7)
% 0.30/0.47 thf(fact_210_inf__sup__aci_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A )
% 0.30/0.47 => ! [X6: A,Y: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) )
% 0.30/0.47 = ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(8)
% 0.30/0.47 thf(fact_211_the__elem__eq,axiom,
% 0.30/0.47 ! [A: $tType,X6: A] :
% 0.30/0.47 ( ( the_elem @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = X6 ) ).
% 0.30/0.47
% 0.30/0.47 % the_elem_eq
% 0.30/0.47 thf(fact_212_Collect__empty__eq__bot,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( collect @ A @ P2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( P2
% 0.30/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_empty_eq_bot
% 0.30/0.47 thf(fact_213_is__singletonI,axiom,
% 0.30/0.47 ! [A: $tType,X6: A] : ( is_singleton @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singletonI
% 0.30/0.47 thf(fact_214_is__singleton__the__elem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( is_singleton @ A )
% 0.30/0.47 = ( ^ [A7: set @ A] :
% 0.30/0.47 ( A7
% 0.30/0.47 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singleton_the_elem
% 0.30/0.47 thf(fact_215_is__singletonI_H,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( A5
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 => ( ! [X2: A,Y3: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A5 )
% 0.30/0.47 => ( ( member @ A @ Y3 @ A5 )
% 0.30/0.47 => ( X2 = Y3 ) ) )
% 0.30/0.47 => ( is_singleton @ A @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singletonI'
% 0.30/0.47 thf(fact_216_is__singletonE,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( is_singleton @ A @ A5 )
% 0.30/0.47 => ~ ! [X2: A] :
% 0.30/0.47 ( A5
% 0.30/0.47 != ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singletonE
% 0.30/0.47 thf(fact_217_is__singleton__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( is_singleton @ A )
% 0.30/0.47 = ( ^ [A7: set @ A] :
% 0.30/0.47 ? [X7: A] :
% 0.30/0.47 ( A7
% 0.30/0.47 = ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singleton_def
% 0.30/0.47 thf(fact_218_labeled__graph_Osplit__sel,axiom,
% 0.30/0.47 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.30/0.47 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F2 @ Labeled_graph ) )
% 0.30/0.47 = ( ( Labeled_graph
% 0.30/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.30/0.47 => ( P2 @ ( F2 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.split_sel
% 0.30/0.47 thf(fact_219_labeled__graph_Osplit__sel__asm,axiom,
% 0.30/0.47 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.30/0.47 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F2 @ Labeled_graph ) )
% 0.30/0.47 = ( ~ ( ( Labeled_graph
% 0.30/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.30/0.47 & ~ ( P2 @ ( F2 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.split_sel_asm
% 0.30/0.47 thf(fact_220_labeled__graph_Ocase,axiom,
% 0.30/0.47 ! [L: $tType,A: $tType,V: $tType,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.30/0.47 ( ( labele1974067554_graph @ L @ V @ A @ F2 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.30/0.47 = ( F2 @ X1 @ X22 ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.case
% 0.30/0.47 thf(fact_221_labeled__graph_Ocase__eq__if,axiom,
% 0.30/0.47 ! [A: $tType,V: $tType,L: $tType] :
% 0.30/0.47 ( ( labele1974067554_graph @ L @ V @ A )
% 0.30/0.47 = ( ^ [F5: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F5 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % labeled_graph.case_eq_if
% 0.30/0.47 thf(fact_222_extensibleI,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G5: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G5 )
% 0.30/0.47 => ( ( agree_on @ A @ B @ C @ R1 @ F2 @ G5 )
% 0.30/0.47 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % extensibleI
% 0.30/0.47 thf(fact_223_total__on__singleton,axiom,
% 0.30/0.47 ! [A: $tType,X6: A] : ( total_on @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ X6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % total_on_singleton
% 0.30/0.47 thf(fact_224_agree__on__refl,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R2 @ F2 @ F2 ) ).
% 0.30/0.47
% 0.30/0.47 % agree_on_refl
% 0.30/0.47 thf(fact_225_agree__on__comm,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( agree_on @ A @ B @ C )
% 0.30/0.47 = ( ^ [X9: labeled_graph @ A @ B,F5: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X9 @ G6 @ F5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % agree_on_comm
% 0.30/0.47 thf(fact_226_agree__on__empty,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,F2: set @ ( product_prod @ B @ C ),G5: 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 ) ) ) @ F2 @ G5 ) ).
% 0.30/0.47
% 0.30/0.47 % agree_on_empty
% 0.30/0.47 thf(fact_227_total__onI,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ! [X2: A,Y3: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A5 )
% 0.30/0.47 => ( ( member @ A @ Y3 @ A5 )
% 0.30/0.47 => ( ( X2 != Y3 )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
% 0.30/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) ) ) ) )
% 0.30/0.47 => ( total_on @ A @ A5 @ R ) ) ).
% 0.30/0.47
% 0.30/0.47 % total_onI
% 0.30/0.47 thf(fact_228_total__on__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( total_on @ A )
% 0.30/0.47 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( member @ A @ X7 @ A7 )
% 0.30/0.47 => ! [Y6: A] :
% 0.30/0.47 ( ( member @ A @ Y6 @ A7 )
% 0.30/0.47 => ( ( X7 != Y6 )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ R3 )
% 0.30/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X7 ) @ R3 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % total_on_def
% 0.30/0.47 thf(fact_229_total__on__empty,axiom,
% 0.30/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 0.30/0.47
% 0.30/0.47 % total_on_empty
% 0.30/0.47 thf(fact_230_agree__on__trans,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,X8: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( agree_on @ A @ B @ C @ X8 @ F2 @ G5 )
% 0.30/0.47 => ( ( agree_on @ A @ B @ C @ X8 @ G5 @ H )
% 0.30/0.47 => ( agree_on @ A @ B @ C @ X8 @ F2 @ H ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % agree_on_trans
% 0.30/0.47 thf(fact_231_agree__on__subg__compose,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G5: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F4: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( agree_on @ A @ B @ C @ R2 @ G5 @ H )
% 0.30/0.47 => ( ( agree_on @ A @ B @ C @ F4 @ F2 @ G5 )
% 0.30/0.47 => ( ( graph_homomorphism @ A @ B @ B @ F4 @ R2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F4 ) ) )
% 0.30/0.47 => ( agree_on @ A @ B @ C @ F4 @ F2 @ H ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % agree_on_subg_compose
% 0.30/0.47 thf(fact_232_Field__insert,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ( field @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field @ A @ R ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Field_insert
% 0.30/0.47 thf(fact_233_subgraph__subset_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.30/0.47 => ( 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 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subgraph_subset(2)
% 0.30/0.47 thf(fact_234_order__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X6: A] : ( ord_less_eq @ A @ X6 @ X6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_235_subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A5 )
% 0.30/0.47 => ( member @ A @ X2 @ B6 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_236_subset__antisym,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
% 0.30/0.47 => ( A5 = B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_237_le__sup__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
% 0.30/0.47 = ( ( ord_less_eq @ A @ X6 @ Z )
% 0.30/0.47 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_sup_iff
% 0.30/0.47 thf(fact_238_sup_Obounded__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [B2: A,C5: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C5 ) @ A2 )
% 0.30/0.47 = ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 & ( ord_less_eq @ A @ C5 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.bounded_iff
% 0.30/0.47 thf(fact_239_empty__subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 0.30/0.47
% 0.30/0.47 % empty_subsetI
% 0.30/0.47 thf(fact_240_subset__empty,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_empty
% 0.30/0.47 thf(fact_241_insert__subset,axiom,
% 0.30/0.47 ! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X6 @ A5 ) @ B6 )
% 0.30/0.47 = ( ( member @ A @ X6 @ B6 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_subset
% 0.30/0.47 thf(fact_242_Un__subset__iff,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) @ C8 )
% 0.30/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A5 @ C8 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_subset_iff
% 0.30/0.47 thf(fact_243_singleton__insert__inj__eq,axiom,
% 0.30/0.47 ! [A: $tType,B2: A,A2: A,A5: set @ A] :
% 0.30/0.47 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( insert @ A @ A2 @ A5 ) )
% 0.30/0.47 = ( ( A2 = B2 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_insert_inj_eq
% 0.30/0.47 thf(fact_244_singleton__insert__inj__eq_H,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A5: set @ A,B2: A] :
% 0.30/0.47 ( ( ( insert @ A @ A2 @ A5 )
% 0.30/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( ( A2 = B2 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_insert_inj_eq'
% 0.30/0.47 thf(fact_245_Field__empty,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( field @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Field_empty
% 0.30/0.47 thf(fact_246_Field__Un,axiom,
% 0.30/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ( field @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ ( field @ A @ R ) @ ( field @ A @ S3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Field_Un
% 0.30/0.47 thf(fact_247_graph__unionI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.30/0.47 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.30/0.47 = G_2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % graph_unionI
% 0.30/0.47 thf(fact_248_inf__sup__ord_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A )
% 0.30/0.47 => ! [Y: A,X6: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_ord(4)
% 0.30/0.47 thf(fact_249_inf__sup__ord_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A )
% 0.30/0.47 => ! [X6: A,Y: A] : ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_ord(3)
% 0.30/0.47 thf(fact_250_le__supE,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A2: A,B2: A,X6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X6 )
% 0.30/0.47 => ~ ( ( ord_less_eq @ A @ A2 @ X6 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ B2 @ X6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_supE
% 0.30/0.47 thf(fact_251_le__supI,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [A2: A,X6: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ X6 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ X6 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_supI
% 0.30/0.47 thf(fact_252_sup__ge1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,Y: A] : ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_ge1
% 0.30/0.47 thf(fact_253_sup__ge2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [Y: A,X6: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_ge2
% 0.30/0.47 thf(fact_254_le__supI1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X6 @ A2 )
% 0.30/0.47 => ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_supI1
% 0.30/0.47 thf(fact_255_le__supI2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A )
% 0.30/0.47 => ! [X6: A,B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X6 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_supI2
% 0.30/0.47
% 0.30/0.47 % Type constructors (22)
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.30/0.47 bounded_lattice @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.30/0.47 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( bounded_lattice @ A9 )
% 0.30/0.47 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( bounded_lattice @ A9 )
% 0.30/0.47 => ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A9 )
% 0.30/0.47 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( preorder @ A9 )
% 0.30/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( lattice @ A9 )
% 0.30/0.47 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( bot @ A9 )
% 0.30/0.47 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
% 0.30/0.47 semilattice_sup @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 0.30/0.47 preorder @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Lattices_Olattice_5,axiom,
% 0.30/0.47 lattice @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Obot_6,axiom,
% 0.30/0.47 bot @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_7,axiom,
% 0.30/0.47 ! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
% 0.30/0.47 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
% 0.30/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Olattice_10,axiom,
% 0.30/0.47 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.30/0.47 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
% 0.30/0.47 bounde1808546759up_bot @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_13,axiom,
% 0.30/0.47 semilattice_sup @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 0.30/0.47 preorder @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
% 0.30/0.47 lattice @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 0.30/0.47 bot @ $o ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ y ) @ g ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.92/1.66 ---- Embedded in THF ---
% 2.92/1.66 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.92/1.66 %%% Generated on Tue Feb 24 08:29:13 EST 2026
% 2.92/1.66 %%% using '$$dhol' embedding, version 1.3.0.
% 2.92/1.66 %%% Logic specification used:
% 2.92/1.66 %%% thf(spec, logic, $$dhol).
% 2.92/1.66
% 2.92/1.66 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.TNBOsnf0Kz/DTF2DTF31760.p
% 2.92/1.66 thf(ty_t_LabeledGraphSemantics_OStandard__Constant, type, standard_Constant: ($tType > $tType)).
% 2.92/1.66 thf(ty_t_LabeledGraphSemantics_Oallegorical__term, type, allegorical_term: ($tType > $tType)).
% 2.92/1.66 thf(ty_t_LabeledGraphs_Olabeled__graph, type, labeled_graph: ($tType > ($tType > $tType))).
% 2.92/1.66 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.92/1.66 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.92/1.66 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.92/1.66 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.92/1.66 thf(ty_tf_V_H, type, v: $tType).
% 2.92/1.66 thf(ty_tf_V, type, v2: $tType).
% 2.92/1.66 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: ($o))).
% 2.92/1.66 thf(sy_c_BNF__Def_OGr, type, bNF_Gr: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (set @ ((product_prod @ A) @ B))))))).
% 2.92/1.66 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.92/1.66 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Bot, type, standard_S_Bot: (!> [V:$tType]: ((standard_Constant @ V)))).
% 2.92/1.66 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Const, type, standard_S_Const: (!> [V:$tType]: ((V > (standard_Constant @ V))))).
% 2.92/1.66 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Top, type, standard_S_Top: (!> [V:$tType]: ((standard_Constant @ V)))).
% 2.92/1.66 thf(sy_c_LabeledGraphSemantics_OgetRel, type, getRel: (!> [B:$tType,A:$tType]: ((B > (((labeled_graph @ B) @ A) > (set @ ((product_prod @ A) @ A))))))).
% 2.92/1.66 thf(sy_c_LabeledGraphSemantics_Osemantics, type, semantics: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((allegorical_term @ A) > (set @ ((product_prod @ B) @ B))))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Oedge__preserving, type, edge_preserving: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ ((product_prod @ C) @ ((product_prod @ A) @ A))) > ((set @ ((product_prod @ C) @ ((product_prod @ B) @ B))) > $o)))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Ograph__homomorphism, type, graph_homomorphism: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ C) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Ograph__union, type, graph_union: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B)))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG, type, labeled_LG: (!> [L:$tType,V:$tType]: (((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > ((labeled_graph @ L) @ V)))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph, type, labele1974067554_graph: (!> [L:$tType,V:$tType,A:$tType]: ((((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)) > (((labeled_graph @ L) @ V) > A))))).
% 2.92/1.66 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))))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices, type, labeled_vertices: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ V))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Omap__graph, type, map_graph: (!> [C:$tType,B:$tType,A:$tType]: (((set @ ((product_prod @ C) @ B)) > (((labeled_graph @ A) @ C) > ((labeled_graph @ A) @ B)))))).
% 2.92/1.66 thf(sy_c_LabeledGraphs_Orestrict, type, restrict: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B))))).
% 2.92/1.66 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.92/1.66 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.92/1.66 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.92/1.66 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.92/1.66 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.92/1.66 thf(sy_c_Relation_OField, type, field: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ A))))).
% 2.92/1.66 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 2.92/1.66 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.92/1.66 thf(sy_c_RulesAndChains_Oagree__on, type, agree_on: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > ((set @ ((product_prod @ B) @ C)) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 2.92/1.66 thf(sy_c_RulesAndChains_Oconsequence__graph, type, consequence_graph: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) > (((labeled_graph @ A) @ C) > $o))))).
% 2.92/1.66 thf(sy_c_RulesAndChains_Oextensible, type, extensible: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > ((set @ ((product_prod @ X) @ V)) > $o)))))).
% 2.92/1.66 thf(sy_c_RulesAndChains_Oleast, type, least: (!> [X:$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) > $o))))))).
% 2.92/1.66 thf(sy_c_RulesAndChains_Oleast__consequence__graph, type, least_559130134_graph: (!> [X:$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) > $o))))))).
% 2.92/1.66 thf(sy_c_RulesAndChains_Omaintained, type, maintained: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > $o))))).
% 2.92/1.66 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.92/1.66 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.92/1.66 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.92/1.66 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.92/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules, type, standa1138209853_rules: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))))))).
% 2.92/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules, type, standa2002409347_rules: (!> [A:$tType]: (((set @ (standard_Constant @ A)) > (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))))))).
% 2.92/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule, type, standa244753842y_rule: (!> [L:$tType]: (((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat))))).
% 2.92/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ostandard__rules, type, standa438229444_rules: (!> [A:$tType]: (((set @ A) > ((set @ (standard_Constant @ A)) > (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)))))))).
% 2.92/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule, type, standa400804411p_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 2.92/1.66 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.92/1.66 thf(sy_v_C, type, c: (set @ v2)).
% 2.92/1.66 thf(sy_v_G_H, type, g: ((labeled_graph @ (standard_Constant @ v2)) @ v)).
% 2.92/1.66 thf(sy_v_L, type, l: (set @ (standard_Constant @ v2))).
% 2.92/1.66 thf(sy_v_h____, type, h: (v > v)).
% 2.92/1.66 thf(sy_v_m____, type, m: (v2 > v)).
% 2.92/1.66 thf(sy_v_x____, type, x: v).
% 2.92/1.66 thf(sy_v_xa____, type, xa: v2).
% 2.92/1.66 thf(sy_v_y____, type, y: v2).
% 2.92/1.66 thf(fact_0_x, axiom, (((member @ v) @ x) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g))).
% 2.92/1.66 thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [X2:v]: ((~ ((((member @ v) @ X2) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g))))))))).
% 2.92/1.66 thf(fact_2__C1_C_I3_J, axiom, ((m @ xa) = (m @ y))).
% 2.92/1.66 thf(fact_3__C1_C_I1_J, axiom, (((member @ v2) @ xa) @ c)).
% 2.92/1.66 thf(fact_4__C1_C_I2_J, axiom, (((member @ v2) @ y) @ c)).
% 2.92/1.66 thf(fact_5__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062, axiom, (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ (m @ xa)) @ (m @ xa))) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ xa)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g)))).
% 2.92/1.66 thf(fact_6__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ay_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062, axiom, (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ (m @ xa)) @ (m @ xa))) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ y)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g)))).
% 2.92/1.66 thf(fact_7_gr_I1_J, axiom, (g = (((restrict @ (standard_Constant @ v2)) @ v) @ g))).
% 2.92/1.66 thf(fact_8_h_I3_J, axiom, ((((((graph_homomorphism @ (standard_Constant @ v2)) @ v) @ v) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g)) @ g) @ ((id_on @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g))))).
% 2.92/1.66 thf(fact_9_Standard__Constant_Oinject, axiom, (! [V:$tType,X4:V,Y4:V]: (((((standard_S_Const @ V) @ X4) = ((standard_S_Const @ V) @ Y4)) = (X4 = Y4))))).
% 2.92/1.66 thf(fact_10_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)))))).
% 2.92/1.66 thf(fact_11_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)))))).
% 2.92/1.66 thf(fact_12_m, axiom, (! [X3:v2]: (((((member @ v2) @ X3) @ c) => (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ (m @ X3)) @ (m @ X3))) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ X3)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g))))))).
% 2.92/1.66 thf(fact_13_top__nonempty, axiom, (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ x) @ x)) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ (standard_S_Top @ v2)) @ g))).
% 2.92/1.66 thf(fact_14_cf, axiom, (((((getRel @ (standard_Constant @ v2)) @ v) @ (standard_S_Bot @ v2)) @ g) = (bot_bot @ (set @ ((product_prod @ v) @ v))))).
% 2.92/1.66 thf(fact_15__092_060open_062_092_060And_062c_O_Ac_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_092_060exists_062v_O_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062, axiom, (! [C2:v2]: (((((member @ v2) @ C2) @ c) => (? [V2:v]: ((((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ V2) @ V2)) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ C2)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g))))))))).
% 2.92/1.66 thf(fact_16__092_060open_062_092_060forall_062c_O_A_092_060exists_062v_O_Ac_A_092_060in_062_AC_A_092_060longrightarrow_062_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062, axiom, (! [C3:v2]: ((? [V2:v]: (((((member @ v2) @ C3) @ c) => (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ V2) @ V2)) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ C3)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g))))))))).
% 2.92/1.66 thf(fact_17__092_060open_062_092_060exists_062f_O_A_092_060forall_062x_O_Ax_A_092_060in_062_AC_A_092_060longrightarrow_062_A_If_Ax_M_Af_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062, axiom, (? [F:(v2 > v)]: ((! [X5:v2]: (((((member @ v2) @ X5) @ c) => (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ (F @ X5)) @ (F @ X5))) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ X5)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g))))))))).
% 2.92/1.66 thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [M:(v2 > v)]: ((~ ((! [X5:v2]: (((((member @ v2) @ X5) @ c) => (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ (M @ X5)) @ (M @ X5))) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ X5)) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g))))))))))))).
% 2.92/1.66 thf(fact_19_getRel__homR, axiom, (! [C:$tType,A:$tType,B:$tType,Y:A,Z:A,L2:B,G:((labeled_graph @ B) @ A),U:C,F2:(set @ ((product_prod @ A) @ C)),V3:C]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ ((((getRel @ B) @ A) @ L2) @ G)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ Y) @ U)) @ F2) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ Z) @ V3)) @ F2) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ U) @ V3)) @ ((((getRel @ B) @ C) @ L2) @ (((((map_graph @ A) @ C) @ B) @ F2) @ G))))))))).
% 2.92/1.66 thf(fact_20_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X2:A,Y3:B]: ((P = ((((product_Pair @ A) @ B) @ X2) @ Y3))))))).
% 2.92/1.66 thf(fact_21_h_I1_J, axiom, ((((((comp @ v) @ v) @ v) @ h) @ h) = h)).
% 2.92/1.66 thf(fact_22_getRel__map__fn, axiom, (! [C:$tType,A:$tType,B:$tType,A22:A,G:((labeled_graph @ B) @ A),B22:A,L2:B,F2:(A > C),A2:C,B2:C]: (((((member @ A) @ A22) @ (((labeled_vertices @ B) @ A) @ G)) => ((((member @ A) @ B22) @ (((labeled_vertices @ B) @ A) @ G)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A22) @ B22)) @ ((((getRel @ B) @ A) @ L2) @ G)) => (((F2 @ A22) = A2) => (((F2 @ B22) = B2) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ A2) @ B2)) @ ((((getRel @ B) @ C) @ L2) @ (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ F2)) @ G))))))))))).
% 2.92/1.66 thf(fact_23_getRel__hom, axiom, (! [C:$tType,A:$tType,B:$tType,Y:A,Z:A,L2:B,G:((labeled_graph @ B) @ A),F2:(A > C)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ ((((getRel @ B) @ A) @ L2) @ G)) => ((((member @ A) @ Y) @ (((labeled_vertices @ B) @ A) @ G)) => ((((member @ A) @ Z) @ (((labeled_vertices @ B) @ A) @ G)) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ (F2 @ Y)) @ (F2 @ Z))) @ ((((getRel @ B) @ C) @ L2) @ (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ F2)) @ G))))))))).
% 2.92/1.66 thf(fact_24_Standard__Constant_Odistinct_I1_J, axiom, (! [V:$tType]: (((standard_S_Top @ V) != (standard_S_Bot @ V))))).
% 2.92/1.66 thf(fact_25_Standard__Constant_Odistinct_I5_J, axiom, (! [V:$tType,X4:V]: (((standard_S_Top @ V) != ((standard_S_Const @ V) @ X4))))).
% 2.92/1.66 thf(fact_26_Standard__Constant_Odistinct_I9_J, axiom, (! [V:$tType,X4:V]: (((standard_S_Bot @ V) != ((standard_S_Const @ V) @ X4))))).
% 2.92/1.66 thf(fact_27_getRel__dom_I1_J, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),A2:B,B2:B,L2:A]: (((G = (((restrict @ A) @ B) @ G)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((getRel @ A) @ B) @ L2) @ G)) => (((member @ B) @ A2) @ (((labeled_vertices @ A) @ B) @ G))))))).
% 2.92/1.66 thf(fact_28_getRel__dom_I2_J, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),A2:B,B2:B,L2:A]: (((G = (((restrict @ A) @ B) @ G)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((getRel @ A) @ B) @ L2) @ G)) => (((member @ B) @ B2) @ (((labeled_vertices @ A) @ B) @ G))))))).
% 2.92/1.66 thf(fact_29_getRel__subgraph, axiom, (! [A:$tType,B:$tType,Y:A,Z:A,L2:B,G:((labeled_graph @ B) @ A),G2:((labeled_graph @ B) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ ((((getRel @ B) @ A) @ L2) @ G)) => (((((((graph_homomorphism @ B) @ A) @ A) @ G) @ G2) @ ((id_on @ A) @ (((labeled_vertices @ B) @ A) @ G))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ ((((getRel @ B) @ A) @ L2) @ G2))))))).
% 2.92/1.66 thf(fact_30_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P2 @ Prod))))).
% 2.92/1.66 thf(fact_31_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 2.92/1.66 thf(fact_32_prod__induct7, axiom, (! [G3:$tType,F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) > $o),X6:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: (((! [A4:A,B4:B,C4:C,D2:D,E2:E,F:F3,G4:G3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F) @ G4))))))))) => (P2 @ X6))))).
% 2.92/1.66 thf(fact_33_prod__induct6, axiom, (! [F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) > $o),X6:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: (((! [A4:A,B4:B,C4:C,D2:D,E2:E,F:F3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F)))))))) => (P2 @ X6))))).
% 2.92/1.66 thf(fact_34_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X6:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C4:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C4) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X6))))).
% 2.92/1.66 thf(fact_35_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X6:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C4:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C4) @ D2)))))) => (P2 @ X6))))).
% 2.92/1.66 thf(fact_36_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X6:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C4:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C4))))) => (P2 @ X6))))).
% 2.92/1.66 thf(fact_37_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: ((~ ((! [A4:A,B4:B,C4:C,D2:D,E2:E,F:F3,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F) @ G4)))))))))))))).
% 2.92/1.66 thf(fact_38_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A4:A,B4:B,C4:C,D2:D,E2:E,F:F3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F))))))))))))).
% 2.92/1.66 thf(fact_39_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C4:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C4) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.92/1.66 thf(fact_40_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C4:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C4) @ D2))))))))))).
% 2.92/1.66 thf(fact_41_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C4:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C4)))))))))).
% 2.92/1.66 thf(fact_42_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)))))))).
% 2.92/1.66 thf(fact_43_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P2 @ P))))).
% 2.92/1.66 thf(fact_44_mg, axiom, (! [B5:$tType,R:((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ B5)) @ ((labeled_graph @ (standard_Constant @ v2)) @ B5))]: (((((((maintained @ (standard_Constant @ v2)) @ B5) @ v) @ R) @ g) => (((((maintained @ (standard_Constant @ v2)) @ B5) @ v) @ R) @ (((((map_graph @ v) @ v) @ (standard_Constant @ v2)) @ ((((bNF_Gr @ v) @ v) @ (((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g)) @ h)) @ g)))))).
% 2.92/1.66 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.92/1.66 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X7:A]: ((((member @ A) @ X7) @ A5)))) = A5)))).
% 2.92/1.66 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) = (Q @ X2)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.92/1.66 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G5:(A > B)]: (((! [X2:A]: (((F2 @ X2) = (G5 @ X2)))) => (F2 = G5))))).
% 2.92/1.66 thf(fact_49_graph__homo, axiom, (! [A:$tType,C:$tType,B:$tType,G:((labeled_graph @ A) @ B),F2:(B > C)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((graph_homomorphism @ A) @ B) @ C) @ G) @ (((((map_graph @ B) @ C) @ A) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ G)) @ F2)) @ G)) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ G)) @ F2)))))).
% 2.92/1.66 thf(fact_50_map__graph__fn__id_I2_J, axiom, (! [B:$tType,A:$tType,X8:((labeled_graph @ A) @ B)]: (((((((map_graph @ B) @ B) @ A) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ X8))) @ X8) = (((restrict @ A) @ B) @ X8))))).
% 2.92/1.66 thf(fact_51_map__graph__fn__graphI, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ C),F2:(C > B)]: (((((((map_graph @ C) @ B) @ A) @ ((((bNF_Gr @ C) @ B) @ (((labeled_vertices @ A) @ C) @ G)) @ F2)) @ G) = (((restrict @ A) @ B) @ (((((map_graph @ C) @ B) @ A) @ ((((bNF_Gr @ C) @ B) @ (((labeled_vertices @ A) @ C) @ G)) @ F2)) @ G)))))).
% 2.92/1.66 thf(fact_52_map__graph__preserves__subgraph, axiom, (! [C:$tType,B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B6) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => ((((((graph_homomorphism @ A) @ C) @ C) @ (((((map_graph @ B) @ C) @ A) @ F2) @ A5)) @ (((((map_graph @ B) @ C) @ A) @ F2) @ B6)) @ ((id_on @ C) @ (((labeled_vertices @ A) @ C) @ (((((map_graph @ B) @ C) @ A) @ F2) @ A5)))))))).
% 2.92/1.66 thf(fact_53_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)))))).
% 2.92/1.66 thf(fact_54_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)))))).
% 2.92/1.66 thf(fact_55_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)))))).
% 2.92/1.66 thf(fact_56_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.92/1.66 thf(fact_57_Gr__empty, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((bNF_Gr @ A) @ B) @ (bot_bot @ (set @ A))) @ F2) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.92/1.66 thf(fact_58_restrict__idemp, axiom, (! [B:$tType,A:$tType,X6:((labeled_graph @ A) @ B)]: (((((restrict @ A) @ B) @ (((restrict @ A) @ B) @ X6)) = (((restrict @ A) @ B) @ X6))))).
% 2.92/1.66 thf(fact_59_in__Gr, axiom, (! [A:$tType,B:$tType,X6:A,Y:B,A5:(set @ A),F2:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Y)) @ ((((bNF_Gr @ A) @ B) @ A5) @ F2)) = ((((member @ A) @ X6) @ A5) & ((F2 @ X6) = Y)))))).
% 2.92/1.66 thf(fact_60_Gr__not__in, axiom, (! [B:$tType,A:$tType,X6:A,F4:(set @ A),F2:(A > B),Y:B]: ((((~ ((((member @ A) @ X6) @ F4))) | ((F2 @ X6) != Y)) => (~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Y)) @ ((((bNF_Gr @ A) @ B) @ F4) @ F2)))))))).
% 2.92/1.66 thf(fact_61_Id__onI, axiom, (! [A:$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)))))).
% 2.92/1.66 thf(fact_62_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))))).
% 2.92/1.66 thf(fact_63_map__graph__preserves__restricted, axiom, (! [C:$tType,B:$tType,A:$tType,G:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C))]: (((G = (((restrict @ A) @ B) @ G)) => ((((((map_graph @ B) @ C) @ A) @ F2) @ G) = (((restrict @ A) @ C) @ (((((map_graph @ B) @ C) @ A) @ F2) @ G))))))).
% 2.92/1.66 thf(fact_64_map__graph__fn__comp, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ C),F2:(D > B),G5:(C > D)]: (((((((map_graph @ C) @ B) @ A) @ ((((bNF_Gr @ C) @ B) @ (((labeled_vertices @ A) @ C) @ G)) @ (((((comp @ D) @ B) @ C) @ F2) @ G5))) @ G) = (((((map_graph @ D) @ B) @ A) @ ((((bNF_Gr @ D) @ B) @ (((labeled_vertices @ A) @ D) @ (((((map_graph @ C) @ D) @ A) @ ((((bNF_Gr @ C) @ D) @ (((labeled_vertices @ A) @ C) @ G)) @ G5)) @ G))) @ F2)) @ (((((map_graph @ C) @ D) @ A) @ ((((bNF_Gr @ C) @ D) @ (((labeled_vertices @ A) @ C) @ G)) @ G5)) @ G)))))).
% 2.92/1.66 thf(fact_65_assms_I1_J, axiom, (! [X5:((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))) @ X5) @ (((standa438229444_rules @ v2) @ c) @ l)) => (((((maintained @ (standard_Constant @ v2)) @ nat) @ v) @ X5) @ g))))).
% 2.92/1.66 thf(fact_66_ne, axiom, (((((maintained @ (standard_Constant @ v2)) @ nat) @ v) @ (standa244753842y_rule @ (standard_Constant @ v2))) @ g)).
% 2.92/1.66 thf(fact_67__092_060open_062maintained_Anonempty__rule_AG_H_A_061_A_Ivertices_AG_H_A_092_060noteq_062_A_123_125_J_092_060close_062, axiom, ((((((maintained @ (standard_Constant @ v2)) @ nat) @ v) @ (standa244753842y_rule @ (standard_Constant @ v2))) @ g) = ((((labeled_vertices @ (standard_Constant @ v2)) @ v) @ g) != (bot_bot @ (set @ v))))).
% 2.92/1.66 thf(fact_68_maintained__preserved__by__isomorphism, axiom, (! [C:$tType,D:$tType,A:$tType,B:$tType,G:((labeled_graph @ B) @ A),F2:(C > A),G5:(A > C),R:((product_prod @ ((labeled_graph @ B) @ D)) @ ((labeled_graph @ B) @ D))]: (((! [X2:A]: (((((member @ A) @ X2) @ (((labeled_vertices @ B) @ A) @ G)) => (((((((comp @ C) @ A) @ A) @ F2) @ G5) @ X2) = X2)))) => ((G = (((restrict @ B) @ A) @ G)) => ((((((maintained @ B) @ D) @ C) @ R) @ (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ G5)) @ G)) => (((((maintained @ B) @ D) @ A) @ R) @ G))))))).
% 2.92/1.66 thf(fact_69_idemp__embedding__maintained__preserved, axiom, (! [C:$tType,B:$tType,A:$tType,G:((labeled_graph @ A) @ B),F2:(B > B),R:((product_prod @ ((labeled_graph @ A) @ C)) @ ((labeled_graph @ A) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ (((((map_graph @ B) @ B) @ A) @ ((((bNF_Gr @ B) @ B) @ (((labeled_vertices @ A) @ B) @ G)) @ F2)) @ G)) @ G) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ (((((map_graph @ B) @ B) @ A) @ ((((bNF_Gr @ B) @ B) @ (((labeled_vertices @ A) @ B) @ G)) @ F2)) @ G)))) => ((! [X2:B]: (((((member @ B) @ X2) @ (((labeled_vertices @ A) @ B) @ G)) => (((((((comp @ B) @ B) @ B) @ F2) @ F2) @ X2) = (F2 @ X2))))) => ((((((maintained @ A) @ C) @ B) @ R) @ G) => (((((maintained @ A) @ C) @ B) @ R) @ (((((map_graph @ B) @ B) @ A) @ ((((bNF_Gr @ B) @ B) @ (((labeled_vertices @ A) @ B) @ G)) @ F2)) @ G)))))))).
% 2.92/1.66 thf(fact_70_Id__onE, axiom, (! [A:$tType,C5:((product_prod @ A) @ A),A5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C5) @ ((id_on @ A) @ A5)) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (C5 != ((((product_Pair @ A) @ A) @ X2) @ X2))))))))))).
% 2.92/1.66 thf(fact_71_Id__on__eqI, axiom, (! [A:$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))))))).
% 2.92/1.66 thf(fact_72_Id__on__iff, axiom, (! [A:$tType,X6:A,Y:A,A5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X6) @ Y)) @ ((id_on @ A) @ A5)) = ((X6 = Y) & (((member @ A) @ X6) @ A5)))))).
% 2.92/1.66 thf(fact_73_subgraph__preserves__hom, axiom, (! [A:$tType,B:$tType,C:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ B),X8:((labeled_graph @ A) @ C),H:(set @ ((product_prod @ C) @ B))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B6) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => (((((((graph_homomorphism @ A) @ C) @ B) @ X8) @ A5) @ H) => ((((((graph_homomorphism @ A) @ C) @ B) @ X8) @ B6) @ H)))))).
% 2.92/1.66 thf(fact_74_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)))))))).
% 2.92/1.66 thf(fact_75_map__graph__fn__eqI, axiom, (! [C:$tType,A:$tType,B:$tType,G:((labeled_graph @ B) @ A),F2:(A > C),G5:(A > C)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((labeled_vertices @ B) @ A) @ G)) => ((F2 @ X2) = (G5 @ X2))))) => ((((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ F2)) @ G) = (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ G5)) @ G)))))).
% 2.92/1.66 thf(fact_76_tr, axiom, (((((maintained @ (standard_Constant @ v2)) @ nat) @ v) @ ((standa400804411p_rule @ (standard_Constant @ v2)) @ (standard_S_Top @ v2))) @ g)).
% 2.92/1.66 thf(fact_77_h_I4_J, axiom, (! [L2:(standard_Constant @ v2),X6:v,Y:v]: (((((member @ ((product_prod @ (standard_Constant @ v2)) @ ((product_prod @ v) @ v))) @ ((((product_Pair @ (standard_Constant @ v2)) @ ((product_prod @ v) @ v)) @ L2) @ ((((product_Pair @ v) @ v) @ X6) @ Y))) @ (((labeled_edges @ (standard_Constant @ v2)) @ v) @ g)) = (((member @ ((product_prod @ (standard_Constant @ v2)) @ ((product_prod @ v) @ v))) @ ((((product_Pair @ (standard_Constant @ v2)) @ ((product_prod @ v) @ v)) @ L2) @ ((((product_Pair @ v) @ v) @ (h @ X6)) @ (h @ Y)))) @ (((labeled_edges @ (standard_Constant @ v2)) @ v) @ g)))))).
% 2.92/1.66 thf(fact_78_old_Oprod_Orec, axiom, (! [A:$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))))).
% 2.92/1.66 thf(fact_79_maintained__refl, axiom, (! [B:$tType,C:$tType,A:$tType,R2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C)]: ((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R2) @ R2)) @ G)))).
% 2.92/1.66 thf(fact_80_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F5:(B > A),G6:(C > B),X7:C]: ((F5 @ (G6 @ X7)))))))).
% 2.92/1.66 thf(fact_81_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X7:A]: ((~ ((P2 @ X7))))))))).
% 2.92/1.66 thf(fact_82_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X7:A]: ((~ ((P2 @ X7))))))))).
% 2.92/1.66 thf(fact_83_empty__iff, axiom, (! [A:$tType,C5:A]: ((~ ((((member @ A) @ C5) @ (bot_bot @ (set @ A)))))))).
% 2.92/1.66 thf(fact_84_all__not__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X7:A]: ((~ ((((member @ A) @ X7) @ A5))))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.92/1.66 thf(fact_85_nonempty__rule, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ A) @ B)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((maintained @ A) @ nat) @ B) @ (standa244753842y_rule @ A)) @ G) = ((((labeled_vertices @ A) @ B) @ G) != (bot_bot @ (set @ B)))))))).
% 2.92/1.66 thf(fact_86__092_060open_062maintainedA_A_Iidentity__rules_AL_J_AG_H_092_060close_062, axiom, (! [X5:((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))) @ X5) @ ((standa2002409347_rules @ v2) @ l)) => (((((maintained @ (standard_Constant @ v2)) @ nat) @ v) @ X5) @ g))))).
% 2.92/1.66 thf(fact_87_labeled__graph_Oexpand, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V),Labeled_graph2:((labeled_graph @ L) @ V)]: (((((((labeled_edges @ L) @ V) @ Labeled_graph) = (((labeled_edges @ L) @ V) @ Labeled_graph2)) & ((((labeled_vertices @ L) @ V) @ Labeled_graph) = (((labeled_vertices @ L) @ V) @ Labeled_graph2))) => (Labeled_graph = Labeled_graph2))))).
% 2.92/1.66 thf(fact_88_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.92/1.66 thf(fact_89_equals0D, axiom, (! [A:$tType,A5:(set @ A),A2:A]: (((A5 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A5))))))).
% 2.92/1.66 thf(fact_90_equals0I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A5))))) => (A5 = (bot_bot @ (set @ A))))))).
% 2.92/1.66 thf(fact_91_ex__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((? [X7:A]: ((((member @ A) @ X7) @ A5))) = (A5 != (bot_bot @ (set @ A))))))).
% 2.92/1.66 thf(fact_92_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F5:(B > C),G6:(A > B),X7:A]: ((F5 @ (G6 @ X7)))))))).
% 2.92/1.66 thf(fact_93_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F2:(D > B),G5:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F2) @ G5)) @ H) = (((((comp @ D) @ B) @ A) @ F2) @ (((((comp @ C) @ D) @ A) @ G5) @ H)))))).
% 2.92/1.66 thf(fact_94_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C5:(D > B),D3:(A > D),V3:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C5) @ D3)) => ((A2 @ (B2 @ V3)) = (C5 @ (D3 @ V3))))))).
% 2.92/1.66 thf(fact_95_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C5:(D > B),D3:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C5) @ D3)) => (! [V4:A]: (((A2 @ (B2 @ V4)) = (C5 @ (D3 @ V4))))))))).
% 2.92/1.66 thf(fact_96_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C5:(A > B),V3:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = C5) => ((A2 @ (B2 @ V3)) = (C5 @ V3)))))).
% 2.92/1.66 thf(fact_97_edge__preserving__on__graphI, axiom, (! [B:$tType,C:$tType,A:$tType,X8:((labeled_graph @ A) @ B),F2:(B > C),Y5:(set @ ((product_prod @ A) @ ((product_prod @ C) @ C)))]: (((! [L3:A,X2:B,Y3:B]: (((((member @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ L3) @ ((((product_Pair @ B) @ B) @ X2) @ Y3))) @ (((labeled_edges @ A) @ B) @ X8)) => ((((member @ B) @ X2) @ (((labeled_vertices @ A) @ B) @ X8)) => ((((member @ B) @ Y3) @ (((labeled_vertices @ A) @ B) @ X8)) => (((member @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ ((((product_Pair @ A) @ ((product_prod @ C) @ C)) @ L3) @ ((((product_Pair @ C) @ C) @ (F2 @ X2)) @ (F2 @ Y3)))) @ Y5)))))) => ((((((edge_preserving @ B) @ C) @ A) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ X8)) @ F2)) @ (((labeled_edges @ A) @ B) @ X8)) @ Y5))))).
% 2.92/1.66 thf(fact_98_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X7:D]: ((bot_bot @ C)))))))).
% 2.92/1.66 thf(fact_99_least__consequence__graphI, axiom, (! [X:$tType,A:$tType,C:$tType,L:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ A)) @ ((labeled_graph @ L) @ A))),G:((labeled_graph @ L) @ C),S:((labeled_graph @ L) @ C),T2:(itself @ X)]: (((((((consequence_graph @ L) @ A) @ C) @ Rs) @ G) => (((((((graph_homomorphism @ L) @ C) @ C) @ S) @ G) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S))) => ((! [C6:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs) @ C6) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S) @ G)) @ C6)))) => ((((((((least_559130134_graph @ X) @ L) @ A) @ C) @ T2) @ Rs) @ S) @ G))))))).
% 2.92/1.66 thf(fact_100_GrD2, axiom, (! [A:$tType,B:$tType,X6:A,Fx:B,A5:(set @ A),F2:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A5) @ F2)) => ((F2 @ X6) = Fx))))).
% 2.92/1.66 thf(fact_101_GrD1, axiom, (! [B:$tType,A:$tType,X6:A,Fx:B,A5:(set @ A),F2:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A5) @ F2)) => (((member @ A) @ X6) @ A5))))).
% 2.92/1.66 thf(fact_102_leastI, axiom, (! [X:$tType,A:$tType,C:$tType,L:$tType,S:((labeled_graph @ L) @ C),G:((labeled_graph @ L) @ C),Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ A)) @ ((labeled_graph @ L) @ A))),T2:(itself @ X)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S) @ G) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S))) => ((! [C6:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs) @ C6) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S) @ G)) @ C6)))) => ((((((((least @ X) @ L) @ A) @ C) @ T2) @ Rs) @ S) @ G)))))).
% 2.92/1.66 thf(fact_103_edge__preserving__Id, axiom, (! [A:$tType,B:$tType,Y:(set @ A),X6:(set @ ((product_prod @ B) @ ((product_prod @ A) @ A)))]: (((((((edge_preserving @ A) @ A) @ B) @ ((id_on @ A) @ Y)) @ X6) @ X6)))).
% 2.92/1.66 thf(fact_104_map__graph__edge__preserving, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(set @ ((product_prod @ A) @ B)),G:((labeled_graph @ C) @ A)]: (((((((edge_preserving @ A) @ B) @ C) @ F2) @ (((labeled_edges @ C) @ A) @ G)) @ (((labeled_edges @ C) @ B) @ (((((map_graph @ A) @ B) @ C) @ F2) @ G)))))).
% 2.92/1.66 thf(fact_105_least__consequence__graph__def, axiom, (! [L:$tType,V:$tType,C:$tType,X:$tType]: ((((((least_559130134_graph @ X) @ L) @ V) @ C) = (^ [T3:(itself @ X),Rs2:(set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))),S2:((labeled_graph @ L) @ C),G7:((labeled_graph @ L) @ C)]: (((((((consequence_graph @ L) @ V) @ C) @ Rs2) @ G7) & ((((((((least @ X) @ L) @ V) @ C) @ T3) @ Rs2) @ S2) @ G7)))))))).
% 2.92/1.66 thf(fact_106_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X7:A]: ((((member @ A) @ X7) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.66 thf(fact_107_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.92/1.66 thf(fact_108_consequence__graphD_I3_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C)]: (((((((consequence_graph @ A) @ B) @ C) @ Rs) @ G) => (G = (((restrict @ A) @ C) @ G)))))).
% 2.92/1.66 thf(fact_109_consequence__graphD_I1_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C),R2:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((((consequence_graph @ A) @ B) @ C) @ Rs) @ G) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R2) @ Rs) => (((((maintained @ A) @ B) @ C) @ R2) @ G)))))).
% 2.92/1.66 thf(fact_110_edge__preserving__atomic, axiom, (! [A:$tType,B:$tType,C:$tType,H1:(set @ ((product_prod @ A) @ B)),E1:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A))),E22:(set @ ((product_prod @ C) @ ((product_prod @ B) @ B))),V1:A,V12:B,V22:A,V23:B,K:C]: ((((((((edge_preserving @ A) @ B) @ C) @ H1) @ E1) @ E22) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ V1) @ V12)) @ H1) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ V22) @ V23)) @ H1) => ((((member @ ((product_prod @ C) @ ((product_prod @ A) @ A))) @ ((((product_Pair @ C) @ ((product_prod @ A) @ A)) @ K) @ ((((product_Pair @ A) @ A) @ V1) @ V22))) @ E1) => (((member @ ((product_prod @ C) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ C) @ ((product_prod @ B) @ B)) @ K) @ ((((product_Pair @ B) @ B) @ V12) @ V23))) @ E22)))))))).
% 2.92/1.66 thf(fact_111_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X7:A]: ((bot_bot @ B)))))))).
% 2.92/1.66 thf(fact_112_least__def, axiom, (! [L:$tType,V:$tType,C:$tType,X:$tType]: ((((((least @ X) @ L) @ V) @ C) = (^ [Uu:(itself @ X),Rs2:(set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))),S2:((labeled_graph @ L) @ C),G7:((labeled_graph @ L) @ C)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S2) @ G7) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S2))) & (! [C7:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ V) @ X) @ Rs2) @ C7) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S2) @ G7)) @ C7))))))))))).
% 2.92/1.66 thf(fact_113_mnt, axiom, (! [X5:((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))) @ X5) @ (((sup_sup @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)))) @ (((sup_sup @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)))) @ ((standa1138209853_rules @ v2) @ c)) @ ((standa2002409347_rules @ v2) @ l))) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))) @ ((standa400804411p_rule @ (standard_Constant @ v2)) @ (standard_S_Top @ v2))) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat))) @ (standa244753842y_rule @ (standard_Constant @ v2))) @ (bot_bot @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)) @ ((labeled_graph @ (standard_Constant @ v2)) @ nat)))))))) => (((((maintained @ (standard_Constant @ v2)) @ nat) @ v) @ X5) @ g))))).
% 2.92/1.66 thf(fact_114_map__graph__in, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A),F2:(B > C)]: (((G = (((restrict @ A) @ B) @ G)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ G) @ E3)) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ (F2 @ A2)) @ (F2 @ B2))) @ ((((semantics @ A) @ C) @ (((((map_graph @ B) @ C) @ A) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ G)) @ F2)) @ G)) @ E3))))))).
% 2.92/1.66 thf(fact_115_maintainedI, axiom, (! [A:$tType,B:$tType,C:$tType,A5:((labeled_graph @ C) @ A),G:((labeled_graph @ C) @ B),B6:((labeled_graph @ C) @ A)]: (((! [F:(set @ ((product_prod @ A) @ B))]: ((((((((graph_homomorphism @ C) @ A) @ B) @ A5) @ G) @ F) => ((((((extensible @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A5) @ B6)) @ G) @ F)))) => (((((maintained @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A5) @ B6)) @ G))))).
% 2.92/1.66 thf(fact_116_graph__homomorphism__empty, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ C),F2:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ (bot_bot @ (set @ B)))) @ G) @ F2) = ((F2 = (bot_bot @ (set @ ((product_prod @ B) @ C)))) & (G = (((restrict @ A) @ C) @ G))))))).
% 2.92/1.66 thf(fact_117_subgraph__def, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) = ((G_1 = (((restrict @ A) @ B) @ G_1)) & ((G_2 = (((restrict @ A) @ B) @ G_2)) & (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2))))))).
% 2.92/1.66 thf(fact_118_insert__absorb2, axiom, (! [A:$tType,X6:A,A5:(set @ A)]: (((((insert @ A) @ X6) @ (((insert @ A) @ X6) @ A5)) = (((insert @ A) @ X6) @ A5))))).
% 2.92/1.66 thf(fact_119_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A5)) = ((A2 = B2) | (((member @ A) @ A2) @ A5)))))).
% 2.92/1.66 thf(fact_120_insertCI, axiom, (! [A:$tType,A2:A,B6:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B6))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B6)))))).
% 2.92/1.66 thf(fact_121_Un__iff, axiom, (! [A:$tType,C5:A,A5:(set @ A),B6:(set @ A)]: (((((member @ A) @ C5) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) = ((((member @ A) @ C5) @ A5) | (((member @ A) @ C5) @ B6)))))).
% 2.92/1.66 thf(fact_122_UnCI, axiom, (! [A:$tType,C5:A,B6:(set @ A),A5:(set @ A)]: ((((~ ((((member @ A) @ C5) @ B6))) => (((member @ A) @ C5) @ A5)) => (((member @ A) @ C5) @ (((sup_sup @ (set @ A)) @ A5) @ B6)))))).
% 2.92/1.66 thf(fact_123_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)))))).
% 2.92/1.66 thf(fact_124_graph__union__idemp_I1_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ A5) = A5)))).
% 2.92/1.66 thf(fact_125_graph__union__idemp_I2_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ ((((graph_union @ A) @ B) @ A5) @ B6)) = ((((graph_union @ A) @ B) @ A5) @ B6))))).
% 2.92/1.66 thf(fact_126_graph__union__idemp_I3_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ ((((graph_union @ A) @ B) @ B6) @ A5)) = ((((graph_union @ A) @ B) @ B6) @ A5))))).
% 2.92/1.66 thf(fact_127_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.92/1.66 thf(fact_128_Un__empty, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A5) @ B6) = (bot_bot @ (set @ A))) = ((A5 = (bot_bot @ (set @ A))) & (B6 = (bot_bot @ (set @ A)))))))).
% 2.92/1.66 thf(fact_129_Un__insert__right, axiom, (! [A:$tType,A5:(set @ A),A2:A,B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A5) @ (((insert @ A) @ A2) @ B6)) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ A5) @ B6)))))).
% 2.92/1.66 thf(fact_130_Un__insert__left, axiom, (! [A:$tType,A2:A,B6:(set @ A),C8:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ B6)) @ C8) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ B6) @ C8)))))).
% 2.92/1.66 thf(fact_131_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)))))))).
% 2.92/1.66 thf(fact_132_graph__union__vertices, axiom, (! [A:$tType,B:$tType,G_1:((labeled_graph @ B) @ A),G_2:((labeled_graph @ B) @ A)]: (((((labeled_vertices @ B) @ A) @ ((((graph_union @ B) @ A) @ G_1) @ G_2)) = (((sup_sup @ (set @ A)) @ (((labeled_vertices @ B) @ A) @ G_1)) @ (((labeled_vertices @ B) @ A) @ G_2)))))).
% 2.92/1.66 thf(fact_133_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)))).
% 2.92/1.66 thf(fact_134_extensible__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F2:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R2) @ G) @ F2) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R2) @ R2)) @ G) @ F2))))).
% 2.92/1.66 thf(fact_135_graph__empty__e, axiom, (! [A:$tType,B:$tType,V3:(set @ B)]: ((((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V3) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V3)))))).
% 2.92/1.66 thf(fact_136_constant__rules__empty, axiom, (! [A:$tType]: ((((standa1138209853_rules @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)))))))).
% 2.92/1.66 thf(fact_137_Gr__insert, axiom, (! [B:$tType,A:$tType,X6:A,F4:(set @ A),F2:(A > B)]: ((((((bNF_Gr @ A) @ B) @ (((insert @ A) @ X6) @ F4)) @ F2) = (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ (F2 @ X6))) @ ((((bNF_Gr @ A) @ B) @ F4) @ F2)))))).
% 2.92/1.66 thf(fact_138_extensible__refl__concr, axiom, (! [A:$tType,C:$tType,B:$tType,E_1:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B))),V3:(set @ B),G:((labeled_graph @ A) @ C),F2:(set @ ((product_prod @ B) @ C)),E_2:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_1) @ V3)) @ G) @ F2) => (((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ ((((labeled_LG @ A) @ B) @ E_1) @ V3)) @ ((((labeled_LG @ A) @ B) @ E_2) @ V3))) @ G) @ F2) = ((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_2) @ V3)) @ G) @ F2)))))).
% 2.92/1.66 thf(fact_139_graph__single, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,C5:B]: ((((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A2) @ ((((product_Pair @ B) @ B) @ B2) @ C5))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B2) @ (((insert @ B) @ C5) @ (bot_bot @ (set @ B))))) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A2) @ ((((product_Pair @ B) @ B) @ B2) @ C5))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B2) @ (((insert @ B) @ C5) @ (bot_bot @ (set @ B)))))))))).
% 2.92/1.66 thf(fact_140_insert__is__Un, axiom, (! [A:$tType]: (((insert @ A) = (^ [A6:A]: (((sup_sup @ (set @ A)) @ (((insert @ A) @ A6) @ (bot_bot @ (set @ A)))))))))).
% 2.92/1.66 thf(fact_141_Un__singleton__iff, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),X6:A]: ((((((sup_sup @ (set @ A)) @ A5) @ B6) = (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) = (((A5 = (bot_bot @ (set @ A))) & (B6 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A))))) | (((A5 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) & (B6 = (bot_bot @ (set @ A)))) | ((A5 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) & (B6 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A))))))))))).
% 2.92/1.66 thf(fact_142_singleton__Un__iff, axiom, (! [A:$tType,X6:A,A5:(set @ A),B6:(set @ A)]: ((((((insert @ A) @ X6) @ (bot_bot @ (set @ A))) = (((sup_sup @ (set @ A)) @ A5) @ B6)) = (((A5 = (bot_bot @ (set @ A))) & (B6 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A))))) | (((A5 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) & (B6 = (bot_bot @ (set @ A)))) | ((A5 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) & (B6 = (((insert @ A) @ X6) @ (bot_bot @ (set @ A))))))))))).
% 2.92/1.66 thf(fact_143_graph__union__def, axiom, (! [B:$tType,A:$tType]: ((((graph_union @ A) @ B) = (^ [G_12:((labeled_graph @ A) @ B),G_22:((labeled_graph @ A) @ B)]: (((((labeled_LG @ A) @ B) @ (((sup_sup @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_12)) @ (((labeled_edges @ A) @ B) @ G_22))) @ (((sup_sup @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_12)) @ (((labeled_vertices @ A) @ B) @ G_22))))))))).
% 2.92/1.66 thf(fact_144_Un__empty__left, axiom, (! [A:$tType,B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B6) = B6)))).
% 2.92/1.66 thf(fact_145_Un__empty__right, axiom, (! [A:$tType,A5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = A5)))).
% 2.92/1.66 thf(fact_146_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 2.92/1.66 thf(fact_147_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 2.92/1.66 thf(fact_148_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C5:A,D3:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C5) @ (((insert @ A) @ D3) @ (bot_bot @ (set @ A))))) = (((A2 = C5) & (B2 = D3)) | ((A2 = D3) & (B2 = C5))))))).
% 2.92/1.66 thf(fact_149_insert__not__empty, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((insert @ A) @ A2) @ A5) != (bot_bot @ (set @ A)))))).
% 2.92/1.66 thf(fact_150_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 2.92/1.66 thf(fact_151_labeled__graph_Oexhaust, axiom, (! [L:$tType,V:$tType,Y:((labeled_graph @ L) @ V)]: ((~ ((! [X12:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X23:(set @ V)]: ((Y != ((((labeled_LG @ L) @ V) @ X12) @ X23))))))))).
% 2.92/1.66 thf(fact_152_labeled__graph_Oinduct, axiom, (! [V:$tType,L:$tType,P2:(((labeled_graph @ L) @ V) > $o),Labeled_graph:((labeled_graph @ L) @ V)]: (((! [X1a:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X2a:(set @ V)]: ((P2 @ ((((labeled_LG @ L) @ V) @ X1a) @ X2a)))) => (P2 @ Labeled_graph))))).
% 2.92/1.66 thf(fact_153_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ A5) => (? [B7:(set @ A)]: (((A5 = (((insert @ A) @ A2) @ B7)) & (~ ((((member @ A) @ A2) @ B7)))))))))).
% 2.92/1.66 thf(fact_154_Un__left__commute, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C8:(set @ A)]: (((((sup_sup @ (set @ A)) @ A5) @ (((sup_sup @ (set @ A)) @ B6) @ C8)) = (((sup_sup @ (set @ A)) @ B6) @ (((sup_sup @ (set @ A)) @ A5) @ C8)))))).
% 2.92/1.66 thf(fact_155_insert__commute, axiom, (! [A:$tType,X6:A,Y:A,A5:(set @ A)]: (((((insert @ A) @ X6) @ (((insert @ A) @ Y) @ A5)) = (((insert @ A) @ Y) @ (((insert @ A) @ X6) @ A5)))))).
% 2.92/1.66 thf(fact_156_Un__left__absorb, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A5) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) = (((sup_sup @ (set @ A)) @ A5) @ B6))))).
% 2.92/1.66 thf(fact_157_insert__eq__iff, axiom, (! [A:$tType,A2:A,A5:(set @ A),B2:A,B6:(set @ A)]: (((~ ((((member @ A) @ A2) @ A5))) => ((~ ((((member @ A) @ B2) @ B6))) => (((((insert @ A) @ A2) @ A5) = (((insert @ A) @ B2) @ B6)) = (((A2 = B2) => (A5 = B6)) & ((A2 != B2) => (? [C7:(set @ A)]: (((A5 = (((insert @ A) @ B2) @ C7)) & ((~ ((((member @ A) @ B2) @ C7))) & ((B6 = (((insert @ A) @ A2) @ C7)) & (~ ((((member @ A) @ A2) @ C7)))))))))))))))).
% 2.92/1.66 thf(fact_158_insert__absorb, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ A5) => ((((insert @ A) @ A2) @ A5) = A5))))).
% 2.92/1.66 thf(fact_159_insert__ident, axiom, (! [A:$tType,X6:A,A5:(set @ A),B6:(set @ A)]: (((~ ((((member @ A) @ X6) @ A5))) => ((~ ((((member @ A) @ X6) @ B6))) => (((((insert @ A) @ X6) @ A5) = (((insert @ A) @ X6) @ B6)) = (A5 = B6))))))).
% 2.92/1.66 thf(fact_160_Set_Oset__insert, axiom, (! [A:$tType,X6:A,A5:(set @ A)]: (((((member @ A) @ X6) @ A5) => (~ ((! [B7:(set @ A)]: (((A5 = (((insert @ A) @ X6) @ B7)) => (((member @ A) @ X6) @ B7)))))))))).
% 2.92/1.66 thf(fact_161_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A7:(set @ A),B8:(set @ A)]: ((((sup_sup @ (set @ A)) @ B8) @ A7))))))).
% 2.92/1.66 thf(fact_162_Un__absorb, axiom, (! [A:$tType,A5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A5) @ A5) = A5)))).
% 2.92/1.66 thf(fact_163_insertI2, axiom, (! [A:$tType,A2:A,B6:(set @ A),B2:A]: (((((member @ A) @ A2) @ B6) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B6)))))).
% 2.92/1.66 thf(fact_164_insertI1, axiom, (! [A:$tType,A2:A,B6:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B6))))).
% 2.92/1.66 thf(fact_165_Un__assoc, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C8:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) @ C8) = (((sup_sup @ (set @ A)) @ A5) @ (((sup_sup @ (set @ A)) @ B6) @ C8)))))).
% 2.92/1.66 thf(fact_166_insertE, axiom, (! [A:$tType,A2:A,B2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A5)) => ((A2 != B2) => (((member @ A) @ A2) @ A5)))))).
% 2.92/1.66 thf(fact_167_ball__Un, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),P2:(A > $o)]: (((! [X7:A]: (((((member @ A) @ X7) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) => (P2 @ X7)))) = ((! [X7:A]: (((((member @ A) @ X7) @ A5) => (P2 @ X7)))) & (! [X7:A]: (((((member @ A) @ X7) @ B6) => (P2 @ X7))))))))).
% 2.92/1.66 thf(fact_168_bex__Un, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),P2:(A > $o)]: (((? [X7:A]: (((((member @ A) @ X7) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) & (P2 @ X7)))) = ((? [X7:A]: (((((member @ A) @ X7) @ A5) & (P2 @ X7)))) | (? [X7:A]: (((((member @ A) @ X7) @ B6) & (P2 @ X7))))))))).
% 2.92/1.66 thf(fact_169_UnI2, axiom, (! [A:$tType,C5:A,B6:(set @ A),A5:(set @ A)]: (((((member @ A) @ C5) @ B6) => (((member @ A) @ C5) @ (((sup_sup @ (set @ A)) @ A5) @ B6)))))).
% 2.92/1.66 thf(fact_170_UnI1, axiom, (! [A:$tType,C5:A,A5:(set @ A),B6:(set @ A)]: (((((member @ A) @ C5) @ A5) => (((member @ A) @ C5) @ (((sup_sup @ (set @ A)) @ A5) @ B6)))))).
% 2.92/1.66 thf(fact_171_UnE, axiom, (! [A:$tType,C5:A,A5:(set @ A),B6:(set @ A)]: (((((member @ A) @ C5) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) => ((~ ((((member @ A) @ C5) @ A5))) => (((member @ A) @ C5) @ B6)))))).
% 2.92/1.66 thf(fact_172_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)))).
% 2.92/1.66 thf(fact_173_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)))).
% 2.92/1.66 thf(fact_174_graph__homomorphism__semantics, axiom, (! [B:$tType,C:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ C),F2:(set @ ((product_prod @ B) @ C)),A2:B,B2:B,E3:(allegorical_term @ A),A3:C,B3:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ B6) @ F2) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ A2) @ A3)) @ F2) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B2) @ B3)) @ F2) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ A3) @ B3)) @ ((((semantics @ A) @ C) @ B6) @ E3))))))))).
% 2.92/1.66 thf(fact_175_graph__homomorphism__nonempty, axiom, (! [B:$tType,A:$tType,C:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ C),F2:(set @ ((product_prod @ B) @ C)),E3:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ B6) @ F2) => ((((((semantics @ A) @ B) @ A5) @ E3) != (bot_bot @ (set @ ((product_prod @ B) @ B)))) => (((((semantics @ A) @ C) @ B6) @ E3) != (bot_bot @ (set @ ((product_prod @ C) @ C))))))))).
% 2.92/1.66 thf(fact_176_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)))))).
% 2.92/1.66 thf(fact_177_standard__rules__def, axiom, (! [A:$tType]: (((standa438229444_rules @ A) = (^ [C7:(set @ A),L4:(set @ (standard_Constant @ A))]: ((((sup_sup @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)))) @ (((sup_sup @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat)))) @ ((standa1138209853_rules @ A) @ C7)) @ ((standa2002409347_rules @ A) @ L4))) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))) @ ((standa400804411p_rule @ (standard_Constant @ A)) @ (standard_S_Top @ A))) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))) @ (standa244753842y_rule @ (standard_Constant @ A))) @ (bot_bot @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))))))))))))).
% 2.92/1.66 thf(fact_178_semantics__in__vertices_I2_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: (((A5 = (((restrict @ A) @ B) @ A5)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ B) @ B2) @ (((labeled_vertices @ A) @ B) @ A5))))))).
% 2.92/1.66 thf(fact_179_semantics__in__vertices_I1_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: (((A5 = (((restrict @ A) @ B) @ A5)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ B) @ A2) @ (((labeled_vertices @ A) @ B) @ A5))))))).
% 2.92/1.66 thf(fact_180_subgraph__semantics, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B6) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ B6) @ E3))))))).
% 2.92/1.66 thf(fact_181_maintainedD, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B6:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F2:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B6)) @ G) => (((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ G) @ F2) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B6)) @ G) @ F2)))))).
% 2.92/1.66 thf(fact_182_sup__bot_Oright__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ (bot_bot @ A)) = A2))))))).
% 2.92/1.66 thf(fact_183_sup__bot_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A,B2:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ A2) @ B2)) = ((A2 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))).
% 2.92/1.66 thf(fact_184_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A2) = A2))))))).
% 2.92/1.66 thf(fact_185_sup__bot_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A,B2:A]: ((((((sup_sup @ A) @ A2) @ B2) = (bot_bot @ A)) = ((A2 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))).
% 2.92/1.66 thf(fact_186_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ B2) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.92/1.66 thf(fact_187_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,Y:A]: (((((sup_sup @ A) @ X6) @ (((sup_sup @ A) @ X6) @ Y)) = (((sup_sup @ A) @ X6) @ Y)))))))).
% 2.92/1.66 thf(fact_188_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.92/1.66 thf(fact_189_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A]: (((((sup_sup @ A) @ X6) @ X6) = X6))))))).
% 2.92/1.66 thf(fact_190_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.92/1.66 thf(fact_191_sup__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F5:(A > B),G6:(A > B),X7:A]: ((((sup_sup @ B) @ (F5 @ X7)) @ (G6 @ X7))))))))).
% 2.92/1.66 thf(fact_192_sup__bot__left, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X6:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X6) = X6))))))).
% 2.92/1.66 thf(fact_193_sup__bot__right, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X6:A]: (((((sup_sup @ A) @ X6) @ (bot_bot @ A)) = X6))))))).
% 2.92/1.66 thf(fact_194_bot__eq__sup__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X6:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X6) @ Y)) = ((X6 = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.92/1.66 thf(fact_195_sup__eq__bot__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X6:A,Y:A]: ((((((sup_sup @ A) @ X6) @ Y) = (bot_bot @ A)) = ((X6 = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.92/1.66 thf(fact_196_edge__preserving__unionI, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(set @ ((product_prod @ A) @ B)),A5:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A))),G:(set @ ((product_prod @ C) @ ((product_prod @ B) @ B))),B6:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A)))]: ((((((((edge_preserving @ A) @ B) @ C) @ F2) @ A5) @ G) => (((((((edge_preserving @ A) @ B) @ C) @ F2) @ B6) @ G) => ((((((edge_preserving @ A) @ B) @ C) @ F2) @ (((sup_sup @ (set @ ((product_prod @ C) @ ((product_prod @ A) @ A)))) @ A5) @ B6)) @ G)))))).
% 2.92/1.66 thf(fact_197_graph__union__edges, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((((labeled_edges @ A) @ B) @ ((((graph_union @ A) @ B) @ G_1) @ G_2)) = (((sup_sup @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2)))))).
% 2.92/1.66 thf(fact_198_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,Y:A,Z:A]: (((((sup_sup @ A) @ X6) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X6) @ Z))))))))).
% 2.92/1.66 thf(fact_199_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A,C5:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A2) @ C5)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C5))))))))).
% 2.92/1.66 thf(fact_200_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X7:A,Y6:A]: ((((sup_sup @ A) @ Y6) @ X7)))))))).
% 2.92/1.66 thf(fact_201_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A6:A,B9:A]: ((((sup_sup @ A) @ B9) @ A6)))))))).
% 2.92/1.66 thf(fact_202_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X6) @ Y)) @ Z) = (((sup_sup @ A) @ X6) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.92/1.66 thf(fact_203_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A,C5:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ C5) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C5))))))))).
% 2.92/1.66 thf(fact_204_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B6:A,K:A,B2:A,A2:A]: (((B6 = (((sup_sup @ A) @ K) @ B2)) => ((((sup_sup @ A) @ A2) @ B6) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B2)))))))))).
% 2.92/1.66 thf(fact_205_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A5:A,K:A,A2:A,B2:A]: (((A5 = (((sup_sup @ A) @ K) @ A2)) => ((((sup_sup @ A) @ A5) @ B2) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B2)))))))))).
% 2.92/1.66 thf(fact_206_sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F5:(A > B),G6:(A > B),X7:A]: ((((sup_sup @ B) @ (F5 @ X7)) @ (G6 @ X7))))))))).
% 2.92/1.66 thf(fact_207_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X7:A,Y6:A]: ((((sup_sup @ A) @ Y6) @ X7)))))))).
% 2.92/1.66 thf(fact_208_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X6:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X6) @ Y)) @ Z) = (((sup_sup @ A) @ X6) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.92/1.66 thf(fact_209_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X6:A,Y:A,Z:A]: (((((sup_sup @ A) @ X6) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X6) @ Z))))))))).
% 2.92/1.66 thf(fact_210_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X6:A,Y:A]: (((((sup_sup @ A) @ X6) @ (((sup_sup @ A) @ X6) @ Y)) = (((sup_sup @ A) @ X6) @ Y)))))))).
% 2.92/1.66 thf(fact_211_the__elem__eq, axiom, (! [A:$tType,X6:A]: ((((the_elem @ A) @ (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) = X6)))).
% 2.92/1.66 thf(fact_212_Collect__empty__eq__bot, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (P2 = (bot_bot @ (A > $o))))))).
% 2.92/1.66 thf(fact_213_is__singletonI, axiom, (! [A:$tType,X6:A]: (((is_singleton @ A) @ (((insert @ A) @ X6) @ (bot_bot @ (set @ A))))))).
% 2.92/1.66 thf(fact_214_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((A7 = (((insert @ A) @ ((the_elem @ A) @ A7)) @ (bot_bot @ (set @ A)))))))))).
% 2.92/1.66 thf(fact_215_is__singletonI_H, axiom, (! [A:$tType,A5:(set @ A)]: (((A5 != (bot_bot @ (set @ A))) => ((! [X2:A,Y3:A]: (((((member @ A) @ X2) @ A5) => ((((member @ A) @ Y3) @ A5) => (X2 = Y3))))) => ((is_singleton @ A) @ A5)))))).
% 2.92/1.66 thf(fact_216_is__singletonE, axiom, (! [A:$tType,A5:(set @ A)]: ((((is_singleton @ A) @ A5) => (~ ((! [X2:A]: ((A5 != (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 2.92/1.66 thf(fact_217_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((? [X7:A]: ((A7 = (((insert @ A) @ X7) @ (bot_bot @ (set @ A)))))))))))).
% 2.92/1.66 thf(fact_218_labeled__graph_Osplit__sel, axiom, (! [A:$tType,V:$tType,L:$tType,P2:(A > $o),F2:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P2 @ (((((labele1974067554_graph @ L) @ V) @ A) @ F2) @ Labeled_graph)) = ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) => (P2 @ ((F2 @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))).
% 2.92/1.66 thf(fact_219_labeled__graph_Osplit__sel__asm, axiom, (! [A:$tType,V:$tType,L:$tType,P2:(A > $o),F2:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P2 @ (((((labele1974067554_graph @ L) @ V) @ A) @ F2) @ Labeled_graph)) = (~ (((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) & (~ ((P2 @ ((F2 @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))))))).
% 2.92/1.66 thf(fact_220_labeled__graph_Ocase, axiom, (! [L:$tType,A:$tType,V:$tType,F2:((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) @ F2) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = ((F2 @ X1) @ X22))))).
% 2.92/1.66 thf(fact_221_labeled__graph_Ocase__eq__if, axiom, (! [A:$tType,V:$tType,L:$tType]: (((((labele1974067554_graph @ L) @ V) @ A) = (^ [F5:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph3:((labeled_graph @ L) @ V)]: (((F5 @ (((labeled_edges @ L) @ V) @ Labeled_graph3)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph3)))))))).
% 2.92/1.66 thf(fact_222_extensibleI, axiom, (! [A:$tType,C:$tType,B:$tType,R22:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),G5:(set @ ((product_prod @ B) @ C)),R1:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R22) @ G) @ G5) => (((((((agree_on @ A) @ B) @ C) @ R1) @ F2) @ G5) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R1) @ R22)) @ G) @ F2)))))).
% 2.92/1.66 thf(fact_223_total__on__singleton, axiom, (! [A:$tType,X6:A]: ((((total_on @ A) @ (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X6) @ X6)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.92/1.66 thf(fact_224_agree__on__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R2:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ R2) @ F2) @ F2)))).
% 2.92/1.66 thf(fact_225_agree__on__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((agree_on @ A) @ B) @ C) = (^ [X9:((labeled_graph @ A) @ B),F5:(set @ ((product_prod @ B) @ C)),G6:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ X9) @ G6) @ F5))))))).
% 2.92/1.66 thf(fact_226_agree__on__empty, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(set @ ((product_prod @ B) @ C)),G5:(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)))) @ F2) @ G5)))).
% 2.92/1.66 thf(fact_227_total__onI, axiom, (! [A:$tType,A5:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X2:A,Y3:A]: (((((member @ A) @ X2) @ A5) => ((((member @ A) @ Y3) @ A5) => ((X2 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y3)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X2)) @ R))))))) => (((total_on @ A) @ A5) @ R))))).
% 2.92/1.66 thf(fact_228_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A7:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X7:A]: (((((member @ A) @ X7) @ A7) => (! [Y6:A]: (((((member @ A) @ Y6) @ A7) => ((X7 != Y6) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X7) @ Y6)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X7)) @ R3))))))))))))))).
% 2.92/1.66 thf(fact_229_total__on__empty, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R)))).
% 2.92/1.66 thf(fact_230_agree__on__trans, axiom, (! [A:$tType,C:$tType,B:$tType,X8:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C)),G5:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ X8) @ F2) @ G5) => (((((((agree_on @ A) @ B) @ C) @ X8) @ G5) @ H) => ((((((agree_on @ A) @ B) @ C) @ X8) @ F2) @ H)))))).
% 2.92/1.66 thf(fact_231_agree__on__subg__compose, axiom, (! [A:$tType,C:$tType,B:$tType,R2:((labeled_graph @ A) @ B),G5:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C)),F4:((labeled_graph @ A) @ B),F2:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ R2) @ G5) @ H) => (((((((agree_on @ A) @ B) @ C) @ F4) @ F2) @ G5) => (((((((graph_homomorphism @ A) @ B) @ B) @ F4) @ R2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ F4))) => ((((((agree_on @ A) @ B) @ C) @ F4) @ F2) @ H))))))).
% 2.92/1.66 thf(fact_232_Field__insert, axiom, (! [A:$tType,A2:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: ((((field @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R)) = (((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((field @ A) @ R)))))).
% 2.92/1.66 thf(fact_233_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)))))).
% 2.92/1.66 thf(fact_234_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X6:A]: ((((ord_less_eq @ A) @ X6) @ X6))))))).
% 2.92/1.66 thf(fact_235_subsetI, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B6))))).
% 2.92/1.66 thf(fact_236_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A5) => (A5 = B6)))))).
% 2.92/1.66 thf(fact_237_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X6) @ Y)) @ Z) = ((((ord_less_eq @ A) @ X6) @ Z) & (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.92/1.66 thf(fact_238_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C5:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C5)) @ A2) = ((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ C5) @ A2))))))))).
% 2.92/1.66 thf(fact_239_empty__subsetI, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A5)))).
% 2.92/1.66 thf(fact_240_subset__empty, axiom, (! [A:$tType,A5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.92/1.66 thf(fact_241_insert__subset, axiom, (! [A:$tType,X6:A,A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X6) @ A5)) @ B6) = ((((member @ A) @ X6) @ B6) & (((ord_less_eq @ (set @ A)) @ A5) @ B6)))))).
% 2.92/1.66 thf(fact_242_Un__subset__iff, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A5) @ B6)) @ C8) = ((((ord_less_eq @ (set @ A)) @ A5) @ C8) & (((ord_less_eq @ (set @ A)) @ B6) @ C8)))))).
% 2.92/1.66 thf(fact_243_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A5:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A5)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.66 thf(fact_244_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A5:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A5) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.66 thf(fact_245_Field__empty, axiom, (! [A:$tType]: ((((field @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) = (bot_bot @ (set @ A)))))).
% 2.92/1.66 thf(fact_246_Field__Un, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: ((((field @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R) @ S3)) = (((sup_sup @ (set @ A)) @ ((field @ A) @ R)) @ ((field @ A) @ S3)))))).
% 2.92/1.66 thf(fact_247_graph__unionI, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2)) => ((((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)) => (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2)))))).
% 2.92/1.66 thf(fact_248_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y:A,X6:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X6) @ Y)))))))).
% 2.92/1.66 thf(fact_249_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X6:A,Y:A]: ((((ord_less_eq @ A) @ X6) @ (((sup_sup @ A) @ X6) @ Y)))))))).
% 2.92/1.66 thf(fact_250_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A,X6:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X6) => (~ (((((ord_less_eq @ A) @ A2) @ X6) => (~ ((((ord_less_eq @ A) @ B2) @ X6))))))))))))).
% 2.92/1.66 thf(fact_251_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,X6:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X6) => ((((ord_less_eq @ A) @ B2) @ X6) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X6))))))))).
% 2.92/1.66 thf(fact_252_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,Y:A]: ((((ord_less_eq @ A) @ X6) @ (((sup_sup @ A) @ X6) @ Y)))))))).
% 2.92/1.66 thf(fact_253_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X6:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X6) @ Y)))))))).
% 2.92/1.66 thf(fact_254_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X6) @ A2) => (((ord_less_eq @ A) @ X6) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.92/1.66 thf(fact_255_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X6:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ X6) @ B2) => (((ord_less_eq @ A) @ X6) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.92/1.66 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 2.92/1.66 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A8:$tType]: ((bounded_lattice @ (set @ A8))))).
% 2.92/1.66 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice @ (A8 > A9)))))).
% 2.92/1.66 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounde1808546759up_bot @ (A8 > A9)))))).
% 2.92/1.66 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_sup @ A9) => (semilattice_sup @ (A8 > A9)))))).
% 2.92/1.66 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.92/1.66 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: (((lattice @ A9) => (lattice @ (A8 > A9)))))).
% 2.92/1.66 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 2.92/1.66 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3, axiom, (semilattice_sup @ nat)).
% 2.92/1.66 thf(tcon_Nat_Onat___Orderings_Opreorder_4, axiom, (preorder @ nat)).
% 2.92/1.66 thf(tcon_Nat_Onat___Lattices_Olattice_5, axiom, (lattice @ nat)).
% 2.92/1.66 thf(tcon_Nat_Onat___Orderings_Obot_6, axiom, (bot @ nat)).
% 2.92/1.66 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_7, axiom, (! [A8:$tType]: ((bounde1808546759up_bot @ (set @ A8))))).
% 2.92/1.66 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8, axiom, (! [A8:$tType]: ((semilattice_sup @ (set @ A8))))).
% 2.92/1.66 thf(tcon_Set_Oset___Orderings_Opreorder_9, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.92/1.66 thf(tcon_Set_Oset___Lattices_Olattice_10, axiom, (! [A8:$tType]: ((lattice @ (set @ A8))))).
% 2.92/1.66 thf(tcon_Set_Oset___Orderings_Obot_11, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 2.92/1.66 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12, axiom, (bounde1808546759up_bot @ $o)).
% 2.92/1.66 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_13, axiom, (semilattice_sup @ $o)).
% 2.92/1.66 thf(tcon_HOL_Obool___Orderings_Opreorder_14, axiom, (preorder @ $o)).
% 2.92/1.66 thf(tcon_HOL_Obool___Lattices_Olattice_15, axiom, (lattice @ $o)).
% 2.92/1.66 thf(tcon_HOL_Obool___Orderings_Obot_16, axiom, (bot @ $o)).
% 2.92/1.66 thf(conj_0, conjecture, (((member @ ((product_prod @ v) @ v)) @ ((((product_Pair @ v) @ v) @ (m @ xa)) @ (m @ xa))) @ ((((getRel @ (standard_Constant @ v2)) @ v) @ ((standard_S_Const @ v2) @ y)) @ g))).
% 2.92/1.66 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.TNBOsnf0Kz/DTF2DTF31760.p
% 2.92/1.66 ------------------------
% 2.92/1.69 ---- Cleaned THF ---
% 2.92/1.69 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 2.92/1.69 standard_Constant: $tType > $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 2.92/1.69 allegorical_term: $tType > $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 2.92/1.69 labeled_graph: $tType > $tType > $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_t_Product__Type_Oprod,type,
% 2.92/1.69 product_prod: $tType > $tType > $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_t_Set_Oset,type,
% 2.92/1.69 set: $tType > $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_t_Nat_Onat,type,
% 2.92/1.69 nat: $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_t_itself,type,
% 2.92/1.69 itself: $tType > $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_tf_V_H,type,
% 2.92/1.69 v: $tType ).
% 2.92/1.69
% 2.92/1.69 thf(ty_tf_V,type,
% 2.92/1.69 v2: $tType ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.92/1.69 bounded_lattice:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_HOL_Otype,type,
% 2.92/1.69 type:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_Orderings_Obot,type,
% 2.92/1.69 bot:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_Lattices_Olattice,type,
% 2.92/1.69 lattice:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_Orderings_Opreorder,type,
% 2.92/1.69 preorder:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.92/1.69 semilattice_sup:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 2.92/1.69 bounde1808546759up_bot:
% 2.92/1.69 !>[A: $tType] : $o ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_BNF__Def_OGr,type,
% 2.92/1.69 bNF_Gr:
% 2.92/1.69 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Fun_Ocomp,type,
% 2.92/1.69 comp:
% 2.92/1.69 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Bot,type,
% 2.92/1.69 standard_S_Bot:
% 2.92/1.69 !>[V: $tType] : ( standard_Constant @ V ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Const,type,
% 2.92/1.69 standard_S_Const:
% 2.92/1.69 !>[V: $tType] : ( V > ( standard_Constant @ V ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Top,type,
% 2.92/1.69 standard_S_Top:
% 2.92/1.69 !>[V: $tType] : ( standard_Constant @ V ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 2.92/1.69 getRel:
% 2.92/1.69 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 2.92/1.69 semantics:
% 2.92/1.69 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Oedge__preserving,type,
% 2.92/1.69 edge_preserving:
% 2.92/1.69 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) > ( set @ ( product_prod @ C @ ( product_prod @ B @ B ) ) ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 2.92/1.69 graph_homomorphism:
% 2.92/1.69 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 2.92/1.69 graph_union:
% 2.92/1.69 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 2.92/1.69 labeled_LG:
% 2.92/1.69 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 2.92/1.69 labele1974067554_graph:
% 2.92/1.69 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 2.92/1.69 labeled_edges:
% 2.92/1.69 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 2.92/1.69 labeled_vertices:
% 2.92/1.69 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Omap__graph,type,
% 2.92/1.69 map_graph:
% 2.92/1.69 !>[C: $tType,B: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ B ) ) > ( labeled_graph @ A @ C ) > ( labeled_graph @ A @ B ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_LabeledGraphs_Orestrict,type,
% 2.92/1.69 restrict:
% 2.92/1.69 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.92/1.69 sup_sup:
% 2.92/1.69 !>[A: $tType] : ( A > A > A ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.92/1.69 bot_bot:
% 2.92/1.69 !>[A: $tType] : A ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.92/1.69 ord_less_eq:
% 2.92/1.69 !>[A: $tType] : ( A > A > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Product__Type_OPair,type,
% 2.92/1.69 product_Pair:
% 2.92/1.69 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.92/1.69 product_rec_prod:
% 2.92/1.69 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Relation_OField,type,
% 2.92/1.69 field:
% 2.92/1.69 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Relation_OId__on,type,
% 2.92/1.69 id_on:
% 2.92/1.69 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Relation_Ototal__on,type,
% 2.92/1.69 total_on:
% 2.92/1.69 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_RulesAndChains_Oagree__on,type,
% 2.92/1.69 agree_on:
% 2.92/1.69 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 2.92/1.69 consequence_graph:
% 2.92/1.69 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_RulesAndChains_Oextensible,type,
% 2.92/1.69 extensible:
% 2.92/1.69 !>[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 ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_RulesAndChains_Oleast,type,
% 2.92/1.69 least:
% 2.92/1.69 !>[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 ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 2.92/1.69 least_559130134_graph:
% 2.92/1.69 !>[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 ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_RulesAndChains_Omaintained,type,
% 2.92/1.69 maintained:
% 2.92/1.69 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Set_OCollect,type,
% 2.92/1.69 collect:
% 2.92/1.69 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Set_Oinsert,type,
% 2.92/1.69 insert:
% 2.92/1.69 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Set_Ois__singleton,type,
% 2.92/1.69 is_singleton:
% 2.92/1.69 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_Set_Othe__elem,type,
% 2.92/1.69 the_elem:
% 2.92/1.69 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oconstant__rules,type,
% 2.92/1.69 standa1138209853_rules:
% 2.92/1.69 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
% 2.92/1.69 standa2002409347_rules:
% 2.92/1.69 !>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 2.92/1.69 standa244753842y_rule:
% 2.92/1.69 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ostandard__rules,type,
% 2.92/1.69 standa438229444_rules:
% 2.92/1.69 !>[A: $tType] : ( ( set @ A ) > ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otop__rule,type,
% 2.92/1.69 standa400804411p_rule:
% 2.92/1.69 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_c_member,type,
% 2.92/1.69 member:
% 2.92/1.69 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_C,type,
% 2.92/1.69 c: set @ v2 ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_G_H,type,
% 2.92/1.69 g: labeled_graph @ ( standard_Constant @ v2 ) @ v ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_L,type,
% 2.92/1.69 l: set @ ( standard_Constant @ v2 ) ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_h____,type,
% 2.92/1.69 h: v > v ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_m____,type,
% 2.92/1.69 m: v2 > v ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_x____,type,
% 2.92/1.69 x: v ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_xa____,type,
% 2.92/1.69 xa: v2 ).
% 2.92/1.69
% 2.92/1.69 thf(sy_v_y____,type,
% 2.92/1.69 y: v2 ).
% 2.92/1.69
% 2.92/1.69 thf(fact_0_x,axiom,
% 2.92/1.69 member @ v @ x @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.92/1.69 ~ ! [X2: v] :
% 2.92/1.69 ~ ( member @ v @ X2 @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_2__C1_C_I3_J,axiom,
% 2.92/1.69 ( ( m @ xa )
% 2.92/1.69 = ( m @ y ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_3__C1_C_I1_J,axiom,
% 2.92/1.69 member @ v2 @ xa @ c ).
% 2.92/1.69
% 2.92/1.69 thf(fact_4__C1_C_I2_J,axiom,
% 2.92/1.69 member @ v2 @ y @ c ).
% 2.92/1.69
% 2.92/1.69 thf(fact_5__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 2.92/1.69 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ xa ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_6__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ay_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 2.92/1.69 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ y ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_7_gr_I1_J,axiom,
% 2.92/1.69 ( g
% 2.92/1.69 = ( restrict @ ( standard_Constant @ v2 ) @ v @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_8_h_I3_J,axiom,
% 2.92/1.69 graph_homomorphism @ ( standard_Constant @ v2 ) @ v @ v @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) @ g @ ( id_on @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_9_Standard__Constant_Oinject,axiom,
% 2.92/1.69 ! [V: $tType,X4: V,Y4: V] :
% 2.92/1.69 ( ( ( standard_S_Const @ V @ X4 )
% 2.92/1.69 = ( standard_S_Const @ V @ Y4 ) )
% 2.92/1.69 = ( X4 = Y4 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_10_prod_Oinject,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.92/1.69 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.92/1.69 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.92/1.69 = ( ( X1 = Y1 )
% 2.92/1.69 & ( X22 = Y2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_11_old_Oprod_Oinject,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.92/1.69 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.92/1.69 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.92/1.69 = ( ( A2 = A3 )
% 2.92/1.69 & ( B2 = B3 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_12_m,axiom,
% 2.92/1.69 ! [X3: v2] :
% 2.92/1.69 ( ( member @ v2 @ X3 @ c )
% 2.92/1.69 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ X3 ) @ ( m @ X3 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X3 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_13_top__nonempty,axiom,
% 2.92/1.69 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ x @ x ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Top @ v2 ) @ g ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_14_cf,axiom,
% 2.92/1.69 ( ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Bot @ v2 ) @ g )
% 2.92/1.69 = ( bot_bot @ ( set @ ( product_prod @ v @ v ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_15__092_060open_062_092_060And_062c_O_Ac_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_092_060exists_062v_O_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 2.92/1.69 ! [C2: v2] :
% 2.92/1.69 ( ( member @ v2 @ C2 @ c )
% 2.92/1.69 => ? [V2: v] : ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ V2 @ V2 ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ C2 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_16__092_060open_062_092_060forall_062c_O_A_092_060exists_062v_O_Ac_A_092_060in_062_AC_A_092_060longrightarrow_062_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 2.92/1.69 ! [C3: v2] :
% 2.92/1.69 ? [V2: v] :
% 2.92/1.69 ( ( member @ v2 @ C3 @ c )
% 2.92/1.69 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ V2 @ V2 ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ C3 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_17__092_060open_062_092_060exists_062f_O_A_092_060forall_062x_O_Ax_A_092_060in_062_AC_A_092_060longrightarrow_062_A_If_Ax_M_Af_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
% 2.92/1.69 ? [F: v2 > v] :
% 2.92/1.69 ! [X5: v2] :
% 2.92/1.69 ( ( member @ v2 @ X5 @ c )
% 2.92/1.69 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( F @ X5 ) @ ( F @ X5 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X5 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.92/1.69 ~ ! [M: v2 > v] :
% 2.92/1.69 ~ ! [X5: v2] :
% 2.92/1.69 ( ( member @ v2 @ X5 @ c )
% 2.92/1.69 => ( member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( M @ X5 ) @ ( M @ X5 ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ X5 ) @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_19_getRel__homR,axiom,
% 2.92/1.69 ! [C: $tType,A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,U: C,F2: set @ ( product_prod @ A @ C ),V3: C] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Y @ U ) @ F2 )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Z @ V3 ) @ F2 )
% 2.92/1.69 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ U @ V3 ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ F2 @ G ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_20_surj__pair,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.92/1.69 ? [X2: A,Y3: B] :
% 2.92/1.69 ( P
% 2.92/1.69 = ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_21_h_I1_J,axiom,
% 2.92/1.69 ( ( comp @ v @ v @ v @ h @ h )
% 2.92/1.69 = h ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_22_getRel__map__fn,axiom,
% 2.92/1.69 ! [C: $tType,A: $tType,B: $tType,A22: A,G: labeled_graph @ B @ A,B22: A,L2: B,F2: A > C,A2: C,B2: C] :
% 2.92/1.69 ( ( member @ A @ A22 @ ( labeled_vertices @ B @ A @ G ) )
% 2.92/1.69 => ( ( member @ A @ B22 @ ( labeled_vertices @ B @ A @ G ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B22 ) @ ( getRel @ B @ A @ L2 @ G ) )
% 2.92/1.69 => ( ( ( F2 @ A22 )
% 2.92/1.69 = A2 )
% 2.92/1.69 => ( ( ( F2 @ B22 )
% 2.92/1.69 = B2 )
% 2.92/1.69 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A2 @ B2 ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_23_getRel__hom,axiom,
% 2.92/1.69 ! [C: $tType,A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,F2: A > C] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
% 2.92/1.69 => ( ( member @ A @ Y @ ( labeled_vertices @ B @ A @ G ) )
% 2.92/1.69 => ( ( member @ A @ Z @ ( labeled_vertices @ B @ A @ G ) )
% 2.92/1.69 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel @ B @ C @ L2 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_24_Standard__Constant_Odistinct_I1_J,axiom,
% 2.92/1.69 ! [V: $tType] :
% 2.92/1.69 ( ( standard_S_Top @ V )
% 2.92/1.69 != ( standard_S_Bot @ V ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_25_Standard__Constant_Odistinct_I5_J,axiom,
% 2.92/1.69 ! [V: $tType,X4: V] :
% 2.92/1.69 ( ( standard_S_Top @ V )
% 2.92/1.69 != ( standard_S_Const @ V @ X4 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_26_Standard__Constant_Odistinct_I9_J,axiom,
% 2.92/1.69 ! [V: $tType,X4: V] :
% 2.92/1.69 ( ( standard_S_Bot @ V )
% 2.92/1.69 != ( standard_S_Const @ V @ X4 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_27_getRel__dom_I1_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 2.92/1.69 ( ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G ) )
% 2.92/1.69 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_28_getRel__dom_I2_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 2.92/1.69 ( ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G ) )
% 2.92/1.69 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_29_getRel__subgraph,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,Y: A,Z: A,L2: B,G: labeled_graph @ B @ A,G2: labeled_graph @ B @ A] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G ) )
% 2.92/1.69 => ( ( graph_homomorphism @ B @ A @ A @ G @ G2 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G ) ) )
% 2.92/1.69 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( getRel @ B @ A @ L2 @ G2 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_30_old_Oprod_Oinducts,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.92/1.69 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.92/1.69 => ( P2 @ Prod ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_31_old_Oprod_Oexhaust,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.92/1.69 ~ ! [A4: A,B4: B] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_32_prod__induct7,axiom,
% 2.92/1.69 ! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 2.92/1.69 ( ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F @ G4 ) ) ) ) ) ) )
% 2.92/1.69 => ( P2 @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_33_prod__induct6,axiom,
% 2.92/1.69 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 2.92/1.69 ( ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) )
% 2.92/1.69 => ( P2 @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_34_prod__induct5,axiom,
% 2.92/1.69 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.92/1.69 ( ! [A4: A,B4: B,C4: C,D2: D,E2: E] : ( P2 @ ( 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 ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.92/1.69 => ( P2 @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_35_prod__induct4,axiom,
% 2.92/1.69 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.92/1.69 ( ! [A4: A,B4: B,C4: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
% 2.92/1.69 => ( P2 @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_36_prod__induct3,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ! [A4: A,B4: B,C4: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
% 2.92/1.69 => ( P2 @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_37_prod__cases7,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 2.92/1.69 ~ ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3,G4: G3] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F @ G4 ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_38_prod__cases6,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 2.92/1.69 ~ ! [A4: A,B4: B,C4: C,D2: D,E2: E,F: F3] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_39_prod__cases5,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.92/1.69 ~ ! [A4: A,B4: B,C4: C,D2: D,E2: E] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( 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 ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_40_prod__cases4,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.92/1.69 ~ ! [A4: A,B4: B,C4: C,D2: D] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_41_prod__cases3,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.92/1.69 ~ ! [A4: A,B4: B,C4: C] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_42_Pair__inject,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.92/1.69 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.92/1.69 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.92/1.69 => ~ ( ( A2 = A3 )
% 2.92/1.69 => ( B2 != B3 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_43_prod__cases,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.92/1.69 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.92/1.69 => ( P2 @ P ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_44_mg,axiom,
% 2.92/1.69 ! [B5: $tType,R: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ B5 ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ B5 )] :
% 2.92/1.69 ( ( maintained @ ( standard_Constant @ v2 ) @ B5 @ v @ R @ g )
% 2.92/1.69 => ( maintained @ ( standard_Constant @ v2 ) @ B5 @ v @ R @ ( map_graph @ v @ v @ ( standard_Constant @ v2 ) @ ( bNF_Gr @ v @ v @ ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g ) @ h ) @ g ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_45_mem__Collect__eq,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,P2: A > $o] :
% 2.92/1.69 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.92/1.69 = ( P2 @ A2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_46_Collect__mem__eq,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( collect @ A
% 2.92/1.69 @ ^ [X7: A] : ( member @ A @ X7 @ A5 ) )
% 2.92/1.69 = A5 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_47_Collect__cong,axiom,
% 2.92/1.69 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.92/1.69 ( ! [X2: A] :
% 2.92/1.69 ( ( P2 @ X2 )
% 2.92/1.69 = ( Q @ X2 ) )
% 2.92/1.69 => ( ( collect @ A @ P2 )
% 2.92/1.69 = ( collect @ A @ Q ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_48_ext,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,F2: A > B,G5: A > B] :
% 2.92/1.69 ( ! [X2: A] :
% 2.92/1.69 ( ( F2 @ X2 )
% 2.92/1.69 = ( G5 @ X2 ) )
% 2.92/1.69 => ( F2 = G5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_49_graph__homo,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,F2: B > C] :
% 2.92/1.69 ( ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) )
% 2.92/1.69 => ( graph_homomorphism @ A @ B @ C @ G @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_50_map__graph__fn__id_I2_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,X8: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( map_graph @ B @ B @ A @ ( id_on @ B @ ( labeled_vertices @ A @ B @ X8 ) ) @ X8 )
% 2.92/1.69 = ( restrict @ A @ B @ X8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_51_map__graph__fn__graphI,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: C > B] :
% 2.92/1.69 ( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F2 ) @ G )
% 2.92/1.69 = ( restrict @ A @ B @ ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F2 ) @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_52_map__graph__preserves__subgraph,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 2.92/1.69 => ( graph_homomorphism @ A @ C @ C @ ( map_graph @ B @ C @ A @ F2 @ A5 ) @ ( map_graph @ B @ C @ A @ F2 @ B6 ) @ ( id_on @ C @ ( labeled_vertices @ A @ C @ ( map_graph @ B @ C @ A @ F2 @ A5 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_53_subgraph__refl,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 2.92/1.69 = ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_54_subgraph__restrict,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 2.92/1.69 = ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_55_graph__homomorphism__Id,axiom,
% 2.92/1.69 ! [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 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_56_Id__on__empty,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_57_Gr__empty,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,F2: A > B] :
% 2.92/1.69 ( ( bNF_Gr @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F2 )
% 2.92/1.69 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_58_restrict__idemp,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,X6: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X6 ) )
% 2.92/1.69 = ( restrict @ A @ B @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_59_in__Gr,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,X6: A,Y: B,A5: set @ A,F2: A > B] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
% 2.92/1.69 = ( ( member @ A @ X6 @ A5 )
% 2.92/1.69 & ( ( F2 @ X6 )
% 2.92/1.69 = Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_60_Gr__not__in,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,X6: A,F4: set @ A,F2: A > B,Y: B] :
% 2.92/1.69 ( ( ~ ( member @ A @ X6 @ F4 )
% 2.92/1.69 | ( ( F2 @ X6 )
% 2.92/1.69 != Y ) )
% 2.92/1.69 => ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y ) @ ( bNF_Gr @ A @ B @ F4 @ F2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_61_Id__onI,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ A2 @ A5 )
% 2.92/1.69 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_62_vertices__restrict,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
% 2.92/1.69 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
% 2.92/1.69 = ( labeled_vertices @ B @ A @ G ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_63_map__graph__preserves__restricted,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) )
% 2.92/1.69 => ( ( map_graph @ B @ C @ A @ F2 @ G )
% 2.92/1.69 = ( restrict @ A @ C @ ( map_graph @ B @ C @ A @ F2 @ G ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_64_map__graph__fn__comp,axiom,
% 2.92/1.69 ! [B: $tType,D: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: D > B,G5: C > D] :
% 2.92/1.69 ( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ ( comp @ D @ B @ C @ F2 @ G5 ) ) @ G )
% 2.92/1.69 = ( map_graph @ D @ B @ A @ ( bNF_Gr @ D @ B @ ( labeled_vertices @ A @ D @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ G ) @ G5 ) @ G ) ) @ F2 ) @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ G ) @ G5 ) @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_65_assms_I1_J,axiom,
% 2.92/1.69 ! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
% 2.92/1.69 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( standa438229444_rules @ v2 @ c @ l ) )
% 2.92/1.69 => ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_66_ne,axiom,
% 2.92/1.69 maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ g ).
% 2.92/1.69
% 2.92/1.69 thf(fact_67__092_060open_062maintained_Anonempty__rule_AG_H_A_061_A_Ivertices_AG_H_A_092_060noteq_062_A_123_125_J_092_060close_062,axiom,
% 2.92/1.69 ( ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ g )
% 2.92/1.69 = ( ( labeled_vertices @ ( standard_Constant @ v2 ) @ v @ g )
% 2.92/1.69 != ( bot_bot @ ( set @ v ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_68_maintained__preserved__by__isomorphism,axiom,
% 2.92/1.69 ! [C: $tType,D: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F2: C > A,G5: A > C,R: product_prod @ ( labeled_graph @ B @ D ) @ ( labeled_graph @ B @ D )] :
% 2.92/1.69 ( ! [X2: A] :
% 2.92/1.69 ( ( member @ A @ X2 @ ( labeled_vertices @ B @ A @ G ) )
% 2.92/1.69 => ( ( comp @ C @ A @ A @ F2 @ G5 @ X2 )
% 2.92/1.69 = X2 ) )
% 2.92/1.69 => ( ( G
% 2.92/1.69 = ( restrict @ B @ A @ G ) )
% 2.92/1.69 => ( ( maintained @ B @ D @ C @ R @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G5 ) @ G ) )
% 2.92/1.69 => ( maintained @ B @ D @ A @ R @ G ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_69_idemp__embedding__maintained__preserved,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F2: B > B,R: product_prod @ ( labeled_graph @ A @ C ) @ ( labeled_graph @ A @ C )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) ) ) )
% 2.92/1.69 => ( ! [X2: B] :
% 2.92/1.69 ( ( member @ B @ X2 @ ( labeled_vertices @ A @ B @ G ) )
% 2.92/1.69 => ( ( comp @ B @ B @ B @ F2 @ F2 @ X2 )
% 2.92/1.69 = ( F2 @ X2 ) ) )
% 2.92/1.69 => ( ( maintained @ A @ C @ B @ R @ G )
% 2.92/1.69 => ( maintained @ A @ C @ B @ R @ ( map_graph @ B @ B @ A @ ( bNF_Gr @ B @ B @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_70_Id__onE,axiom,
% 2.92/1.69 ! [A: $tType,C5: product_prod @ A @ A,A5: set @ A] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ A ) @ C5 @ ( id_on @ A @ A5 ) )
% 2.92/1.69 => ~ ! [X2: A] :
% 2.92/1.69 ( ( member @ A @ X2 @ A5 )
% 2.92/1.69 => ( C5
% 2.92/1.69 != ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_71_Id__on__eqI,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 2.92/1.69 ( ( A2 = B2 )
% 2.92/1.69 => ( ( member @ A @ A2 @ A5 )
% 2.92/1.69 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_72_Id__on__iff,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,Y: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Y ) @ ( id_on @ A @ A5 ) )
% 2.92/1.69 = ( ( X6 = Y )
% 2.92/1.69 & ( member @ A @ X6 @ A5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_73_subgraph__preserves__hom,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,X8: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 2.92/1.69 => ( ( graph_homomorphism @ A @ C @ B @ X8 @ A5 @ H )
% 2.92/1.69 => ( graph_homomorphism @ A @ C @ B @ X8 @ B6 @ H ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_74_subgraph__trans,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.92/1.69 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 2.92/1.69 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_75_map__graph__fn__eqI,axiom,
% 2.92/1.69 ! [C: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F2: A > C,G5: A > C] :
% 2.92/1.69 ( ! [X2: A] :
% 2.92/1.69 ( ( member @ A @ X2 @ ( labeled_vertices @ B @ A @ G ) )
% 2.92/1.69 => ( ( F2 @ X2 )
% 2.92/1.69 = ( G5 @ X2 ) ) )
% 2.92/1.69 => ( ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F2 ) @ G )
% 2.92/1.69 = ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G5 ) @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_76_tr,axiom,
% 2.92/1.69 maintained @ ( standard_Constant @ v2 ) @ nat @ v @ ( standa400804411p_rule @ ( standard_Constant @ v2 ) @ ( standard_S_Top @ v2 ) ) @ g ).
% 2.92/1.69
% 2.92/1.69 thf(fact_77_h_I4_J,axiom,
% 2.92/1.69 ! [L2: standard_Constant @ v2,X6: v,Y: v] :
% 2.92/1.69 ( ( member @ ( product_prod @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) ) @ ( product_Pair @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) @ L2 @ ( product_Pair @ v @ v @ X6 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ v2 ) @ v @ g ) )
% 2.92/1.69 = ( member @ ( product_prod @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) ) @ ( product_Pair @ ( standard_Constant @ v2 ) @ ( product_prod @ v @ v ) @ L2 @ ( product_Pair @ v @ v @ ( h @ X6 ) @ ( h @ Y ) ) ) @ ( labeled_edges @ ( standard_Constant @ v2 ) @ v @ g ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_78_old_Oprod_Orec,axiom,
% 2.92/1.69 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.92/1.69 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.92/1.69 = ( F1 @ A2 @ B2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_79_maintained__refl,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_80_comp__apply,axiom,
% 2.92/1.69 ! [C: $tType,A: $tType,B: $tType] :
% 2.92/1.69 ( ( comp @ B @ A @ C )
% 2.92/1.69 = ( ^ [F5: B > A,G6: C > B,X7: C] : ( F5 @ ( G6 @ X7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_81_empty__Collect__eq,axiom,
% 2.92/1.69 ! [A: $tType,P2: A > $o] :
% 2.92/1.69 ( ( ( bot_bot @ ( set @ A ) )
% 2.92/1.69 = ( collect @ A @ P2 ) )
% 2.92/1.69 = ( ! [X7: A] :
% 2.92/1.69 ~ ( P2 @ X7 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_82_Collect__empty__eq,axiom,
% 2.92/1.69 ! [A: $tType,P2: A > $o] :
% 2.92/1.69 ( ( ( collect @ A @ P2 )
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( ! [X7: A] :
% 2.92/1.69 ~ ( P2 @ X7 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_83_empty__iff,axiom,
% 2.92/1.69 ! [A: $tType,C5: A] :
% 2.92/1.69 ~ ( member @ A @ C5 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_84_all__not__in__conv,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( ! [X7: A] :
% 2.92/1.69 ~ ( member @ A @ X7 @ A5 ) )
% 2.92/1.69 = ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_85_nonempty__rule,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) )
% 2.92/1.69 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
% 2.92/1.69 = ( ( labeled_vertices @ A @ B @ G )
% 2.92/1.69 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_86__092_060open_062maintainedA_A_Iidentity__rules_AL_J_AG_H_092_060close_062,axiom,
% 2.92/1.69 ! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
% 2.92/1.69 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( standa2002409347_rules @ v2 @ l ) )
% 2.92/1.69 => ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_87_labeled__graph_Oexpand,axiom,
% 2.92/1.69 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 2.92/1.69 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 2.92/1.69 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 2.92/1.69 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 2.92/1.69 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 2.92/1.69 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_88_emptyE,axiom,
% 2.92/1.69 ! [A: $tType,A2: A] :
% 2.92/1.69 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_89_equals0D,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,A2: A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 => ~ ( member @ A @ A2 @ A5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_90_equals0I,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ! [Y3: A] :
% 2.92/1.69 ~ ( member @ A @ Y3 @ A5 )
% 2.92/1.69 => ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_91_ex__in__conv,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( ? [X7: A] : ( member @ A @ X7 @ A5 ) )
% 2.92/1.69 = ( A5
% 2.92/1.69 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_92_comp__def,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType] :
% 2.92/1.69 ( ( comp @ B @ C @ A )
% 2.92/1.69 = ( ^ [F5: B > C,G6: A > B,X7: A] : ( F5 @ ( G6 @ X7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_93_comp__assoc,axiom,
% 2.92/1.69 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G5: C > D,H: A > C] :
% 2.92/1.69 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G5 ) @ H )
% 2.92/1.69 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G5 @ H ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_94_comp__eq__dest,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C5: D > B,D3: A > D,V3: A] :
% 2.92/1.69 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.92/1.69 = ( comp @ D @ B @ A @ C5 @ D3 ) )
% 2.92/1.69 => ( ( A2 @ ( B2 @ V3 ) )
% 2.92/1.69 = ( C5 @ ( D3 @ V3 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_95_comp__eq__elim,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C5: D > B,D3: A > D] :
% 2.92/1.69 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.92/1.69 = ( comp @ D @ B @ A @ C5 @ D3 ) )
% 2.92/1.69 => ! [V4: A] :
% 2.92/1.69 ( ( A2 @ ( B2 @ V4 ) )
% 2.92/1.69 = ( C5 @ ( D3 @ V4 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_96_comp__eq__dest__lhs,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C5: A > B,V3: A] :
% 2.92/1.69 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.92/1.69 = C5 )
% 2.92/1.69 => ( ( A2 @ ( B2 @ V3 ) )
% 2.92/1.69 = ( C5 @ V3 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_97_edge__preserving__on__graphI,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,X8: labeled_graph @ A @ B,F2: B > C,Y5: set @ ( product_prod @ A @ ( product_prod @ C @ C ) )] :
% 2.92/1.69 ( ! [L3: A,X2: B,Y3: B] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L3 @ ( product_Pair @ B @ B @ X2 @ Y3 ) ) @ ( labeled_edges @ A @ B @ X8 ) )
% 2.92/1.69 => ( ( member @ B @ X2 @ ( labeled_vertices @ A @ B @ X8 ) )
% 2.92/1.69 => ( ( member @ B @ Y3 @ ( labeled_vertices @ A @ B @ X8 ) )
% 2.92/1.69 => ( member @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L3 @ ( product_Pair @ C @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
% 2.92/1.69 => ( edge_preserving @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ X8 ) @ F2 ) @ ( labeled_edges @ A @ B @ X8 ) @ Y5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_98_bot__apply,axiom,
% 2.92/1.69 ! [C: $tType,D: $tType] :
% 2.92/1.69 ( ( bot @ C )
% 2.92/1.69 => ( ( bot_bot @ ( D > C ) )
% 2.92/1.69 = ( ^ [X7: D] : ( bot_bot @ C ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_99_least__consequence__graphI,axiom,
% 2.92/1.69 ! [X: $tType,A: $tType,C: $tType,L: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G: labeled_graph @ L @ C,S: labeled_graph @ L @ C,T2: itself @ X] :
% 2.92/1.69 ( ( consequence_graph @ L @ A @ C @ Rs @ G )
% 2.92/1.69 => ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 2.92/1.69 => ( ! [C6: labeled_graph @ L @ X] :
% 2.92/1.69 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 2.92/1.69 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C6 ) )
% 2.92/1.69 => ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_100_GrD2,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,X6: A,Fx: B,A5: set @ A,F2: A > B] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
% 2.92/1.69 => ( ( F2 @ X6 )
% 2.92/1.69 = Fx ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_101_GrD1,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,X6: A,Fx: B,A5: set @ A,F2: A > B] :
% 2.92/1.69 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F2 ) )
% 2.92/1.69 => ( member @ A @ X6 @ A5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_102_leastI,axiom,
% 2.92/1.69 ! [X: $tType,A: $tType,C: $tType,L: $tType,S: labeled_graph @ L @ C,G: labeled_graph @ L @ C,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T2: itself @ X] :
% 2.92/1.69 ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 2.92/1.69 => ( ! [C6: labeled_graph @ L @ X] :
% 2.92/1.69 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 2.92/1.69 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C6 ) )
% 2.92/1.69 => ( least @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_103_edge__preserving__Id,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,Y: set @ A,X6: set @ ( product_prod @ B @ ( product_prod @ A @ A ) )] : ( edge_preserving @ A @ A @ B @ ( id_on @ A @ Y ) @ X6 @ X6 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_104_map__graph__edge__preserving,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,C: $tType,F2: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] : ( edge_preserving @ A @ B @ C @ F2 @ ( labeled_edges @ C @ A @ G ) @ ( labeled_edges @ C @ B @ ( map_graph @ A @ B @ C @ F2 @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_105_least__consequence__graph__def,axiom,
% 2.92/1.69 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 2.92/1.69 ( ( least_559130134_graph @ X @ L @ V @ C )
% 2.92/1.69 = ( ^ [T3: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 2.92/1.69 ( ( consequence_graph @ L @ V @ C @ Rs2 @ G7 )
% 2.92/1.69 & ( least @ X @ L @ V @ C @ T3 @ Rs2 @ S2 @ G7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_106_bot__empty__eq,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bot_bot @ ( A > $o ) )
% 2.92/1.69 = ( ^ [X7: A] : ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_107_bot__set__def,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bot_bot @ ( set @ A ) )
% 2.92/1.69 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_108_consequence__graphD_I3_J,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 2.92/1.69 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 2.92/1.69 => ( G
% 2.92/1.69 = ( restrict @ A @ C @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_109_consequence__graphD_I1_J,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C,R2: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 2.92/1.69 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 2.92/1.69 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
% 2.92/1.69 => ( maintained @ A @ B @ C @ R2 @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_110_edge__preserving__atomic,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E22: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),V1: A,V12: B,V22: A,V23: B,K: C] :
% 2.92/1.69 ( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E22 )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V1 @ V12 ) @ H1 )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V22 @ V23 ) @ H1 )
% 2.92/1.69 => ( ( member @ ( product_prod @ C @ ( product_prod @ A @ A ) ) @ ( product_Pair @ C @ ( product_prod @ A @ A ) @ K @ ( product_Pair @ A @ A @ V1 @ V22 ) ) @ E1 )
% 2.92/1.69 => ( member @ ( product_prod @ C @ ( product_prod @ B @ B ) ) @ ( product_Pair @ C @ ( product_prod @ B @ B ) @ K @ ( product_Pair @ B @ B @ V12 @ V23 ) ) @ E22 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_111_bot__fun__def,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType] :
% 2.92/1.69 ( ( bot @ B )
% 2.92/1.69 => ( ( bot_bot @ ( A > B ) )
% 2.92/1.69 = ( ^ [X7: A] : ( bot_bot @ B ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_112_least__def,axiom,
% 2.92/1.69 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 2.92/1.69 ( ( least @ X @ L @ V @ C )
% 2.92/1.69 = ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 2.92/1.69 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G7 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 2.92/1.69 & ! [C7: labeled_graph @ L @ X] :
% 2.92/1.69 ( ( consequence_graph @ L @ V @ X @ Rs2 @ C7 )
% 2.92/1.69 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G7 ) @ C7 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_113_mnt,axiom,
% 2.92/1.69 ! [X5: product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat )] :
% 2.92/1.69 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ X5 @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) @ ( standa1138209853_rules @ v2 @ c ) @ ( standa2002409347_rules @ v2 @ l ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ ( standa400804411p_rule @ ( standard_Constant @ v2 ) @ ( standard_S_Top @ v2 ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) @ ( standa244753842y_rule @ ( standard_Constant @ v2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ v2 ) @ nat ) ) ) ) ) ) ) )
% 2.92/1.69 => ( maintained @ ( standard_Constant @ v2 ) @ nat @ v @ X5 @ g ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_114_map__graph__in,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A,F2: B > C] :
% 2.92/1.69 ( ( G
% 2.92/1.69 = ( restrict @ A @ B @ G ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ G @ E3 ) )
% 2.92/1.69 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) @ ( semantics @ A @ C @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F2 ) @ G ) @ E3 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_115_maintainedI,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B6: labeled_graph @ C @ A] :
% 2.92/1.69 ( ! [F: set @ ( product_prod @ A @ B )] :
% 2.92/1.69 ( ( graph_homomorphism @ C @ A @ B @ A5 @ G @ F )
% 2.92/1.69 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B6 ) @ G @ F ) )
% 2.92/1.69 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B6 ) @ G ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_116_graph__homomorphism__empty,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F2 )
% 2.92/1.69 = ( ( F2
% 2.92/1.69 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 2.92/1.69 & ( G
% 2.92/1.69 = ( restrict @ A @ C @ G ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_117_subgraph__def,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.92/1.69 = ( ( G_1
% 2.92/1.69 = ( restrict @ A @ B @ G_1 ) )
% 2.92/1.69 & ( G_2
% 2.92/1.69 = ( restrict @ A @ B @ G_2 ) )
% 2.92/1.69 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.92/1.69 = G_2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_118_insert__absorb2,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,A5: set @ A] :
% 2.92/1.69 ( ( insert @ A @ X6 @ ( insert @ A @ X6 @ A5 ) )
% 2.92/1.69 = ( insert @ A @ X6 @ A5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_119_insert__iff,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 2.92/1.69 = ( ( A2 = B2 )
% 2.92/1.69 | ( member @ A @ A2 @ A5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_120_insertCI,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B6: set @ A,B2: A] :
% 2.92/1.69 ( ( ~ ( member @ A @ A2 @ B6 )
% 2.92/1.69 => ( A2 = B2 ) )
% 2.92/1.69 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_121_Un__iff,axiom,
% 2.92/1.69 ! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 2.92/1.69 = ( ( member @ A @ C5 @ A5 )
% 2.92/1.69 | ( member @ A @ C5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_122_UnCI,axiom,
% 2.92/1.69 ! [A: $tType,C5: A,B6: set @ A,A5: set @ A] :
% 2.92/1.69 ( ( ~ ( member @ A @ C5 @ B6 )
% 2.92/1.69 => ( member @ A @ C5 @ A5 ) )
% 2.92/1.69 => ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_123_labeled__graph_Oinject,axiom,
% 2.92/1.69 ! [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] :
% 2.92/1.69 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 2.92/1.69 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 2.92/1.69 = ( ( X1 = Y1 )
% 2.92/1.69 & ( X22 = Y2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_124_graph__union__idemp_I1_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_union @ A @ B @ A5 @ A5 )
% 2.92/1.69 = A5 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_125_graph__union__idemp_I2_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B6 ) )
% 2.92/1.69 = ( graph_union @ A @ B @ A5 @ B6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_126_graph__union__idemp_I3_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B6 @ A5 ) )
% 2.92/1.69 = ( graph_union @ A @ B @ B6 @ A5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_127_singletonI,axiom,
% 2.92/1.69 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_128_Un__empty,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_129_Un__insert__right,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,A2: A,B6: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) )
% 2.92/1.69 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_130_Un__insert__left,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B6: set @ A,C8: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B6 ) @ C8 )
% 2.92/1.69 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_131_graph__union__preserves__restrict,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( G_1
% 2.92/1.69 = ( restrict @ A @ B @ G_1 ) )
% 2.92/1.69 => ( ( G_2
% 2.92/1.69 = ( restrict @ A @ B @ G_2 ) )
% 2.92/1.69 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.92/1.69 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_132_graph__union__vertices,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,G_1: labeled_graph @ B @ A,G_2: labeled_graph @ B @ A] :
% 2.92/1.69 ( ( labeled_vertices @ B @ A @ ( graph_union @ B @ A @ G_1 @ G_2 ) )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ ( labeled_vertices @ B @ A @ G_1 ) @ ( labeled_vertices @ B @ A @ G_2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_133_labeled__graph_Ocollapse,axiom,
% 2.92/1.69 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.92/1.69 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 2.92/1.69 = Labeled_graph ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_134_extensible__refl,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ C @ R2 @ G @ F2 )
% 2.92/1.69 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G @ F2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_135_graph__empty__e,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,V3: set @ B] :
% 2.92/1.69 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
% 2.92/1.69 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_136_constant__rules__empty,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( standa1138209853_rules @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_137_Gr__insert,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,X6: A,F4: set @ A,F2: A > B] :
% 2.92/1.69 ( ( bNF_Gr @ A @ B @ ( insert @ A @ X6 @ F4 ) @ F2 )
% 2.92/1.69 = ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ ( F2 @ X6 ) ) @ ( bNF_Gr @ A @ B @ F4 @ F2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_138_extensible__refl__concr,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V3: set @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V3 ) @ G @ F2 )
% 2.92/1.69 => ( ( 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 @ F2 )
% 2.92/1.69 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V3 ) @ G @ F2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_139_graph__single,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,A2: A,B2: B,C5: B] :
% 2.92/1.69 ( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C5 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C5 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.92/1.69 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C5 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C5 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_140_insert__is__Un,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( insert @ A )
% 2.92/1.69 = ( ^ [A6: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_141_Un__singleton__iff,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A,X6: A] :
% 2.92/1.69 ( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 = ( ( ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.69 | ( ( A5
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 | ( ( A5
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_142_singleton__Un__iff,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 2.92/1.69 = ( ( ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.69 | ( ( A5
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 | ( ( A5
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_143_graph__union__def,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType] :
% 2.92/1.69 ( ( graph_union @ A @ B )
% 2.92/1.69 = ( ^ [G_12: labeled_graph @ A @ B,G_22: labeled_graph @ A @ B] : ( labeled_LG @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_12 ) @ ( labeled_edges @ A @ B @ G_22 ) ) @ ( sup_sup @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_12 ) @ ( labeled_vertices @ A @ B @ G_22 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_144_Un__empty__left,axiom,
% 2.92/1.69 ! [A: $tType,B6: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 2.92/1.69 = B6 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_145_Un__empty__right,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = A5 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_146_singletonD,axiom,
% 2.92/1.69 ! [A: $tType,B2: A,A2: A] :
% 2.92/1.69 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 => ( B2 = A2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_147_singleton__iff,axiom,
% 2.92/1.69 ! [A: $tType,B2: A,A2: A] :
% 2.92/1.69 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 = ( B2 = A2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_148_doubleton__eq__iff,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B2: A,C5: A,D3: A] :
% 2.92/1.69 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 = ( insert @ A @ C5 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.69 = ( ( ( A2 = C5 )
% 2.92/1.69 & ( B2 = D3 ) )
% 2.92/1.69 | ( ( A2 = D3 )
% 2.92/1.69 & ( B2 = C5 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_149_insert__not__empty,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,A5: set @ A] :
% 2.92/1.69 ( ( insert @ A @ A2 @ A5 )
% 2.92/1.69 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_150_singleton__inject,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B2: A] :
% 2.92/1.69 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 => ( A2 = B2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_151_labeled__graph_Oexhaust,axiom,
% 2.92/1.69 ! [L: $tType,V: $tType,Y: labeled_graph @ L @ V] :
% 2.92/1.69 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 2.92/1.69 ( Y
% 2.92/1.69 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_152_labeled__graph_Oinduct,axiom,
% 2.92/1.69 ! [V: $tType,L: $tType,P2: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 2.92/1.69 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P2 @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 2.92/1.69 => ( P2 @ Labeled_graph ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_153_mk__disjoint__insert,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ A2 @ A5 )
% 2.92/1.69 => ? [B7: set @ A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( insert @ A @ A2 @ B7 ) )
% 2.92/1.69 & ~ ( member @ A @ A2 @ B7 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_154_Un__left__commute,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A5 @ C8 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_155_insert__commute,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,Y: A,A5: set @ A] :
% 2.92/1.69 ( ( insert @ A @ X6 @ ( insert @ A @ Y @ A5 ) )
% 2.92/1.69 = ( insert @ A @ Y @ ( insert @ A @ X6 @ A5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_156_Un__left__absorb,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_157_insert__eq__iff,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,A5: set @ A,B2: A,B6: set @ A] :
% 2.92/1.69 ( ~ ( member @ A @ A2 @ A5 )
% 2.92/1.69 => ( ~ ( member @ A @ B2 @ B6 )
% 2.92/1.69 => ( ( ( insert @ A @ A2 @ A5 )
% 2.92/1.69 = ( insert @ A @ B2 @ B6 ) )
% 2.92/1.69 = ( ( ( A2 = B2 )
% 2.92/1.69 => ( A5 = B6 ) )
% 2.92/1.69 & ( ( A2 != B2 )
% 2.92/1.69 => ? [C7: set @ A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( insert @ A @ B2 @ C7 ) )
% 2.92/1.69 & ~ ( member @ A @ B2 @ C7 )
% 2.92/1.69 & ( B6
% 2.92/1.69 = ( insert @ A @ A2 @ C7 ) )
% 2.92/1.69 & ~ ( member @ A @ A2 @ C7 ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_158_insert__absorb,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ A2 @ A5 )
% 2.92/1.69 => ( ( insert @ A @ A2 @ A5 )
% 2.92/1.69 = A5 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_159_insert__ident,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ~ ( member @ A @ X6 @ A5 )
% 2.92/1.69 => ( ~ ( member @ A @ X6 @ B6 )
% 2.92/1.69 => ( ( ( insert @ A @ X6 @ A5 )
% 2.92/1.69 = ( insert @ A @ X6 @ B6 ) )
% 2.92/1.69 = ( A5 = B6 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_160_Set_Oset__insert,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ X6 @ A5 )
% 2.92/1.69 => ~ ! [B7: set @ A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( insert @ A @ X6 @ B7 ) )
% 2.92/1.69 => ( member @ A @ X6 @ B7 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_161_Un__commute,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) )
% 2.92/1.69 = ( ^ [A7: set @ A,B8: set @ A] : ( sup_sup @ ( set @ A ) @ B8 @ A7 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_162_Un__absorb,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
% 2.92/1.69 = A5 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_163_insertI2,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B6: set @ A,B2: A] :
% 2.92/1.69 ( ( member @ A @ A2 @ B6 )
% 2.92/1.69 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_164_insertI1,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B6: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_165_Un__assoc,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
% 2.92/1.69 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) @ C8 )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_166_insertE,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 2.92/1.69 => ( ( A2 != B2 )
% 2.92/1.69 => ( member @ A @ A2 @ A5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_167_ball__Un,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A,P2: A > $o] :
% 2.92/1.69 ( ( ! [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 2.92/1.69 => ( P2 @ X7 ) ) )
% 2.92/1.69 = ( ! [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ A5 )
% 2.92/1.69 => ( P2 @ X7 ) )
% 2.92/1.69 & ! [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ B6 )
% 2.92/1.69 => ( P2 @ X7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_168_bex__Un,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A,P2: A > $o] :
% 2.92/1.69 ( ( ? [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 2.92/1.69 & ( P2 @ X7 ) ) )
% 2.92/1.69 = ( ? [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ A5 )
% 2.92/1.69 & ( P2 @ X7 ) )
% 2.92/1.69 | ? [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ B6 )
% 2.92/1.69 & ( P2 @ X7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_169_UnI2,axiom,
% 2.92/1.69 ! [A: $tType,C5: A,B6: set @ A,A5: set @ A] :
% 2.92/1.69 ( ( member @ A @ C5 @ B6 )
% 2.92/1.69 => ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_170_UnI1,axiom,
% 2.92/1.69 ! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( member @ A @ C5 @ A5 )
% 2.92/1.69 => ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_171_UnE,axiom,
% 2.92/1.69 ! [A: $tType,C5: A,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( member @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
% 2.92/1.69 => ( ~ ( member @ A @ C5 @ A5 )
% 2.92/1.69 => ( member @ A @ C5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_172_labeled__graph_Osel_I2_J,axiom,
% 2.92/1.69 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.92/1.69 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.92/1.69 = X22 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_173_labeled__graph_Osel_I1_J,axiom,
% 2.92/1.69 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.92/1.69 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.92/1.69 = X1 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_174_graph__homomorphism__semantics,axiom,
% 2.92/1.69 ! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B6 @ F2 )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F2 )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F2 )
% 2.92/1.69 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B6 @ E3 ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_175_graph__homomorphism__nonempty,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B6 @ F2 )
% 2.92/1.69 => ( ( ( semantics @ A @ B @ A5 @ E3 )
% 2.92/1.69 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 2.92/1.69 => ( ( semantics @ A @ C @ B6 @ E3 )
% 2.92/1.69 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_176_labeled__graph_Oexhaust__sel,axiom,
% 2.92/1.69 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.92/1.69 ( Labeled_graph
% 2.92/1.69 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_177_standard__rules__def,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( standa438229444_rules @ A )
% 2.92/1.69 = ( ^ [C7: set @ A,L4: set @ ( standard_Constant @ A )] : ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) @ ( sup_sup @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) @ ( standa1138209853_rules @ A @ C7 ) @ ( standa2002409347_rules @ A @ L4 ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa400804411p_rule @ ( standard_Constant @ A ) @ ( standard_S_Top @ A ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) @ ( standa244753842y_rule @ ( standard_Constant @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_178_semantics__in__vertices_I2_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( restrict @ A @ B @ A5 ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.92/1.69 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_179_semantics__in__vertices_I1_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( restrict @ A @ B @ A5 ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.92/1.69 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_180_subgraph__semantics,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B6 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 2.92/1.69 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 2.92/1.69 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B6 @ E3 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_181_maintainedD,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B6: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B6 ) @ G )
% 2.92/1.69 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G @ F2 )
% 2.92/1.69 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B6 ) @ G @ F2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_182_sup__bot_Oright__neutral,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [A2: A] :
% 2.92/1.69 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 2.92/1.69 = A2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_183_sup__bot_Oneutr__eq__iff,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [A2: A,B2: A] :
% 2.92/1.69 ( ( ( bot_bot @ A )
% 2.92/1.69 = ( sup_sup @ A @ A2 @ B2 ) )
% 2.92/1.69 = ( ( A2
% 2.92/1.69 = ( bot_bot @ A ) )
% 2.92/1.69 & ( B2
% 2.92/1.69 = ( bot_bot @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_184_sup__bot_Oleft__neutral,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [A2: A] :
% 2.92/1.69 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 2.92/1.69 = A2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_185_sup__bot_Oeq__neutr__iff,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [A2: A,B2: A] :
% 2.92/1.69 ( ( ( sup_sup @ A @ A2 @ B2 )
% 2.92/1.69 = ( bot_bot @ A ) )
% 2.92/1.69 = ( ( A2
% 2.92/1.69 = ( bot_bot @ A ) )
% 2.92/1.69 & ( B2
% 2.92/1.69 = ( bot_bot @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_186_sup_Oright__idem,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A2: A,B2: A] :
% 2.92/1.69 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 2.92/1.69 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_187_sup__left__idem,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,Y: A] :
% 2.92/1.69 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) )
% 2.92/1.69 = ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_188_sup_Oleft__idem,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A2: A,B2: A] :
% 2.92/1.69 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 2.92/1.69 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_189_sup__idem,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A] :
% 2.92/1.69 ( ( sup_sup @ A @ X6 @ X6 )
% 2.92/1.69 = X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_190_sup_Oidem,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A2: A] :
% 2.92/1.69 ( ( sup_sup @ A @ A2 @ A2 )
% 2.92/1.69 = A2 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_191_sup__apply,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ B )
% 2.92/1.69 => ( ( sup_sup @ ( A > B ) )
% 2.92/1.69 = ( ^ [F5: A > B,G6: A > B,X7: A] : ( sup_sup @ B @ ( F5 @ X7 ) @ ( G6 @ X7 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_192_sup__bot__left,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [X6: A] :
% 2.92/1.69 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X6 )
% 2.92/1.69 = X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_193_sup__bot__right,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [X6: A] :
% 2.92/1.69 ( ( sup_sup @ A @ X6 @ ( bot_bot @ A ) )
% 2.92/1.69 = X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_194_bot__eq__sup__iff,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [X6: A,Y: A] :
% 2.92/1.69 ( ( ( bot_bot @ A )
% 2.92/1.69 = ( sup_sup @ A @ X6 @ Y ) )
% 2.92/1.69 = ( ( X6
% 2.92/1.69 = ( bot_bot @ A ) )
% 2.92/1.69 & ( Y
% 2.92/1.69 = ( bot_bot @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_195_sup__eq__bot__iff,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( bounde1808546759up_bot @ A )
% 2.92/1.69 => ! [X6: A,Y: A] :
% 2.92/1.69 ( ( ( sup_sup @ A @ X6 @ Y )
% 2.92/1.69 = ( bot_bot @ A ) )
% 2.92/1.69 = ( ( X6
% 2.92/1.69 = ( bot_bot @ A ) )
% 2.92/1.69 & ( Y
% 2.92/1.69 = ( bot_bot @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_196_edge__preserving__unionI,axiom,
% 2.92/1.69 ! [A: $tType,B: $tType,C: $tType,F2: set @ ( product_prod @ A @ B ),A5: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),G: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),B6: set @ ( product_prod @ C @ ( product_prod @ A @ A ) )] :
% 2.92/1.69 ( ( edge_preserving @ A @ B @ C @ F2 @ A5 @ G )
% 2.92/1.69 => ( ( edge_preserving @ A @ B @ C @ F2 @ B6 @ G )
% 2.92/1.69 => ( edge_preserving @ A @ B @ C @ F2 @ ( sup_sup @ ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) @ A5 @ B6 ) @ G ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_197_graph__union__edges,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( labeled_edges @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) )
% 2.92/1.69 = ( sup_sup @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_198_sup__left__commute,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,Y: A,Z: A] :
% 2.92/1.69 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) )
% 2.92/1.69 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X6 @ Z ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_199_sup_Oleft__commute,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [B2: A,A2: A,C5: A] :
% 2.92/1.69 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C5 ) )
% 2.92/1.69 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C5 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_200_sup__commute,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ( ( sup_sup @ A )
% 2.92/1.69 = ( ^ [X7: A,Y6: A] : ( sup_sup @ A @ Y6 @ X7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_201_sup_Ocommute,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ( ( sup_sup @ A )
% 2.92/1.69 = ( ^ [A6: A,B9: A] : ( sup_sup @ A @ B9 @ A6 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_202_sup__assoc,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,Y: A,Z: A] :
% 2.92/1.69 ( ( sup_sup @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
% 2.92/1.69 = ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_203_sup_Oassoc,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A2: A,B2: A,C5: A] :
% 2.92/1.69 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C5 )
% 2.92/1.69 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C5 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_204_boolean__algebra__cancel_Osup2,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [B6: A,K: A,B2: A,A2: A] :
% 2.92/1.69 ( ( B6
% 2.92/1.69 = ( sup_sup @ A @ K @ B2 ) )
% 2.92/1.69 => ( ( sup_sup @ A @ A2 @ B6 )
% 2.92/1.69 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_205_boolean__algebra__cancel_Osup1,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A5: A,K: A,A2: A,B2: A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 = ( sup_sup @ A @ K @ A2 ) )
% 2.92/1.69 => ( ( sup_sup @ A @ A5 @ B2 )
% 2.92/1.69 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_206_sup__fun__def,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ B )
% 2.92/1.69 => ( ( sup_sup @ ( A > B ) )
% 2.92/1.69 = ( ^ [F5: A > B,G6: A > B,X7: A] : ( sup_sup @ B @ ( F5 @ X7 ) @ ( G6 @ X7 ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_207_inf__sup__aci_I5_J,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( lattice @ A )
% 2.92/1.69 => ( ( sup_sup @ A )
% 2.92/1.69 = ( ^ [X7: A,Y6: A] : ( sup_sup @ A @ Y6 @ X7 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_208_inf__sup__aci_I6_J,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( lattice @ A )
% 2.92/1.69 => ! [X6: A,Y: A,Z: A] :
% 2.92/1.69 ( ( sup_sup @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
% 2.92/1.69 = ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_209_inf__sup__aci_I7_J,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( lattice @ A )
% 2.92/1.69 => ! [X6: A,Y: A,Z: A] :
% 2.92/1.69 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ Y @ Z ) )
% 2.92/1.69 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X6 @ Z ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_210_inf__sup__aci_I8_J,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( lattice @ A )
% 2.92/1.69 => ! [X6: A,Y: A] :
% 2.92/1.69 ( ( sup_sup @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) )
% 2.92/1.69 = ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_211_the__elem__eq,axiom,
% 2.92/1.69 ! [A: $tType,X6: A] :
% 2.92/1.69 ( ( the_elem @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 = X6 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_212_Collect__empty__eq__bot,axiom,
% 2.92/1.69 ! [A: $tType,P2: A > $o] :
% 2.92/1.69 ( ( ( collect @ A @ P2 )
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( P2
% 2.92/1.69 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_213_is__singletonI,axiom,
% 2.92/1.69 ! [A: $tType,X6: A] : ( is_singleton @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_214_is__singleton__the__elem,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( is_singleton @ A )
% 2.92/1.69 = ( ^ [A7: set @ A] :
% 2.92/1.69 ( A7
% 2.92/1.69 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_215_is__singletonI_H,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( A5
% 2.92/1.69 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 => ( ! [X2: A,Y3: A] :
% 2.92/1.69 ( ( member @ A @ X2 @ A5 )
% 2.92/1.69 => ( ( member @ A @ Y3 @ A5 )
% 2.92/1.69 => ( X2 = Y3 ) ) )
% 2.92/1.69 => ( is_singleton @ A @ A5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_216_is__singletonE,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( is_singleton @ A @ A5 )
% 2.92/1.69 => ~ ! [X2: A] :
% 2.92/1.69 ( A5
% 2.92/1.69 != ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_217_is__singleton__def,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( is_singleton @ A )
% 2.92/1.69 = ( ^ [A7: set @ A] :
% 2.92/1.69 ? [X7: A] :
% 2.92/1.69 ( A7
% 2.92/1.69 = ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_218_labeled__graph_Osplit__sel,axiom,
% 2.92/1.69 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.92/1.69 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F2 @ Labeled_graph ) )
% 2.92/1.69 = ( ( Labeled_graph
% 2.92/1.69 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.92/1.69 => ( P2 @ ( F2 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_219_labeled__graph_Osplit__sel__asm,axiom,
% 2.92/1.69 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.92/1.69 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F2 @ Labeled_graph ) )
% 2.92/1.69 = ( ~ ( ( Labeled_graph
% 2.92/1.69 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.92/1.69 & ~ ( P2 @ ( F2 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_220_labeled__graph_Ocase,axiom,
% 2.92/1.69 ! [L: $tType,A: $tType,V: $tType,F2: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.92/1.69 ( ( labele1974067554_graph @ L @ V @ A @ F2 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.92/1.69 = ( F2 @ X1 @ X22 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_221_labeled__graph_Ocase__eq__if,axiom,
% 2.92/1.69 ! [A: $tType,V: $tType,L: $tType] :
% 2.92/1.69 ( ( labele1974067554_graph @ L @ V @ A )
% 2.92/1.69 = ( ^ [F5: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F5 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_222_extensibleI,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G5: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G5 )
% 2.92/1.69 => ( ( agree_on @ A @ B @ C @ R1 @ F2 @ G5 )
% 2.92/1.69 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_223_total__on__singleton,axiom,
% 2.92/1.69 ! [A: $tType,X6: A] : ( total_on @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ X6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_224_agree__on__refl,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R2 @ F2 @ F2 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_225_agree__on__comm,axiom,
% 2.92/1.69 ! [C: $tType,B: $tType,A: $tType] :
% 2.92/1.69 ( ( agree_on @ A @ B @ C )
% 2.92/1.69 = ( ^ [X9: labeled_graph @ A @ B,F5: set @ ( product_prod @ B @ C ),G6: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X9 @ G6 @ F5 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_226_agree__on__empty,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,F2: set @ ( product_prod @ B @ C ),G5: 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 ) ) ) @ F2 @ G5 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_227_total__onI,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
% 2.92/1.69 ( ! [X2: A,Y3: A] :
% 2.92/1.69 ( ( member @ A @ X2 @ A5 )
% 2.92/1.69 => ( ( member @ A @ Y3 @ A5 )
% 2.92/1.69 => ( ( X2 != Y3 )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
% 2.92/1.69 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) ) ) ) )
% 2.92/1.69 => ( total_on @ A @ A5 @ R ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_228_total__on__def,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( total_on @ A )
% 2.92/1.69 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 2.92/1.69 ! [X7: A] :
% 2.92/1.69 ( ( member @ A @ X7 @ A7 )
% 2.92/1.69 => ! [Y6: A] :
% 2.92/1.69 ( ( member @ A @ Y6 @ A7 )
% 2.92/1.69 => ( ( X7 != Y6 )
% 2.92/1.69 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ R3 )
% 2.92/1.69 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X7 ) @ R3 ) ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_229_total__on__empty,axiom,
% 2.92/1.69 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_230_agree__on__trans,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,X8: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( agree_on @ A @ B @ C @ X8 @ F2 @ G5 )
% 2.92/1.69 => ( ( agree_on @ A @ B @ C @ X8 @ G5 @ H )
% 2.92/1.69 => ( agree_on @ A @ B @ C @ X8 @ F2 @ H ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_231_agree__on__subg__compose,axiom,
% 2.92/1.69 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G5: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F4: labeled_graph @ A @ B,F2: set @ ( product_prod @ B @ C )] :
% 2.92/1.69 ( ( agree_on @ A @ B @ C @ R2 @ G5 @ H )
% 2.92/1.69 => ( ( agree_on @ A @ B @ C @ F4 @ F2 @ G5 )
% 2.92/1.69 => ( ( graph_homomorphism @ A @ B @ B @ F4 @ R2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F4 ) ) )
% 2.92/1.69 => ( agree_on @ A @ B @ C @ F4 @ F2 @ H ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_232_Field__insert,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 2.92/1.69 ( ( field @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field @ A @ R ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_233_subgraph__subset_I2_J,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.92/1.69 => ( 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 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_234_order__refl,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( preorder @ A )
% 2.92/1.69 => ! [X6: A] : ( ord_less_eq @ A @ X6 @ X6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_235_subsetI,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ! [X2: A] :
% 2.92/1.69 ( ( member @ A @ X2 @ A5 )
% 2.92/1.69 => ( member @ A @ X2 @ B6 ) )
% 2.92/1.69 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_236_subset__antisym,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 2.92/1.69 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
% 2.92/1.69 => ( A5 = B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_237_le__sup__iff,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,Y: A,Z: A] :
% 2.92/1.69 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X6 @ Y ) @ Z )
% 2.92/1.69 = ( ( ord_less_eq @ A @ X6 @ Z )
% 2.92/1.69 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_238_sup_Obounded__iff,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [B2: A,C5: A,A2: A] :
% 2.92/1.69 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C5 ) @ A2 )
% 2.92/1.69 = ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.92/1.69 & ( ord_less_eq @ A @ C5 @ A2 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_239_empty__subsetI,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_240_subset__empty,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A] :
% 2.92/1.69 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( A5
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_241_insert__subset,axiom,
% 2.92/1.69 ! [A: $tType,X6: A,A5: set @ A,B6: set @ A] :
% 2.92/1.69 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X6 @ A5 ) @ B6 )
% 2.92/1.69 = ( ( member @ A @ X6 @ B6 )
% 2.92/1.69 & ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_242_Un__subset__iff,axiom,
% 2.92/1.69 ! [A: $tType,A5: set @ A,B6: set @ A,C8: set @ A] :
% 2.92/1.69 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) @ C8 )
% 2.92/1.69 = ( ( ord_less_eq @ ( set @ A ) @ A5 @ C8 )
% 2.92/1.69 & ( ord_less_eq @ ( set @ A ) @ B6 @ C8 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_243_singleton__insert__inj__eq,axiom,
% 2.92/1.69 ! [A: $tType,B2: A,A2: A,A5: set @ A] :
% 2.92/1.69 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.69 = ( insert @ A @ A2 @ A5 ) )
% 2.92/1.69 = ( ( A2 = B2 )
% 2.92/1.69 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_244_singleton__insert__inj__eq_H,axiom,
% 2.92/1.69 ! [A: $tType,A2: A,A5: set @ A,B2: A] :
% 2.92/1.69 ( ( ( insert @ A @ A2 @ A5 )
% 2.92/1.69 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.69 = ( ( A2 = B2 )
% 2.92/1.69 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_245_Field__empty,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( field @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 2.92/1.69 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_246_Field__Un,axiom,
% 2.92/1.69 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 2.92/1.69 ( ( field @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 ) )
% 2.92/1.69 = ( sup_sup @ ( set @ A ) @ ( field @ A @ R ) @ ( field @ A @ S3 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_247_graph__unionI,axiom,
% 2.92/1.69 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.92/1.69 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 2.92/1.69 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 2.92/1.69 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.92/1.69 = G_2 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_248_inf__sup__ord_I4_J,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( lattice @ A )
% 2.92/1.69 => ! [Y: A,X6: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_249_inf__sup__ord_I3_J,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( lattice @ A )
% 2.92/1.69 => ! [X6: A,Y: A] : ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_250_le__supE,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A2: A,B2: A,X6: A] :
% 2.92/1.69 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X6 )
% 2.92/1.69 => ~ ( ( ord_less_eq @ A @ A2 @ X6 )
% 2.92/1.69 => ~ ( ord_less_eq @ A @ B2 @ X6 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_251_le__supI,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [A2: A,X6: A,B2: A] :
% 2.92/1.69 ( ( ord_less_eq @ A @ A2 @ X6 )
% 2.92/1.69 => ( ( ord_less_eq @ A @ B2 @ X6 )
% 2.92/1.69 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X6 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_252_sup__ge1,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,Y: A] : ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_253_sup__ge2,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [Y: A,X6: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X6 @ Y ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_254_le__supI1,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,A2: A,B2: A] :
% 2.92/1.69 ( ( ord_less_eq @ A @ X6 @ A2 )
% 2.92/1.69 => ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(fact_255_le__supI2,axiom,
% 2.92/1.69 ! [A: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A )
% 2.92/1.69 => ! [X6: A,B2: A,A2: A] :
% 2.92/1.69 ( ( ord_less_eq @ A @ X6 @ B2 )
% 2.92/1.69 => ( ord_less_eq @ A @ X6 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.92/1.69 bounded_lattice @ $o ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.92/1.69 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.92/1.69 ! [A8: $tType,A9: $tType] :
% 2.92/1.69 ( ( bounded_lattice @ A9 )
% 2.92/1.69 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 2.92/1.69 ! [A8: $tType,A9: $tType] :
% 2.92/1.69 ( ( bounded_lattice @ A9 )
% 2.92/1.69 => ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.92/1.69 ! [A8: $tType,A9: $tType] :
% 2.92/1.69 ( ( semilattice_sup @ A9 )
% 2.92/1.69 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.92/1.69 ! [A8: $tType,A9: $tType] :
% 2.92/1.69 ( ( preorder @ A9 )
% 2.92/1.69 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.92/1.69 ! [A8: $tType,A9: $tType] :
% 2.92/1.69 ( ( lattice @ A9 )
% 2.92/1.69 => ( lattice @ ( A8 > A9 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_fun___Orderings_Obot,axiom,
% 2.92/1.69 ! [A8: $tType,A9: $tType] :
% 2.92/1.69 ( ( bot @ A9 )
% 2.92/1.69 => ( bot @ ( A8 > A9 ) ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
% 2.92/1.69 semilattice_sup @ nat ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 2.92/1.69 preorder @ nat ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Nat_Onat___Lattices_Olattice_5,axiom,
% 2.92/1.69 lattice @ nat ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Nat_Onat___Orderings_Obot_6,axiom,
% 2.92/1.69 bot @ nat ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_7,axiom,
% 2.92/1.69 ! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
% 2.92/1.69 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
% 2.92/1.69 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Set_Oset___Lattices_Olattice_10,axiom,
% 2.92/1.69 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 2.92/1.69 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
% 2.92/1.69 bounde1808546759up_bot @ $o ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_13,axiom,
% 2.92/1.69 semilattice_sup @ $o ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 2.92/1.69 preorder @ $o ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
% 2.92/1.69 lattice @ $o ).
% 2.92/1.69
% 2.92/1.69 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 2.92/1.69 bot @ $o ).
% 2.92/1.69
% 2.92/1.69 thf(conj_0,conjecture,
% 2.92/1.69 member @ ( product_prod @ v @ v ) @ ( product_Pair @ v @ v @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel @ ( standard_Constant @ v2 ) @ v @ ( standard_S_Const @ v2 @ y ) @ g ) ).
% 2.92/1.69 ------------------------
% 3.08/1.71 % (31866)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31760 for (2999ds/183Mi)
% 3.08/1.71 % (31872)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31760 for (2999ds/18Mi)
% 3.08/1.71 % (31871)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_31760 for (2999ds/275Mi)
% 3.08/1.71 % (31870)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_31760 for (2999ds/2Mi)
% 3.08/1.71 % (31870)Instruction limit reached!
% 3.08/1.71 % (31870)------------------------------
% 3.08/1.71 % (31870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.71 % (31870)Termination reason: Unknown
% 3.08/1.71 % (31870)Termination phase: shuffling
% 3.08/1.71
% 3.08/1.71 % (31870)Memory used [KB]: 1407
% 3.08/1.71 % (31870)Time elapsed: 0.003 s
% 3.08/1.71 % (31870)Instructions burned: 3 (million)
% 3.08/1.71 % (31870)------------------------------
% 3.08/1.71 % (31870)------------------------------
% 3.08/1.72 % (31872)Instruction limit reached!
% 3.08/1.72 % (31872)------------------------------
% 3.08/1.72 % (31872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.72 % (31872)Termination reason: Unknown
% 3.08/1.72 % (31872)Termination phase: shuffling
% 3.08/1.72
% 3.08/1.72 % (31872)Memory used [KB]: 1791
% 3.08/1.72 % (31872)Time elapsed: 0.007 s
% 3.08/1.72 % (31872)Instructions burned: 18 (million)
% 3.08/1.72 % (31872)------------------------------
% 3.08/1.72 % (31872)------------------------------
% 3.08/1.72 % (31867)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_31760 for (2999ds/4Mi)
% 3.08/1.72 % (31869)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31760 for (2999ds/2Mi)
% 3.08/1.72 % (31868)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_31760 for (2999ds/27Mi)
% 3.08/1.72 % (31869)Instruction limit reached!
% 3.08/1.72 % (31869)------------------------------
% 3.08/1.72 % (31869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.72 % (31869)Termination reason: Unknown
% 3.08/1.72 % (31869)Termination phase: shuffling
% 3.08/1.72
% 3.08/1.72 % (31869)Memory used [KB]: 1407
% 3.08/1.72 % (31869)Time elapsed: 0.004 s
% 3.08/1.72 % (31869)Instructions burned: 3 (million)
% 3.08/1.72 % (31869)------------------------------
% 3.08/1.72 % (31869)------------------------------
% 3.08/1.72 % (31867)Instruction limit reached!
% 3.08/1.72 % (31867)------------------------------
% 3.08/1.72 % (31867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.72 % (31867)Termination reason: Unknown
% 3.08/1.72 % (31867)Termination phase: shuffling
% 3.08/1.72
% 3.08/1.72 % (31867)Memory used [KB]: 1535
% 3.08/1.72 % (31867)Time elapsed: 0.005 s
% 3.08/1.72 % (31867)Instructions burned: 5 (million)
% 3.08/1.72 % (31867)------------------------------
% 3.08/1.72 % (31867)------------------------------
% 3.08/1.73 % (31874)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_31760 for (2999ds/37Mi)
% 3.08/1.73 % (31873)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31760 for (2999ds/3Mi)
% 3.08/1.73 % (31873)Instruction limit reached!
% 3.08/1.73 % (31873)------------------------------
% 3.08/1.73 % (31873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.73 % (31873)Termination reason: Unknown
% 3.08/1.73 % (31873)Termination phase: shuffling
% 3.08/1.73
% 3.08/1.73 % (31873)Memory used [KB]: 1535
% 3.08/1.73 % (31873)Time elapsed: 0.003 s
% 3.08/1.73 % (31873)Instructions burned: 5 (million)
% 3.08/1.73 % (31873)------------------------------
% 3.08/1.73 % (31873)------------------------------
% 3.08/1.74 % (31868)Instruction limit reached!
% 3.08/1.74 % (31868)------------------------------
% 3.08/1.74 % (31868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.74 % (31868)Termination reason: Unknown
% 3.08/1.74 % (31868)Termination phase: shuffling
% 3.08/1.74
% 3.08/1.74 % (31868)Memory used [KB]: 1918
% 3.08/1.74 % (31868)Time elapsed: 0.019 s
% 3.08/1.74 % (31868)Instructions burned: 27 (million)
% 3.08/1.74 % (31868)------------------------------
% 3.08/1.74 % (31868)------------------------------
% 3.08/1.74 % (31874)Instruction limit reached!
% 3.08/1.74 % (31874)------------------------------
% 3.08/1.74 % (31874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.74 % (31874)Termination reason: Unknown
% 3.08/1.74 % (31874)Termination phase: shuffling
% 3.08/1.74
% 3.08/1.74 % (31874)Memory used [KB]: 2174
% 3.08/1.74 % (31874)Time elapsed: 0.011 s
% 3.08/1.74 % (31874)Instructions burned: 37 (million)
% 3.08/1.74 % (31874)------------------------------
% 3.08/1.74 % (31874)------------------------------
% 3.08/1.74 % (31875)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_31760 for (2999ds/15Mi)
% 3.08/1.74 % (31876)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_31760 for (2999ds/3Mi)
% 3.08/1.74 % (31877)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_31760 for (2999ds/1041Mi)
% 3.08/1.74 % (31871)First to succeed.
% 3.08/1.74 % (31876)Instruction limit reached!
% 3.08/1.74 % (31876)------------------------------
% 3.08/1.74 % (31876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.74 % (31876)Termination reason: Unknown
% 3.08/1.74 % (31876)Termination phase: shuffling
% 3.08/1.74
% 3.08/1.74 % (31876)Memory used [KB]: 1407
% 3.08/1.74 % (31876)Time elapsed: 0.004 s
% 3.08/1.74 % (31876)Instructions burned: 4 (million)
% 3.08/1.74 % (31876)------------------------------
% 3.08/1.74 % (31876)------------------------------
% 3.08/1.74 % (31871)Refutation found. Thanks to Tanya!
% 3.08/1.74 % SZS status Theorem for DTF2THF_31760
% 3.08/1.74 % SZS output start Proof for DTF2THF_31760
% 3.08/1.74 thf(type_def_5, type, set: ($tType) > $tType).
% 3.08/1.74 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 3.08/1.74 thf(type_def_8, type, standard_Constant: ($tType) > $tType).
% 3.08/1.74 thf(type_def_9, type, labeled_graph: ($tType * $tType) > $tType).
% 3.08/1.74 thf(type_def_10, type, allegorical_term: ($tType) > $tType).
% 3.08/1.74 thf(type_def_11, type, itself: ($tType) > $tType).
% 3.08/1.74 thf(type_def_12, type, nat: $tType).
% 3.08/1.74 thf(type_def_13, type, v2: $tType).
% 3.08/1.74 thf(type_def_14, type, v: $tType).
% 3.08/1.74 thf(func_def_0, type, standard_Constant: ($tType) > $tType).
% 3.08/1.74 thf(func_def_1, type, allegorical_term: ($tType) > $tType).
% 3.08/1.74 thf(func_def_2, type, labeled_graph: ($tType * $tType) > $tType).
% 3.08/1.74 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 3.08/1.74 thf(func_def_4, type, set: ($tType) > $tType).
% 3.08/1.74 thf(func_def_5, type, nat: $tType).
% 3.08/1.74 thf(func_def_6, type, itself: ($tType) > $tType).
% 3.08/1.74 thf(func_def_7, type, v: $tType).
% 3.08/1.74 thf(func_def_8, type, v2: $tType).
% 3.08/1.74 thf(func_def_9, type, bounded_lattice: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_10, type, type: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_11, type, bot: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_12, type, lattice: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_13, type, preorder: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_14, type, semilattice_sup: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_15, type, bounde1808546759up_bot: !>[X0: $tType]:($o)).
% 3.08/1.74 thf(func_def_16, type, bNF_Gr: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > set @ product_prod @ X0 @ X1)).
% 3.08/1.74 thf(func_def_17, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 3.08/1.74 thf(func_def_18, type, standard_S_Bot: !>[X0: $tType]:(standard_Constant @ X0)).
% 3.08/1.74 thf(func_def_19, type, standard_S_Const: !>[X0: $tType]:(X0 > standard_Constant @ X0)).
% 3.08/1.74 thf(func_def_20, type, standard_S_Top: !>[X0: $tType]:(standard_Constant @ X0)).
% 3.08/1.74 thf(func_def_21, type, getRel: !>[X0: $tType, X1: $tType]:(X0 > labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X1)).
% 3.08/1.74 thf(func_def_22, type, semantics: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > allegorical_term @ X0 > set @ product_prod @ X1 @ X1)).
% 3.08/1.74 thf(func_def_23, type, edge_preserving: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X2 @ product_prod @ X0 @ X0 > set @ product_prod @ X2 @ product_prod @ X1 @ X1 > $o)).
% 3.08/1.74 thf(func_def_24, type, graph_homomorphism: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.08/1.74 thf(func_def_25, type, graph_union: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.08/1.74 thf(func_def_26, type, labeled_LG: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > labeled_graph @ X0 @ X1)).
% 3.08/1.74 thf(func_def_27, 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)).
% 3.08/1.74 thf(func_def_28, type, labeled_edges: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X0 @ product_prod @ X1 @ X1)).
% 3.08/1.74 thf(func_def_29, type, labeled_vertices: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ X1)).
% 3.08/1.74 thf(func_def_30, type, map_graph: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > labeled_graph @ X2 @ X0 > labeled_graph @ X2 @ X1)).
% 3.08/1.74 thf(func_def_31, type, restrict: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.08/1.74 thf(func_def_32, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.08/1.74 thf(func_def_33, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.08/1.74 thf(func_def_34, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.08/1.74 thf(func_def_35, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.08/1.74 thf(func_def_36, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.08/1.74 thf(func_def_37, type, field: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0)).
% 3.08/1.74 thf(func_def_38, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 3.08/1.74 thf(func_def_39, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.08/1.74 thf(func_def_40, type, agree_on: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.08/1.74 thf(func_def_41, 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)).
% 3.08/1.74 thf(func_def_42, 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)).
% 3.08/1.74 thf(func_def_43, 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)).
% 3.08/1.74 thf(func_def_44, 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)).
% 3.08/1.74 thf(func_def_45, type, maintained: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.08/1.74 thf(func_def_46, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.08/1.74 thf(func_def_47, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.08/1.74 thf(func_def_48, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 3.08/1.74 thf(func_def_49, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.08/1.74 thf(func_def_50, type, standa1138209853_rules: !>[X0: $tType]:(set @ X0 > set @ product_prod @ labeled_graph @ standard_Constant @ X0 @ nat @ labeled_graph @ standard_Constant @ X0 @ nat)).
% 3.08/1.74 thf(func_def_51, type, standa2002409347_rules: !>[X0: $tType]:(set @ standard_Constant @ X0 > set @ product_prod @ labeled_graph @ standard_Constant @ X0 @ nat @ labeled_graph @ standard_Constant @ X0 @ nat)).
% 3.08/1.74 thf(func_def_52, type, standa244753842y_rule: !>[X0: $tType]:(product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.08/1.74 thf(func_def_53, type, standa438229444_rules: !>[X0: $tType]:(set @ X0 > set @ standard_Constant @ X0 > set @ product_prod @ labeled_graph @ standard_Constant @ X0 @ nat @ labeled_graph @ standard_Constant @ X0 @ nat)).
% 3.08/1.74 thf(func_def_54, type, standa400804411p_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.08/1.74 thf(func_def_55, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.08/1.74 thf(func_def_56, type, c: set @ v2).
% 3.08/1.74 thf(func_def_57, type, g: labeled_graph @ standard_Constant @ v2 @ v).
% 3.08/1.74 thf(func_def_58, type, l: set @ standard_Constant @ v2).
% 3.08/1.74 thf(func_def_59, type, h: v > v).
% 3.08/1.74 thf(func_def_60, type, m: v2 > v).
% 3.08/1.74 thf(func_def_61, type, x: v).
% 3.08/1.74 thf(func_def_62, type, xa: v2).
% 3.08/1.74 thf(func_def_63, type, y: v2).
% 3.08/1.74 thf(func_def_79, type, sK0: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.08/1.74 thf(func_def_80, type, sK1: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.08/1.74 thf(func_def_81, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.08/1.74 thf(func_def_82, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.08/1.74 thf(func_def_83, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.08/1.74 thf(func_def_84, type, sK5: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.08/1.74 thf(func_def_85, type, sK6: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.08/1.74 thf(func_def_86, type, sK7: !>[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)).
% 3.08/1.74 thf(func_def_87, type, sK8: !>[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)).
% 3.08/1.74 thf(func_def_88, type, sK9: !>[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)).
% 3.08/1.74 thf(func_def_89, type, sK10: !>[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)).
% 3.08/1.74 thf(func_def_90, type, sK11: !>[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)).
% 3.08/1.74 thf(func_def_91, type, sK12: !>[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)).
% 3.08/1.74 thf(func_def_92, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.08/1.74 thf(func_def_93, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.08/1.74 thf(func_def_94, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.08/1.74 thf(func_def_95, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.08/1.74 thf(func_def_96, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.08/1.74 thf(func_def_97, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.08/1.74 thf(func_def_98, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.08/1.74 thf(func_def_99, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.08/1.74 thf(func_def_100, type, sK21: v2 > v).
% 3.08/1.74 thf(func_def_101, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.08/1.74 thf(func_def_102, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.08/1.74 thf(func_def_103, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.08/1.74 thf(func_def_104, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.08/1.74 thf(func_def_105, type, sK26: !>[X0: $tType]:(set @ X0 > product_prod @ X0 @ X0 > X0)).
% 3.08/1.74 thf(func_def_106, type, sK27: !>[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)).
% 3.08/1.74 thf(func_def_107, type, sK28: !>[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)).
% 3.08/1.74 thf(func_def_108, type, sK29: !>[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)).
% 3.08/1.74 thf(func_def_109, type, sK30: !>[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)).
% 3.08/1.74 thf(func_def_110, type, sK31: !>[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)).
% 3.08/1.74 thf(func_def_111, type, sK32: !>[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)).
% 3.08/1.74 thf(func_def_112, type, sK33: !>[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)).
% 3.08/1.74 thf(func_def_113, type, sK34: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.08/1.74 thf(func_def_114, type, sK35: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.08/1.74 thf(func_def_115, type, sK36: !>[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)).
% 3.08/1.74 thf(func_def_116, type, sK37: !>[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)).
% 3.08/1.74 thf(func_def_117, type, sK38: !>[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)).
% 3.08/1.74 thf(func_def_118, type, sK39: !>[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)).
% 3.08/1.74 thf(func_def_119, type, sK40: !>[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)).
% 3.08/1.74 thf(func_def_120, type, sK41: !>[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)).
% 3.08/1.74 thf(func_def_121, type, sK42: v2 > v).
% 3.08/1.74 thf(func_def_122, type, sK43: v2 > v).
% 3.08/1.74 thf(func_def_123, type, sK44: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.08/1.74 thf(func_def_124, type, sK45: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.08/1.74 thf(func_def_125, 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) > X6)).
% 3.08/1.74 thf(func_def_126, 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) > X5)).
% 3.08/1.74 thf(func_def_127, 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) > X4)).
% 3.08/1.74 thf(func_def_128, 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) > X3)).
% 3.08/1.74 thf(func_def_129, 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) > X2)).
% 3.08/1.74 thf(func_def_130, type, sK51: !>[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)).
% 3.08/1.74 thf(func_def_131, type, sK52: !>[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)).
% 3.08/1.74 thf(func_def_132, type, sK53: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.08/1.74 thf(func_def_133, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > labeled_graph @ X2 @ X1 > X1)).
% 3.08/1.74 thf(func_def_134, 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) > X4)).
% 3.08/1.74 thf(func_def_135, type, sK56: !>[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)).
% 3.08/1.74 thf(func_def_136, type, sK57: !>[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)).
% 3.08/1.74 thf(func_def_137, type, sK58: !>[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)).
% 3.08/1.74 thf(func_def_138, type, sK59: !>[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)).
% 3.08/1.74 thf(func_def_139, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.08/1.74 thf(func_def_140, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.08/1.74 thf(func_def_141, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.08/1.74 thf(func_def_142, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.08/1.74 thf(func_def_143, type, sK64: v2 > v).
% 3.08/1.74 thf(func_def_144, type, sK65: v).
% 3.08/1.74 thf(func_def_146, type, ph67: !>[X0: $tType]:(X0)).
% 3.08/1.74 thf(f935,plain,(
% 3.08/1.74 $false),
% 3.08/1.74 inference(subsumption_resolution,[status(thm)],[f934,f848])).
% 3.08/1.74 thf(f848,plain,(
% 3.08/1.74 ((graph_homomorphism @ standard_Constant @ v2 @ v @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g) @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g)))) = $true)),
% 3.08/1.74 inference(cnf_transformation,[status(thm)],[f390])).
% 3.08/1.74 thf(f390,plain,(
% 3.08/1.74 ((graph_homomorphism @ standard_Constant @ v2 @ v @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g) @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g)))) = $true)),
% 3.08/1.74 inference(fool_elimination,[status(thm)],[f389])).
% 3.08/1.74 thf(f389,plain,(
% 3.08/1.74 (graph_homomorphism @ standard_Constant @ v2 @ v @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g) @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g))))),
% 3.08/1.74 inference(rectify,[status(thm)],[f9])).
% 3.08/1.74 thf(f9,axiom,(
% 3.08/1.74 (graph_homomorphism @ standard_Constant @ v2 @ v @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g) @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g))))),
% 3.08/1.74 file('/export/starexec/sandbox/tmp/tmp.TNBOsnf0Kz/DTF2THF_31760.p',fact_8_h_I3_J)).
% 3.08/1.74 thf(f934,plain,(
% 3.08/1.74 ((graph_homomorphism @ standard_Constant @ v2 @ v @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g) @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g)))) != $true)),
% 3.08/1.74 inference(trivial_inequality_removal,[status(thm)],[f924])).
% 3.08/1.74 thf(f924,plain,(
% 3.08/1.74 ((graph_homomorphism @ standard_Constant @ v2 @ v @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g) @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g)))) != $true) | ($true != $true)),
% 3.08/1.74 inference(superposition,[status(thm)],[f922,f835])).
% 3.08/1.74 thf(f835,plain,(
% 3.08/1.74 ((member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g))) = $true)),
% 3.08/1.74 inference(cnf_transformation,[status(thm)],[f518])).
% 3.08/1.74 thf(f518,plain,(
% 3.08/1.74 ((member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g))) = $true)),
% 3.08/1.74 inference(fool_elimination,[status(thm)],[f517])).
% 3.08/1.74 thf(f517,plain,(
% 3.08/1.74 (member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g)))),
% 3.08/1.74 inference(rectify,[status(thm)],[f7])).
% 3.08/1.74 thf(f7,axiom,(
% 3.08/1.74 (member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ (map_graph @ v @ v @ standard_Constant @ v2 @ (bNF_Gr @ v @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ g) @ h) @ g)))),
% 3.08/1.74 file('/export/starexec/sandbox/tmp/tmp.TNBOsnf0Kz/DTF2THF_31760.p',fact_6__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ay_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062)).
% 3.08/1.74 thf(f922,plain,(
% 3.08/1.74 ( ! [X0 : labeled_graph @ standard_Constant @ v2 @ v] : (($true != (member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ X0))) | ((graph_homomorphism @ standard_Constant @ v2 @ v @ v @ X0 @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ X0))) != $true)) )),
% 3.08/1.74 inference(trivial_inequality_removal,[status(thm)],[f921])).
% 3.08/1.74 thf(f921,plain,(
% 3.08/1.74 ( ! [X0 : labeled_graph @ standard_Constant @ v2 @ v] : (($true != $true) | ($true != (member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ X0))) | ((graph_homomorphism @ standard_Constant @ v2 @ v @ v @ X0 @ g @ (id_on @ v @ (labeled_vertices @ standard_Constant @ v2 @ v @ X0))) != $true)) )),
% 3.08/1.74 inference(superposition,[status(thm)],[f880,f875])).
% 3.08/1.74 thf(f875,plain,(
% 3.08/1.74 ( ! [X1 : $tType,X0 : $tType,X2 : X0,X3 : X0,X6 : labeled_graph @ X1 @ X0,X4 : X1,X5 : labeled_graph @ X1 @ X0] : (((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X6)) = $true) | ((graph_homomorphism @ X1 @ X0 @ X0 @ X5 @ X6 @ (id_on @ X0 @ (labeled_vertices @ X1 @ X0 @ X5))) != $true) | ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X5)) != $true)) )),
% 3.08/1.74 inference(cnf_transformation,[status(thm)],[f740])).
% 3.08/1.74 thf(f740,plain,(
% 3.08/1.74 ! [X0 : $tType,X1 : $tType,X2 : X0,X3 : X0,X4 : X1,X5 : labeled_graph @ X1 @ X0,X6 : labeled_graph @ X1 @ X0] : (((graph_homomorphism @ X1 @ X0 @ X0 @ X5 @ X6 @ (id_on @ X0 @ (labeled_vertices @ X1 @ X0 @ X5))) != $true) | ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X6)) = $true) | ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X5)) != $true))),
% 3.08/1.74 inference(flattening,[status(thm)],[f739])).
% 3.08/1.74 thf(f739,plain,(
% 3.08/1.74 ! [X0 : $tType,X1 : $tType,X2 : X0,X3 : X0,X4 : X1,X5 : labeled_graph @ X1 @ X0,X6 : labeled_graph @ X1 @ X0] : ((((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X6)) = $true) | ((graph_homomorphism @ X1 @ X0 @ X0 @ X5 @ X6 @ (id_on @ X0 @ (labeled_vertices @ X1 @ X0 @ X5))) != $true)) | ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X5)) != $true))),
% 3.08/1.74 inference(ennf_transformation,[status(thm)],[f458])).
% 3.08/1.74 thf(f458,plain,(
% 3.08/1.74 ! [X0 : $tType,X1 : $tType,X2 : X0,X3 : X0,X4 : X1,X5 : labeled_graph @ X1 @ X0,X6 : labeled_graph @ X1 @ X0] : (((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X5)) = $true) => (((graph_homomorphism @ X1 @ X0 @ X0 @ X5 @ X6 @ (id_on @ X0 @ (labeled_vertices @ X1 @ X0 @ X5))) = $true) => ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X6)) = $true)))),
% 3.08/1.74 inference(fool_elimination,[status(thm)],[f457])).
% 3.08/1.74 thf(f457,plain,(
% 3.08/1.74 ! [X0 : $tType,X1 : $tType,X2 : X0,X3 : X0,X4 : X1,X5 : labeled_graph @ X1 @ X0,X6 : labeled_graph @ X1 @ X0] : ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X5)) => ((graph_homomorphism @ X1 @ X0 @ X0 @ X5 @ X6 @ (id_on @ X0 @ (labeled_vertices @ X1 @ X0 @ X5))) => (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X2 @ X3) @ (getRel @ X1 @ X0 @ X4 @ X6))))),
% 3.08/1.74 inference(rectify,[status(thm)],[f30])).
% 3.08/1.74 thf(f30,axiom,(
% 3.08/1.74 ! [X0 : $tType,X1 : $tType,X25 : X0,X26 : X0,X27 : X1,X28 : labeled_graph @ X1 @ X0,X36 : labeled_graph @ X1 @ X0] : ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X25 @ X26) @ (getRel @ X1 @ X0 @ X27 @ X28)) => ((graph_homomorphism @ X1 @ X0 @ X0 @ X28 @ X36 @ (id_on @ X0 @ (labeled_vertices @ X1 @ X0 @ X28))) => (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X25 @ X26) @ (getRel @ X1 @ X0 @ X27 @ X36))))),
% 3.08/1.74 file('/export/starexec/sandbox/tmp/tmp.TNBOsnf0Kz/DTF2THF_31760.p',fact_29_getRel__subgraph)).
% 3.08/1.74 thf(f880,plain,(
% 3.08/1.74 ((member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ g)) != $true)),
% 3.08/1.74 inference(cnf_transformation,[status(thm)],[f707])).
% 3.08/1.74 thf(f707,plain,(
% 3.08/1.74 ((member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ g)) != $true)),
% 3.08/1.74 inference(flattening,[status(thm)],[f702])).
% 3.08/1.74 thf(f702,plain,(
% 3.08/1.74 ~((member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ g)) = $true)),
% 3.08/1.74 inference(fool_elimination,[status(thm)],[f701])).
% 3.08/1.74 thf(f701,plain,(
% 3.08/1.74 ~(member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ g))),
% 3.08/1.74 inference(rectify,[status(thm)],[f280])).
% 3.08/1.74 thf(f280,negated_conjecture,(
% 3.08/1.74 ~(member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ g))),
% 3.08/1.74 inference(negated_conjecture,[status(cth)],[f279])).
% 3.08/1.74 thf(f279,conjecture,(
% 3.08/1.74 (member @ product_prod @ v @ v @ (product_Pair @ v @ v @ (m @ xa) @ (m @ xa)) @ (getRel @ standard_Constant @ v2 @ v @ (standard_S_Const @ v2 @ y) @ g))),
% 3.08/1.74 file('/export/starexec/sandbox/tmp/tmp.TNBOsnf0Kz/DTF2THF_31760.p',conj_0)).
% 3.08/1.74 % SZS output end Proof for DTF2THF_31760
% 3.08/1.74 % (31871)------------------------------
% 3.08/1.74 % (31871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.74 % (31871)Termination reason: Refutation
% 3.08/1.74
% 3.08/1.74 % (31871)Memory used [KB]: 7036
% 3.08/1.74 % (31871)Time elapsed: 0.031 s
% 3.08/1.74 % (31871)Instructions burned: 87 (million)
% 3.08/1.74 % (31871)------------------------------
% 3.08/1.74 % (31871)------------------------------
% 3.08/1.74 % (31865)Success in time 0.037 s
%------------------------------------------------------------------------------