%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP177^2 : 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 : n022.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:56:11 AM UTC 2026
% Result : Theorem 3.05s 1.75s
% Output : Refutation 3.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : ITP177^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n022.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:28:51 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.26/0.45 ---- Original DTF file ---
% 0.26/0.45 thf(spec,logic,$$dhol).
% 0.26/0.45 %------------------------------------------------------------------------------
% 0.26/0.45 % File : ITP177^2 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.45 % Domain : Interactive Theorem Proving
% 0.26/0.45 % Problem : Sledgehammer StandardRules problem prob_340__5390244_1
% 0.26/0.45 % Version : Especial.
% 0.26/0.45 % English :
% 0.26/0.45
% 0.26/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.45 % Source : [Des21]
% 0.26/0.45 % Names : StandardRules/prob_340__5390244_1 [Des21]
% 0.26/0.45
% 0.26/0.45 % Status : Theorem
% 0.26/0.45 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.26/0.45 % Syntax : Number of formulae : 318 ( 126 unt; 58 typ; 0 def)
% 0.26/0.45 % Number of atoms : 655 ( 255 equ; 0 cnn)
% 0.26/0.45 % Maximal formula atoms : 6 ( 2 avg)
% 0.26/0.45 % Number of connectives : 6616 ( 64 ~; 8 |; 42 &;6248 @)
% 0.26/0.45 % ( 0 <=>; 254 =>; 0 <=; 0 <~>)
% 0.26/0.45 % Maximal formula depth : 28 ( 10 avg)
% 0.26/0.45 % Number of types : 4 ( 3 usr)
% 0.26/0.45 % Number of type conns : 158 ( 158 >; 0 *; 0 +; 0 <<)
% 0.26/0.45 % Number of symbols : 58 ( 55 usr; 7 con; 0-8 aty)
% 0.26/0.45 % Number of variables : 1337 ( 61 ^;1186 !; 9 ?;1337 :)
% 0.26/0.45 % ( 81 !>; 0 ?*; 0 @-; 0 @+)
% 0.26/0.45 % SPC : TH1_THM_EQU_NAR
% 0.26/0.45
% 0.26/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:23:02.646
% 0.26/0.45 %------------------------------------------------------------------------------
% 0.26/0.45 % Could-be-implicit typings (9)
% 0.26/0.45 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 0.26/0.45 standard_Constant: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 0.26/0.45 allegorical_term: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 0.26/0.45 labeled_graph: $tType > $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_Product__Type_Oprod,type,
% 0.26/0.45 product_prod: $tType > $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_Set_Oset,type,
% 0.26/0.45 set: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_Nat_Onat,type,
% 0.26/0.45 nat: $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_t_itself,type,
% 0.26/0.45 itself: $tType > $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_tf_b,type,
% 0.26/0.45 b: $tType ).
% 0.26/0.45
% 0.26/0.45 thf(ty_tf_a,type,
% 0.26/0.45 a: $tType ).
% 0.26/0.45
% 0.26/0.45 % Explicit typings (49)
% 0.26/0.45 thf(sy_c_BNF__Def_OGr,type,
% 0.26/0.45 bNF_Gr:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 0.26/0.45 equiv_congruent:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Equiv__Relations_Oequiv,type,
% 0.26/0.45 equiv_equiv:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Equiv__Relations_Oproj,type,
% 0.26/0.45 equiv_proj:
% 0.26/0.45 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Hilbert__Choice_OEps,type,
% 0.26/0.45 hilbert_Eps:
% 0.26/0.45 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_If,type,
% 0.26/0.45 if:
% 0.26/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt,type,
% 0.26/0.45 standard_S_Idt:
% 0.26/0.45 !>[V: $tType] : ( standard_Constant @ V ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 0.26/0.45 getRel:
% 0.26/0.45 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 0.26/0.45 semantics:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Oedge__preserving,type,
% 0.26/0.45 edge_preserving:
% 0.26/0.45 !>[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 ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 0.26/0.45 graph_homomorphism:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 0.26/0.45 graph_union:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 0.26/0.45 labeled_LG:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 0.26/0.45 labele1974067554_graph:
% 0.26/0.45 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 0.26/0.45 labeled_edges:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 0.26/0.45 labeled_vertices:
% 0.26/0.45 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Omap__graph,type,
% 0.26/0.45 map_graph:
% 0.26/0.45 !>[C: $tType,B: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ B ) ) > ( labeled_graph @ A @ C ) > ( labeled_graph @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_LabeledGraphs_Orestrict,type,
% 0.26/0.45 restrict:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.26/0.45 bot_bot:
% 0.26/0.45 !>[A: $tType] : A ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.26/0.45 ord_less_eq:
% 0.26/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Product__Type_OPair,type,
% 0.26/0.45 product_Pair:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.26/0.45 product_rec_prod:
% 0.26/0.45 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_OId__on,type,
% 0.26/0.45 id_on:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_OImage,type,
% 0.26/0.45 image:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_Orefl__on,type,
% 0.26/0.45 refl_on:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Relation_Orelcomp,type,
% 0.26/0.45 relcomp:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oagree__on,type,
% 0.26/0.45 agree_on:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 0.26/0.45 consequence_graph:
% 0.26/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oextensible,type,
% 0.26/0.45 extensible:
% 0.26/0.45 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ X @ V ) ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oleast,type,
% 0.26/0.45 least:
% 0.26/0.45 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 0.26/0.45 least_559130134_graph:
% 0.26/0.45 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_RulesAndChains_Omaintained,type,
% 0.26/0.45 maintained:
% 0.26/0.45 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Set_OCollect,type,
% 0.26/0.45 collect:
% 0.26/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Set_Oimage,type,
% 0.26/0.45 image2:
% 0.26/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Set_Oinsert,type,
% 0.26/0.45 insert:
% 0.26/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_Set_Othe__elem,type,
% 0.26/0.45 the_elem:
% 0.26/0.45 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
% 0.26/0.45 standa2002409347_rules:
% 0.26/0.45 !>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 0.26/0.45 standa244753842y_rule:
% 0.26/0.45 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule,type,
% 0.26/0.45 standa245363227y_rule:
% 0.26/0.45 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule,type,
% 0.26/0.45 standa1805702094y_rule:
% 0.26/0.45 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule,type,
% 0.26/0.45 standa2114124375e_rule:
% 0.26/0.45 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_c_member,type,
% 0.26/0.45 member:
% 0.26/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_G,type,
% 0.26/0.45 g: labeled_graph @ ( standard_Constant @ a ) @ b ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_L,type,
% 0.26/0.45 l: set @ ( standard_Constant @ a ) ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_P____,type,
% 0.26/0.45 p: b > b > $o ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_f____,type,
% 0.26/0.45 f: b > b ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_l____,type,
% 0.26/0.45 l2: standard_Constant @ a ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_x____,type,
% 0.26/0.45 x: b ).
% 0.26/0.45
% 0.26/0.45 thf(sy_v_y____,type,
% 0.26/0.45 y: b ).
% 0.26/0.45
% 0.26/0.45 % Relevant facts (256)
% 0.26/0.45 thf(fact_0__092_060open_062_092_060And_062xa_O_A_092_060lbrakk_062f_Axa_A_092_060in_062_Avertices_AG_059_Axa_A_092_060notin_062_Avertices_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
% 0.26/0.45 ! [Xa: b] :
% 0.26/0.45 ( ( member @ b @ ( f @ Xa ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( member @ b @ Xa @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % \<open>\<And>xa. \<lbrakk>f xa \<in> vertices G; xa \<notin> vertices G\<rbrakk> \<Longrightarrow> False\<close>
% 0.26/0.45 thf(fact_1_fv_I2_J,axiom,
% 0.26/0.45 member @ b @ ( f @ y ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ).
% 0.26/0.45
% 0.26/0.45 % fv(2)
% 0.26/0.45 thf(fact_2_fv_I1_J,axiom,
% 0.26/0.45 member @ b @ ( f @ x ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ).
% 0.26/0.45
% 0.26/0.45 % fv(1)
% 0.26/0.45 thf(fact_3_gu2,axiom,
% 0.26/0.45 ! [X2: b] :
% 0.26/0.45 ( ( member @ b @ X2 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( member @ b @ ( f @ X2 ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % gu2
% 0.26/0.45 thf(fact_4_assms_I1_J,axiom,
% 0.26/0.45 ( g
% 0.26/0.45 = ( restrict @ ( standard_Constant @ a ) @ b @ g ) ) ).
% 0.26/0.45
% 0.26/0.45 % assms(1)
% 0.26/0.45 thf(fact_5_r1,axiom,
% 0.26/0.45 ! [X3: b] :
% 0.26/0.45 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 = ( p @ X3 @ X3 ) ) ).
% 0.26/0.45
% 0.26/0.45 % r1
% 0.26/0.45 thf(fact_6_assms_I2_J,axiom,
% 0.26/0.45 ! [X4: product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat )] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ X4 @ ( standa2002409347_rules @ a @ l ) )
% 0.26/0.45 => ( maintained @ ( standard_Constant @ a ) @ nat @ b @ X4 @ g ) ) ).
% 0.26/0.45
% 0.26/0.45 % assms(2)
% 0.26/0.45 thf(fact_7_gr,axiom,
% 0.26/0.45 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ ( f @ x ) @ ( f @ y ) ) @ ( getRel @ ( standard_Constant @ a ) @ b @ l2 @ g ) ).
% 0.26/0.45
% 0.26/0.45 % gr
% 0.26/0.45 thf(fact_8_a,axiom,
% 0.26/0.45 member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ l2 @ ( product_Pair @ b @ b @ ( f @ x ) @ ( f @ y ) ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) ).
% 0.26/0.45
% 0.26/0.45 % a
% 0.26/0.45 thf(fact_9_local_Orefl,axiom,
% 0.26/0.45 refl_on @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 0.26/0.45
% 0.26/0.45 % local.refl
% 0.26/0.45 thf(fact_10_equiv,axiom,
% 0.26/0.45 equiv_equiv @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 0.26/0.45
% 0.26/0.45 % equiv
% 0.26/0.45 thf(fact_11__092_060open_062graph__union_A_Imap__graph__fn_AG_Af_J_AG_A_061_AG_092_060close_062,axiom,
% 0.26/0.45 ( ( graph_union @ ( standard_Constant @ a ) @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) @ g )
% 0.26/0.45 = g ) ).
% 0.26/0.45
% 0.26/0.45 % \<open>graph_union (map_graph_fn G f) G = G\<close>
% 0.26/0.45 thf(fact_12_gu1,axiom,
% 0.26/0.45 ! [L2: standard_Constant @ a,X2: b,Y: b] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ L2 @ ( product_Pair @ b @ b @ X2 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( ( member @ b @ X2 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( ( member @ b @ Y @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ L2 @ ( product_Pair @ b @ b @ ( f @ X2 ) @ ( f @ Y ) ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % gu1
% 0.26/0.45 thf(fact_13__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 0.26/0.45 ! [X3: b] :
% 0.26/0.45 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ X3 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % \<open>\<And>x. x \<in> vertices G \<Longrightarrow> (x, x) \<in> getRel S_Idt G\<close>
% 0.26/0.45 thf(fact_14_labeled__graph_Osel_I2_J,axiom,
% 0.26/0.45 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.26/0.45 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.26/0.45 = X22 ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.sel(2)
% 0.26/0.45 thf(fact_15_subg,axiom,
% 0.26/0.45 graph_homomorphism @ ( standard_Constant @ a ) @ b @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) @ g @ ( id_on @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subg
% 0.26/0.45 thf(fact_16_vertices__restrict,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
% 0.26/0.45 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
% 0.26/0.45 = ( labeled_vertices @ B @ A @ G ) ) ).
% 0.26/0.45
% 0.26/0.45 % vertices_restrict
% 0.26/0.45 thf(fact_17_restrict__idemp,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X3: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X3 ) )
% 0.26/0.45 = ( restrict @ A @ B @ X3 ) ) ).
% 0.26/0.45
% 0.26/0.45 % restrict_idemp
% 0.26/0.45 thf(fact_18_labeled__graph_Oinject,axiom,
% 0.26/0.45 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V,Y1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),Y2: set @ V] :
% 0.26/0.45 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 0.26/0.45 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 0.26/0.45 = ( ( X1 = Y1 )
% 0.26/0.45 & ( X22 = Y2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.inject
% 0.26/0.45 thf(fact_19_graph__union__idemp_I3_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_union @ A @ B @ A2 @ ( graph_union @ A @ B @ B2 @ A2 ) )
% 0.26/0.45 = ( graph_union @ A @ B @ B2 @ A2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_union_idemp(3)
% 0.26/0.45 thf(fact_20_graph__union__idemp_I2_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_union @ A @ B @ A2 @ ( graph_union @ A @ B @ A2 @ B2 ) )
% 0.26/0.45 = ( graph_union @ A @ B @ A2 @ B2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_union_idemp(2)
% 0.26/0.45 thf(fact_21_graph__union__idemp_I1_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_union @ A @ B @ A2 @ A2 )
% 0.26/0.45 = A2 ) ).
% 0.26/0.45
% 0.26/0.45 % graph_union_idemp(1)
% 0.26/0.45 thf(fact_22_map__graph__preserves__restricted,axiom,
% 0.26/0.45 ! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) )
% 0.26/0.45 => ( ( map_graph @ B @ C @ A @ F @ G )
% 0.26/0.45 = ( restrict @ A @ C @ ( map_graph @ B @ C @ A @ F @ G ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % map_graph_preserves_restricted
% 0.26/0.45 thf(fact_23_graph__union__preserves__restrict,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( G_1
% 0.26/0.45 = ( restrict @ A @ B @ G_1 ) )
% 0.26/0.45 => ( ( G_2
% 0.26/0.45 = ( restrict @ A @ B @ G_2 ) )
% 0.26/0.45 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.45 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_union_preserves_restrict
% 0.26/0.45 thf(fact_24_labeled__graph_Ocollapse,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.45 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 0.26/0.45 = Labeled_graph ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.collapse
% 0.26/0.45 thf(fact_25_subgraph__refl,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 0.26/0.45 = ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_refl
% 0.26/0.45 thf(fact_26_subgraph__restrict,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 0.26/0.45 = ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_restrict
% 0.26/0.45 thf(fact_27_graph__homomorphism__Id,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A3 ) @ ( restrict @ A @ B @ A3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A3 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_homomorphism_Id
% 0.26/0.45 thf(fact_28_map__graph__preserves__subgraph,axiom,
% 0.26/0.45 ! [C: $tType,B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ A2 @ B2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) )
% 0.26/0.45 => ( graph_homomorphism @ A @ C @ C @ ( map_graph @ B @ C @ A @ F @ A2 ) @ ( map_graph @ B @ C @ A @ F @ B2 ) @ ( id_on @ C @ ( labeled_vertices @ A @ C @ ( map_graph @ B @ C @ A @ F @ A2 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % map_graph_preserves_subgraph
% 0.26/0.45 thf(fact_29_map__graph__fn__id_I2_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X5: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( map_graph @ B @ B @ A @ ( id_on @ B @ ( labeled_vertices @ A @ B @ X5 ) ) @ X5 )
% 0.26/0.45 = ( restrict @ A @ B @ X5 ) ) ).
% 0.26/0.45
% 0.26/0.45 % map_graph_fn_id(2)
% 0.26/0.45 thf(fact_30_map__graph__fn__graphI,axiom,
% 0.26/0.45 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F: C > B] :
% 0.26/0.45 ( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F ) @ G )
% 0.26/0.45 = ( restrict @ A @ B @ ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F ) @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % map_graph_fn_graphI
% 0.26/0.45 thf(fact_31_graph__homo,axiom,
% 0.26/0.45 ! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,F: B > C] :
% 0.26/0.45 ( ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) )
% 0.26/0.45 => ( graph_homomorphism @ A @ B @ C @ G @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F ) @ G ) @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % graph_homo
% 0.26/0.45 thf(fact_32_labeled__graph_Osel_I1_J,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.26/0.45 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.26/0.45 = X1 ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.sel(1)
% 0.26/0.45 thf(fact_33_subgraph__def,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.45 = ( ( G_1
% 0.26/0.45 = ( restrict @ A @ B @ G_1 ) )
% 0.26/0.45 & ( G_2
% 0.26/0.45 = ( restrict @ A @ B @ G_2 ) )
% 0.26/0.45 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.45 = G_2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_def
% 0.26/0.45 thf(fact_34_subgraph__trans,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 0.26/0.45 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_trans
% 0.26/0.45 thf(fact_35_map__graph__fn__eqI,axiom,
% 0.26/0.45 ! [C: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F: A > C,G2: A > C] :
% 0.26/0.45 ( ! [X6: A] :
% 0.26/0.45 ( ( member @ A @ X6 @ ( labeled_vertices @ B @ A @ G ) )
% 0.26/0.45 => ( ( F @ X6 )
% 0.26/0.45 = ( G2 @ X6 ) ) )
% 0.26/0.45 => ( ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F ) @ G )
% 0.26/0.45 = ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G2 ) @ G ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % map_graph_fn_eqI
% 0.26/0.45 thf(fact_36_labeled__graph_Oinduct,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType,P: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.45 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 0.26/0.45 => ( P @ Labeled_graph ) ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.induct
% 0.26/0.45 thf(fact_37_labeled__graph_Oexhaust,axiom,
% 0.26/0.45 ! [L: $tType,V: $tType,Y3: labeled_graph @ L @ V] :
% 0.26/0.45 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 0.26/0.45 ( Y3
% 0.26/0.45 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.exhaust
% 0.26/0.45 thf(fact_38_subgraph__preserves__hom,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,X5: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 0.26/0.45 ( ( graph_homomorphism @ A @ B @ B @ A2 @ B2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) )
% 0.26/0.45 => ( ( graph_homomorphism @ A @ C @ B @ X5 @ A2 @ H )
% 0.26/0.45 => ( graph_homomorphism @ A @ C @ B @ X5 @ B2 @ H ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % subgraph_preserves_hom
% 0.26/0.45 thf(fact_39_labeled__graph_Oexhaust__sel,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.45 ( Labeled_graph
% 0.26/0.45 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.exhaust_sel
% 0.26/0.45 thf(fact_40_labeled__graph_Oexpand,axiom,
% 0.26/0.45 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 0.26/0.45 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 0.26/0.45 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 0.26/0.45 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 0.26/0.45 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 0.26/0.45 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.26/0.45
% 0.26/0.45 % labeled_graph.expand
% 0.26/0.45 thf(fact_41_getRel__hom,axiom,
% 0.26/0.45 ! [C: $tType,A: $tType,B: $tType,Y3: A,Z: A,L3: B,G: labeled_graph @ B @ A,F: A > C] :
% 0.26/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G ) )
% 0.26/0.45 => ( ( member @ A @ Y3 @ ( labeled_vertices @ B @ A @ G ) )
% 0.26/0.45 => ( ( member @ A @ Z @ ( labeled_vertices @ B @ A @ G ) )
% 0.26/0.45 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F @ Y3 ) @ ( F @ Z ) ) @ ( getRel @ B @ C @ L3 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F ) @ G ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % getRel_hom
% 0.26/0.45 thf(fact_42_getRel__map__fn,axiom,
% 0.26/0.45 ! [C: $tType,A: $tType,B: $tType,A22: A,G: labeled_graph @ B @ A,B22: A,L3: B,F: A > C,A3: C,B3: C] :
% 0.26/0.45 ( ( member @ A @ A22 @ ( labeled_vertices @ B @ A @ G ) )
% 0.26/0.45 => ( ( member @ A @ B22 @ ( labeled_vertices @ B @ A @ G ) )
% 0.26/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B22 ) @ ( getRel @ B @ A @ L3 @ G ) )
% 0.26/0.45 => ( ( ( F @ A22 )
% 0.26/0.45 = A3 )
% 0.26/0.45 => ( ( ( F @ B22 )
% 0.26/0.45 = B3 )
% 0.26/0.45 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( getRel @ B @ C @ L3 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F ) @ G ) ) ) ) ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % getRel_map_fn
% 0.26/0.45 thf(fact_43_idt__eq,axiom,
% 0.26/0.45 ! [X2: b,Y: b] :
% 0.26/0.45 ( ( member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ ( standard_S_Idt @ a ) @ ( product_Pair @ b @ b @ X2 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( ( hilbert_Eps @ b @ ( p @ X2 ) )
% 0.26/0.45 = ( hilbert_Eps @ b @ ( p @ Y ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % idt_eq
% 0.26/0.45 thf(fact_44_vert__P,axiom,
% 0.26/0.45 ! [X3: b] :
% 0.26/0.45 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.26/0.45 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ ( hilbert_Eps @ b @ ( p @ X3 ) ) ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % vert_P
% 0.26/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.26/0.45 ! [A: $tType,A3: A,P: A > $o] :
% 0.26/0.45 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.26/0.45 = ( P @ A3 ) ) ).
% 0.26/0.45
% 0.26/0.45 % mem_Collect_eq
% 0.26/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.26/0.45 ! [A: $tType,A2: set @ A] :
% 0.26/0.45 ( ( collect @ A
% 0.26/0.45 @ ^ [X7: A] : ( member @ A @ X7 @ A2 ) )
% 0.26/0.45 = A2 ) ).
% 0.26/0.45
% 0.26/0.45 % Collect_mem_eq
% 0.26/0.45 thf(fact_47_Collect__cong,axiom,
% 0.26/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.26/0.45 ( ! [X6: A] :
% 0.26/0.45 ( ( P @ X6 )
% 0.26/0.45 = ( Q @ X6 ) )
% 0.26/0.45 => ( ( collect @ A @ P )
% 0.26/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Collect_cong
% 0.26/0.45 thf(fact_48_getRel__subgraph,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,Y3: A,Z: A,L3: B,G: labeled_graph @ B @ A,G3: labeled_graph @ B @ A] :
% 0.26/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G ) )
% 0.26/0.45 => ( ( graph_homomorphism @ B @ A @ A @ G @ G3 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G ) ) )
% 0.26/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G3 ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % getRel_subgraph
% 0.26/0.45 thf(fact_49_in__Gr,axiom,
% 0.26/0.45 ! [A: $tType,B: $tType,X3: A,Y3: B,A2: set @ A,F: A > B] :
% 0.26/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bNF_Gr @ A @ B @ A2 @ F ) )
% 0.26/0.45 = ( ( member @ A @ X3 @ A2 )
% 0.26/0.45 & ( ( F @ X3 )
% 0.26/0.45 = Y3 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % in_Gr
% 0.26/0.45 thf(fact_50_Gr__not__in,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,X3: A,F2: set @ A,F: A > B,Y3: B] :
% 0.26/0.45 ( ( ~ ( member @ A @ X3 @ F2 )
% 0.26/0.45 | ( ( F @ X3 )
% 0.26/0.45 != Y3 ) )
% 0.26/0.45 => ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bNF_Gr @ A @ B @ F2 @ F ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Gr_not_in
% 0.26/0.45 thf(fact_51_Id__onI,axiom,
% 0.26/0.45 ! [A: $tType,A3: A,A2: set @ A] :
% 0.26/0.45 ( ( member @ A @ A3 @ A2 )
% 0.26/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A2 ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % Id_onI
% 0.26/0.45 thf(fact_52_getRel__dom_I2_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A3: B,B3: B,L3: A] :
% 0.26/0.45 ( ( G
% 0.26/0.45 = ( restrict @ A @ B @ G ) )
% 0.26/0.45 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( getRel @ A @ B @ L3 @ G ) )
% 0.26/0.45 => ( member @ B @ B3 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 0.26/0.45
% 0.26/0.45 % getRel_dom(2)
% 0.26/0.45 thf(fact_53_getRel__dom_I1_J,axiom,
% 0.26/0.45 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A3: B,B3: B,L3: A] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( getRel @ A @ B @ L3 @ G ) )
% 0.26/0.46 => ( member @ B @ A3 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % getRel_dom(1)
% 0.26/0.46 thf(fact_54_Id__on__iff,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Y3: A,A2: set @ A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( id_on @ A @ A2 ) )
% 0.26/0.46 = ( ( X3 = Y3 )
% 0.26/0.46 & ( member @ A @ X3 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_iff
% 0.26/0.46 thf(fact_55_Id__on__eqI,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.26/0.46 ( ( A3 = B3 )
% 0.26/0.46 => ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id_on @ A @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_eqI
% 0.26/0.46 thf(fact_56_Id__onE,axiom,
% 0.26/0.46 ! [A: $tType,C2: product_prod @ A @ A,A2: set @ A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A2 ) )
% 0.26/0.46 => ~ ! [X6: A] :
% 0.26/0.46 ( ( member @ A @ X6 @ A2 )
% 0.26/0.46 => ( C2
% 0.26/0.46 != ( product_Pair @ A @ A @ X6 @ X6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_onE
% 0.26/0.46 thf(fact_57_refl__onD2,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 0.26/0.46 ( ( refl_on @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 0.26/0.46 => ( member @ A @ Y3 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_onD2
% 0.26/0.46 thf(fact_58_refl__onD1,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 0.26/0.46 ( ( refl_on @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 0.26/0.46 => ( member @ A @ X3 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_onD1
% 0.26/0.46 thf(fact_59_refl__onD,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.26/0.46 ( ( refl_on @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_onD
% 0.26/0.46 thf(fact_60_refl__on__Id__on,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] : ( refl_on @ A @ A2 @ ( id_on @ A @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_on_Id_on
% 0.26/0.46 thf(fact_61_getRel__homR,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,B: $tType,Y3: A,Z: A,L3: B,G: labeled_graph @ B @ A,U: C,F: set @ ( product_prod @ A @ C ),V2: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Y3 @ U ) @ F )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Z @ V2 ) @ F )
% 0.26/0.46 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ U @ V2 ) @ ( getRel @ B @ C @ L3 @ ( map_graph @ A @ C @ B @ F @ G ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % getRel_homR
% 0.26/0.46 thf(fact_62_maintained__refl,axiom,
% 0.26/0.46 ! [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 ) ).
% 0.26/0.46
% 0.26/0.46 % maintained_refl
% 0.26/0.46 thf(fact_63_edge__preserving__on__graphI,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,X5: labeled_graph @ A @ B,F: B > C,Y4: set @ ( product_prod @ A @ ( product_prod @ C @ C ) )] :
% 0.26/0.46 ( ! [L4: A,X6: B,Y5: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L4 @ ( product_Pair @ B @ B @ X6 @ Y5 ) ) @ ( labeled_edges @ A @ B @ X5 ) )
% 0.26/0.46 => ( ( member @ B @ X6 @ ( labeled_vertices @ A @ B @ X5 ) )
% 0.26/0.46 => ( ( member @ B @ Y5 @ ( labeled_vertices @ A @ B @ X5 ) )
% 0.26/0.46 => ( member @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L4 @ ( product_Pair @ C @ C @ ( F @ X6 ) @ ( F @ Y5 ) ) ) @ Y4 ) ) ) )
% 0.26/0.46 => ( edge_preserving @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ X5 ) @ F ) @ ( labeled_edges @ A @ B @ X5 ) @ Y4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % edge_preserving_on_graphI
% 0.26/0.46 thf(fact_64_reflexivity__rule,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,L3: A] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( maintained @ A @ nat @ B @ ( standa245363227y_rule @ A @ L3 ) @ G )
% 0.26/0.46 => ( refl_on @ B @ ( labeled_vertices @ A @ B @ G ) @ ( getRel @ A @ B @ L3 @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % reflexivity_rule
% 0.26/0.46 thf(fact_65_old_Oprod_Oinject,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.26/0.46 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.26/0.46 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.26/0.46 = ( ( A3 = A4 )
% 0.26/0.46 & ( B3 = B4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.inject
% 0.26/0.46 thf(fact_66_prod_Oinject,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.26/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.26/0.46 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.26/0.46 = ( ( X1 = Y1 )
% 0.26/0.46 & ( X22 = Y2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod.inject
% 0.26/0.46 thf(fact_67_refl__on__domain,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 0.26/0.46 ( ( refl_on @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
% 0.26/0.46 => ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 & ( member @ A @ B3 @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_on_domain
% 0.26/0.46 thf(fact_68_GrD2,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X3: A,Fx: B,A2: set @ A,F: A > B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A2 @ F ) )
% 0.26/0.46 => ( ( F @ X3 )
% 0.26/0.46 = Fx ) ) ).
% 0.26/0.46
% 0.26/0.46 % GrD2
% 0.26/0.46 thf(fact_69_GrD1,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,X3: A,Fx: B,A2: set @ A,F: A > B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A2 @ F ) )
% 0.26/0.46 => ( member @ A @ X3 @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % GrD1
% 0.26/0.46 thf(fact_70_edge__preserving__Id,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,Y3: set @ A,X3: set @ ( product_prod @ B @ ( product_prod @ A @ A ) )] : ( edge_preserving @ A @ A @ B @ ( id_on @ A @ Y3 ) @ X3 @ X3 ) ).
% 0.26/0.46
% 0.26/0.46 % edge_preserving_Id
% 0.26/0.46 thf(fact_71_map__graph__edge__preserving,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C: $tType,F: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] : ( edge_preserving @ A @ B @ C @ F @ ( labeled_edges @ C @ A @ G ) @ ( labeled_edges @ C @ B @ ( map_graph @ A @ B @ C @ F @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % map_graph_edge_preserving
% 0.26/0.46 thf(fact_72_edge__preserving__atomic,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E2: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),V1: A,V12: B,V22: A,V23: B,K: C] :
% 0.26/0.46 ( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E2 )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V1 @ V12 ) @ H1 )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V22 @ V23 ) @ H1 )
% 0.26/0.46 => ( ( member @ ( product_prod @ C @ ( product_prod @ A @ A ) ) @ ( product_Pair @ C @ ( product_prod @ A @ A ) @ K @ ( product_Pair @ A @ A @ V1 @ V22 ) ) @ E1 )
% 0.26/0.46 => ( member @ ( product_prod @ C @ ( product_prod @ B @ B ) ) @ ( product_Pair @ C @ ( product_prod @ B @ B ) @ K @ ( product_Pair @ B @ B @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % edge_preserving_atomic
% 0.26/0.46 thf(fact_73_surj__pair,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.26/0.46 ? [X6: A,Y5: B] :
% 0.26/0.46 ( P2
% 0.26/0.46 = ( product_Pair @ A @ B @ X6 @ Y5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % surj_pair
% 0.26/0.46 thf(fact_74_prod__cases,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.26/0.46 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.26/0.46 => ( P @ P2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases
% 0.26/0.46 thf(fact_75_Pair__inject,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.26/0.46 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.26/0.46 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.26/0.46 => ~ ( ( A3 = A4 )
% 0.26/0.46 => ( B3 != B4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Pair_inject
% 0.26/0.46 thf(fact_76_old_Oprod_Oexhaust,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.26/0.46 ~ ! [A5: A,B5: B] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.exhaust
% 0.26/0.46 thf(fact_77_old_Oprod_Oinducts,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.26/0.46 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.26/0.46 => ( P @ Prod ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.inducts
% 0.26/0.46 thf(fact_78_prod__cases3,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.46 ~ ! [A5: A,B5: B,C3: C] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases3
% 0.26/0.46 thf(fact_79_prod__cases4,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.46 ~ ! [A5: A,B5: B,C3: C,D2: D] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases4
% 0.26/0.46 thf(fact_80_prod__cases5,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.46 ~ ! [A5: A,B5: B,C3: C,D2: D,E3: E] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases5
% 0.26/0.46 thf(fact_81_prod__cases6,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.26/0.46 ~ ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases6
% 0.26/0.46 thf(fact_82_prod__cases7,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G4: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) )] :
% 0.26/0.46 ~ ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3,G5: G4] :
% 0.26/0.46 ( Y3
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G4 ) @ E3 @ ( product_Pair @ F3 @ G4 @ F4 @ G5 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases7
% 0.26/0.46 thf(fact_83_prod__induct3,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 0.26/0.46 => ( P @ X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct3
% 0.26/0.46 thf(fact_84_prod__induct4,axiom,
% 0.26/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.46 ( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.26/0.46 => ( P @ X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct4
% 0.26/0.46 thf(fact_85_prod__induct5,axiom,
% 0.26/0.46 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.46 ( ! [A5: A,B5: B,C3: C,D2: D,E3: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) )
% 0.26/0.46 => ( P @ X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct5
% 0.26/0.46 thf(fact_86_prod__induct6,axiom,
% 0.26/0.46 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.26/0.46 ( ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E3 @ F4 ) ) ) ) ) )
% 0.26/0.46 => ( P @ X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct6
% 0.26/0.46 thf(fact_87_prod__induct7,axiom,
% 0.26/0.46 ! [G4: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) )] :
% 0.26/0.46 ( ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G4 ) @ E3 @ ( product_Pair @ F3 @ G4 @ F4 @ G5 ) ) ) ) ) ) )
% 0.26/0.46 => ( P @ X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct7
% 0.26/0.46 thf(fact_88_old_Oprod_Orec,axiom,
% 0.26/0.46 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.26/0.46 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.26/0.46 = ( F1 @ A3 @ B3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.rec
% 0.26/0.46 thf(fact_89_least__consequence__graphI,axiom,
% 0.26/0.46 ! [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] :
% 0.26/0.46 ( ( consequence_graph @ L @ A @ C @ Rs @ G )
% 0.26/0.46 => ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 0.26/0.46 => ( ! [C4: labeled_graph @ L @ X] :
% 0.26/0.46 ( ( consequence_graph @ L @ A @ X @ Rs @ C4 )
% 0.26/0.46 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C4 ) )
% 0.26/0.46 => ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % least_consequence_graphI
% 0.26/0.46 thf(fact_90_leastI,axiom,
% 0.26/0.46 ! [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] :
% 0.26/0.46 ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 0.26/0.46 => ( ! [C4: labeled_graph @ L @ X] :
% 0.26/0.46 ( ( consequence_graph @ L @ A @ X @ Rs @ C4 )
% 0.26/0.46 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C4 ) )
% 0.26/0.46 => ( least @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % leastI
% 0.26/0.46 thf(fact_91_map__graph__in,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A,F: B > C] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ G @ E4 ) )
% 0.26/0.46 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F @ A3 ) @ ( F @ B3 ) ) @ ( semantics @ A @ C @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F ) @ G ) @ E4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % map_graph_in
% 0.26/0.46 thf(fact_92_maintainedI,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,A2: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B2: labeled_graph @ C @ A] :
% 0.26/0.46 ( ! [F4: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( graph_homomorphism @ C @ A @ B @ A2 @ G @ F4 )
% 0.26/0.46 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A2 @ B2 ) @ G @ F4 ) )
% 0.26/0.46 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A2 @ B2 ) @ G ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintainedI
% 0.26/0.46 thf(fact_93_extensible__refl__concr,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V2: set @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V2 ) @ G @ F )
% 0.26/0.46 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V2 ) @ ( labeled_LG @ A @ B @ E_2 @ V2 ) ) @ G @ F )
% 0.26/0.46 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V2 ) @ G @ F ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensible_refl_concr
% 0.26/0.46 thf(fact_94_labeled__graph_Osplit__sel,axiom,
% 0.26/0.46 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 0.26/0.46 = ( ( Labeled_graph
% 0.26/0.46 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.26/0.46 => ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.split_sel
% 0.26/0.46 thf(fact_95_extensible__refl,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ R2 @ G @ F )
% 0.26/0.46 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G @ F ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensible_refl
% 0.26/0.46 thf(fact_96_least__consequence__graph__def,axiom,
% 0.26/0.46 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.26/0.46 ( ( least_559130134_graph @ X @ L @ V @ C )
% 0.26/0.46 = ( ^ [T3: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G6: labeled_graph @ L @ C] :
% 0.26/0.46 ( ( consequence_graph @ L @ V @ C @ Rs2 @ G6 )
% 0.26/0.46 & ( least @ X @ L @ V @ C @ T3 @ Rs2 @ S2 @ G6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % least_consequence_graph_def
% 0.26/0.46 thf(fact_97_consequence__graphD_I3_J,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 0.26/0.46 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 0.26/0.46 => ( G
% 0.26/0.46 = ( restrict @ A @ C @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % consequence_graphD(3)
% 0.26/0.46 thf(fact_98_graph__homomorphism__semantics,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),A3: B,B3: B,E4: allegorical_term @ A,A4: C,B4: C] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A3 @ A4 ) @ F )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ B4 ) @ F )
% 0.26/0.46 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A4 @ B4 ) @ ( semantics @ A @ C @ B2 @ E4 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_semantics
% 0.26/0.46 thf(fact_99_consequence__graphD_I1_J,axiom,
% 0.26/0.46 ! [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 )] :
% 0.26/0.46 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 0.26/0.46 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
% 0.26/0.46 => ( maintained @ A @ B @ C @ R2 @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % consequence_graphD(1)
% 0.26/0.46 thf(fact_100_labeled__graph_Ocase,axiom,
% 0.26/0.46 ! [L: $tType,A: $tType,V: $tType,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.26/0.46 ( ( labele1974067554_graph @ L @ V @ A @ F @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.26/0.46 = ( F @ X1 @ X22 ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.case
% 0.26/0.46 thf(fact_101_semantics__in__vertices_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( restrict @ A @ B @ A2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 0.26/0.46 => ( member @ B @ A3 @ ( labeled_vertices @ A @ B @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % semantics_in_vertices(1)
% 0.26/0.46 thf(fact_102_semantics__in__vertices_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( restrict @ A @ B @ A2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 0.26/0.46 => ( member @ B @ B3 @ ( labeled_vertices @ A @ B @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % semantics_in_vertices(2)
% 0.26/0.46 thf(fact_103_labeled__graph_Ocase__eq__if,axiom,
% 0.26/0.46 ! [A: $tType,V: $tType,L: $tType] :
% 0.26/0.46 ( ( labele1974067554_graph @ L @ V @ A )
% 0.26/0.46 = ( ^ [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 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.case_eq_if
% 0.26/0.46 thf(fact_104_subgraph__semantics,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ A2 @ B2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ B2 @ E4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_semantics
% 0.26/0.46 thf(fact_105_maintainedD,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A2 @ B2 ) @ G )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ C @ A2 @ G @ F )
% 0.26/0.46 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A2 @ B2 ) @ G @ F ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintainedD
% 0.26/0.46 thf(fact_106_least__def,axiom,
% 0.26/0.46 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.26/0.46 ( ( least @ X @ L @ V @ C )
% 0.26/0.46 = ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G6: labeled_graph @ L @ C] :
% 0.26/0.46 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G6 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 0.26/0.46 & ! [C5: labeled_graph @ L @ X] :
% 0.26/0.46 ( ( consequence_graph @ L @ V @ X @ Rs2 @ C5 )
% 0.26/0.46 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G6 ) @ C5 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % least_def
% 0.26/0.46 thf(fact_107_labeled__graph_Osplit__sel__asm,axiom,
% 0.26/0.46 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 0.26/0.46 = ( ~ ( ( Labeled_graph
% 0.26/0.46 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.26/0.46 & ~ ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.split_sel_asm
% 0.26/0.46 thf(fact_108_extensibleI,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G2: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G2 )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ R1 @ F @ G2 )
% 0.26/0.46 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % extensibleI
% 0.26/0.46 thf(fact_109_subgraph__subset_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G_1 ) ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_subset(2)
% 0.26/0.46 thf(fact_110_graph__homo__union__id_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A2 @ B2 ) @ G @ F )
% 0.26/0.46 => ( ( A2
% 0.26/0.46 = ( restrict @ A @ B @ A2 ) )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ C @ A2 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) @ F ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homo_union_id(1)
% 0.26/0.46 thf(fact_111_graph__homo__union__id_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A2 @ B2 ) @ G @ F )
% 0.26/0.46 => ( ( B2
% 0.26/0.46 = ( restrict @ A @ B @ B2 ) )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ C @ B2 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ B2 ) ) @ F ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homo_union_id(2)
% 0.26/0.46 thf(fact_112__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
% 0.26/0.46 ( f
% 0.26/0.46 = ( ^ [X7: b] :
% 0.26/0.46 ( if @ b
% 0.26/0.46 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X7 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ b ) ) )
% 0.26/0.46 @ X7
% 0.26/0.46 @ ( hilbert_Eps @ b @ ( p @ X7 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % \<open>f \<equiv> \<lambda>x. if getRel S_Idt G `` {x} = {} then x else Eps (P x)\<close>
% 0.26/0.46 thf(fact_113_agree__on__refl,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R2 @ F @ F ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_refl
% 0.26/0.46 thf(fact_114_agree__on__comm,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C )
% 0.26/0.46 = ( ^ [X8: labeled_graph @ A @ B,F5: set @ ( product_prod @ B @ C ),G7: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X8 @ G7 @ F5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_comm
% 0.26/0.46 thf(fact_115_ImageI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),A2: set @ A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 0.26/0.46 => ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( member @ B @ B3 @ ( image @ A @ B @ R @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ImageI
% 0.26/0.46 thf(fact_116_Image__empty2,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.46 ( ( image @ B @ A @ R2 @ ( bot_bot @ ( set @ B ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_empty2
% 0.26/0.46 thf(fact_117_Image__empty1,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,X5: set @ B] :
% 0.26/0.46 ( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X5 )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_empty1
% 0.26/0.46 thf(fact_118_Id__on__empty,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_empty
% 0.26/0.46 thf(fact_119_Gr__empty,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: A > B] :
% 0.26/0.46 ( ( bNF_Gr @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Gr_empty
% 0.26/0.46 thf(fact_120_graph__homomorphism__composes,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ C,X3: set @ ( product_prod @ B @ C ),C2: labeled_graph @ A @ D,Y3: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A3 @ B3 @ X3 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ C @ D @ B3 @ C2 @ Y3 )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ D @ A3 @ C2 @ ( relcomp @ B @ C @ D @ X3 @ Y3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_composes
% 0.26/0.46 thf(fact_121_P__eq,axiom,
% 0.26/0.46 ! [X2: b] :
% 0.26/0.46 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.26/0.46 != ( bot_bot @ ( set @ b ) ) )
% 0.26/0.46 => ( ( hilbert_Eps @ b @ ( p @ ( hilbert_Eps @ b @ ( p @ X2 ) ) ) )
% 0.26/0.46 = ( hilbert_Eps @ b @ ( p @ X2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % P_eq
% 0.26/0.46 thf(fact_122_Gr__insert,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,X3: A,F2: set @ A,F: A > B] :
% 0.26/0.46 ( ( bNF_Gr @ A @ B @ ( insert @ A @ X3 @ F2 ) @ F )
% 0.26/0.46 = ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ ( F @ X3 ) ) @ ( bNF_Gr @ A @ B @ F2 @ F ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Gr_insert
% 0.26/0.46 thf(fact_123_map__graph__selectors_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,F: set @ ( product_prod @ C @ A ),G: labeled_graph @ B @ C] :
% 0.26/0.46 ( ( labeled_vertices @ B @ A @ ( map_graph @ C @ A @ B @ F @ G ) )
% 0.26/0.46 = ( image @ C @ A @ F @ ( labeled_vertices @ B @ C @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % map_graph_selectors(1)
% 0.26/0.46 thf(fact_124_agree__on__empty,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,F: set @ ( product_prod @ B @ C ),G2: 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 ) ) ) @ F @ G2 ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_empty
% 0.26/0.46 thf(fact_125_Image__singleton__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A ),A3: B] :
% 0.26/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ R @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.26/0.46 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B3 ) @ R ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_singleton_iff
% 0.26/0.46 thf(fact_126_in__on__graph,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,X3: A,G: labeled_graph @ B @ A,A3: A > C,Y3: D,B3: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( member @ A @ X3 @ ( labeled_vertices @ B @ A @ G ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( A3 @ X3 ) @ Y3 ) @ B3 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ D ) @ ( product_Pair @ A @ D @ X3 @ Y3 ) @ ( relcomp @ A @ C @ D @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ A3 ) @ B3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % in_on_graph
% 0.26/0.46 thf(fact_127_graph__homomorphism__empty,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F )
% 0.26/0.46 = ( ( F
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 0.26/0.46 & ( G
% 0.26/0.46 = ( restrict @ A @ C @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_empty
% 0.26/0.46 thf(fact_128_graph__unionI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.46 = G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_unionI
% 0.26/0.46 thf(fact_129_graph__single,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A3: A,B3: B,C2: B] :
% 0.26/0.46 ( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A3 @ ( product_Pair @ B @ B @ B3 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B3 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.26/0.46 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A3 @ ( product_Pair @ B @ B @ B3 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B3 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_single
% 0.26/0.46 thf(fact_130_r2,axiom,
% 0.26/0.46 ! [X3: b] :
% 0.26/0.46 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X3 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ b ) ) )
% 0.26/0.46 = ( ~ ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % r2
% 0.26/0.46 thf(fact_131_refl__on__singleton,axiom,
% 0.26/0.46 ! [A: $tType,X3: A] : ( refl_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_on_singleton
% 0.26/0.46 thf(fact_132_rev__ImageI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 0.26/0.46 => ( member @ B @ B3 @ ( image @ A @ B @ R @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rev_ImageI
% 0.26/0.46 thf(fact_133_Image__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A ),A2: set @ B] :
% 0.26/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ R @ A2 ) )
% 0.26/0.46 = ( ? [X7: B] :
% 0.26/0.46 ( ( member @ B @ X7 @ A2 )
% 0.26/0.46 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X7 @ B3 ) @ R ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_iff
% 0.26/0.46 thf(fact_134_ImageE,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A ),A2: set @ B] :
% 0.26/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ R @ A2 ) )
% 0.26/0.46 => ~ ! [X6: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X6 @ B3 ) @ R )
% 0.26/0.46 => ~ ( member @ B @ X6 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ImageE
% 0.26/0.46 thf(fact_135_relcomp_OrelcompI,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),C2: C,S3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C2 ) @ S3 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.relcompI
% 0.26/0.46 thf(fact_136_relcomp_Oinducts,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C: $tType,X1: A,X22: C,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C ),P: A > C > $o] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X1 @ X22 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) )
% 0.26/0.46 => ( ! [A5: A,B5: B,C3: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C3 ) @ S3 )
% 0.26/0.46 => ( P @ A5 @ C3 ) ) )
% 0.26/0.46 => ( P @ X1 @ X22 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.inducts
% 0.26/0.46 thf(fact_137_relcomp_Osimps,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) )
% 0.26/0.46 = ( ? [A6: A,B6: B,C6: C] :
% 0.26/0.46 ( ( A1 = A6 )
% 0.26/0.46 & ( A22 = C6 )
% 0.26/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B6 ) @ R )
% 0.26/0.46 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B6 @ C6 ) @ S3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.simps
% 0.26/0.46 thf(fact_138_relcomp_Ocases,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) )
% 0.26/0.46 => ~ ! [B5: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B5 ) @ R )
% 0.26/0.46 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A22 ) @ S3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp.cases
% 0.26/0.46 thf(fact_139_relcompEpair,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R @ S3 ) )
% 0.26/0.46 => ~ ! [B5: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B5 ) @ R )
% 0.26/0.46 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcompEpair
% 0.26/0.46 thf(fact_140_relcompE,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S3 ) )
% 0.26/0.46 => ~ ! [X6: A,Y5: C,Z2: B] :
% 0.26/0.46 ( ( Xz
% 0.26/0.46 = ( product_Pair @ A @ B @ X6 @ Z2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X6 @ Y5 ) @ R )
% 0.26/0.46 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y5 @ Z2 ) @ S3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcompE
% 0.26/0.46 thf(fact_141_relcomp__Image,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,X5: set @ ( product_prod @ B @ C ),Y4: set @ ( product_prod @ C @ A ),Z3: set @ B] :
% 0.26/0.46 ( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X5 @ Y4 ) @ Z3 )
% 0.26/0.46 = ( image @ C @ A @ Y4 @ ( image @ B @ C @ X5 @ Z3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_Image
% 0.26/0.46 thf(fact_142_relcomp__mono,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C ),S3: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ R )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S4 @ S3 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R3 @ S4 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_mono
% 0.26/0.46 thf(fact_143_Image__mono,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),A7: set @ A,A2: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ R )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R3 @ A7 ) @ ( image @ A @ B @ R @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_mono
% 0.26/0.46 thf(fact_144_O__assoc,axiom,
% 0.26/0.46 ! [A: $tType,D: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ A @ D ),S: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R2 @ S ) @ T4 )
% 0.26/0.46 = ( relcomp @ A @ D @ B @ R2 @ ( relcomp @ D @ C @ B @ S @ T4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % O_assoc
% 0.26/0.46 thf(fact_145_agree__on__def,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C )
% 0.26/0.46 = ( ^ [G6: labeled_graph @ A @ B,F_1: set @ ( product_prod @ B @ C ),F_2: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ! [X7: B] :
% 0.26/0.46 ( ( member @ B @ X7 @ ( labeled_vertices @ A @ B @ G6 ) )
% 0.26/0.46 => ( ( image @ B @ C @ F_1 @ ( insert @ B @ X7 @ ( bot_bot @ ( set @ B ) ) ) )
% 0.26/0.46 = ( image @ B @ C @ F_2 @ ( insert @ B @ X7 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_def
% 0.26/0.46 thf(fact_146_edge__preserving__subset,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,R_1: set @ ( product_prod @ A @ B ),R_2: set @ ( product_prod @ A @ B ),E_12: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E_22: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),G: set @ ( product_prod @ C @ ( product_prod @ B @ B ) )] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R_1 @ R_2 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) @ E_12 @ E_22 )
% 0.26/0.46 => ( ( edge_preserving @ A @ B @ C @ R_2 @ E_22 @ G )
% 0.26/0.46 => ( edge_preserving @ A @ B @ C @ R_1 @ E_12 @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % edge_preserving_subset
% 0.26/0.46 thf(fact_147_agree__on__trans,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,X5: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C ),G2: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C @ X5 @ F @ G2 )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ X5 @ G2 @ H )
% 0.26/0.46 => ( agree_on @ A @ B @ C @ X5 @ F @ H ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_trans
% 0.26/0.46 thf(fact_148_agree__on__ext,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,G: labeled_graph @ A @ B,F_12: set @ ( product_prod @ B @ C ),F_22: set @ ( product_prod @ B @ C ),G2: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C @ G @ F_12 @ F_22 )
% 0.26/0.46 => ( agree_on @ A @ B @ D @ G @ ( relcomp @ B @ C @ D @ F_12 @ G2 ) @ ( relcomp @ B @ C @ D @ F_22 @ G2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_ext
% 0.26/0.46 thf(fact_149_refl__on__empty,axiom,
% 0.26/0.46 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refl_on_empty
% 0.26/0.46 thf(fact_150_subrelI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ! [X6: A,Y5: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y5 ) @ R )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y5 ) @ S3 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % subrelI
% 0.26/0.46 thf(fact_151_compose__preserves__edge__preserving,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E2: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),H2: set @ ( product_prod @ B @ D ),E32: set @ ( product_prod @ C @ ( product_prod @ D @ D ) )] :
% 0.26/0.46 ( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E2 )
% 0.26/0.46 => ( ( edge_preserving @ B @ D @ C @ H2 @ E2 @ E32 )
% 0.26/0.46 => ( edge_preserving @ A @ D @ C @ ( relcomp @ A @ B @ D @ H1 @ H2 ) @ E1 @ E32 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % compose_preserves_edge_preserving
% 0.26/0.46 thf(fact_152_graph__union__iff,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.26/0.46 = G_2 )
% 0.26/0.46 = ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.26/0.46 & ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_union_iff
% 0.26/0.46 thf(fact_153_restrict__subsD,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G ) ) )
% 0.26/0.46 => ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % restrict_subsD
% 0.26/0.46 thf(fact_154_subgraph__def2,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( G_1
% 0.26/0.46 = ( restrict @ A @ B @ G_1 ) )
% 0.26/0.46 => ( ( G_2
% 0.26/0.46 = ( restrict @ A @ B @ G_2 ) )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 = ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.26/0.46 & ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_def2
% 0.26/0.46 thf(fact_155_subgraph__subset_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subgraph_subset(1)
% 0.26/0.46 thf(fact_156_maintainedD2,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A2 @ B2 ) @ G )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ C @ A2 @ G @ F )
% 0.26/0.46 => ~ ! [G5: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ B2 @ G @ G5 )
% 0.26/0.46 => ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F @ G5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % maintainedD2
% 0.26/0.46 thf(fact_157_Id__on__vertices__identity_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,C: $tType,A: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),Aa: B,Ba: C] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A3 @ B3 @ F )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ F )
% 0.26/0.46 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ ( relcomp @ B @ C @ C @ F @ ( id_on @ C @ ( labeled_vertices @ A @ C @ B3 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_vertices_identity(2)
% 0.26/0.46 thf(fact_158_Id__on__vertices__identity_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),Aa: B,Ba: C] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A3 @ B3 @ F )
% 0.26/0.46 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ F )
% 0.26/0.46 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A3 ) ) @ F ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Id_on_vertices_identity(1)
% 0.26/0.46 thf(fact_159_graph__homomorphism__on__graph,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,D: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,R2: set @ ( product_prod @ B @ C ),F: C > D] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ R2 )
% 0.26/0.46 => ( graph_homomorphism @ A @ B @ D @ A2 @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ B2 ) @ F ) @ B2 ) @ ( relcomp @ B @ C @ D @ R2 @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ B2 ) @ F ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_on_graph
% 0.26/0.46 thf(fact_160_agree__on__subg__compose,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G2: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F2: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ( agree_on @ A @ B @ C @ R2 @ G2 @ H )
% 0.26/0.46 => ( ( agree_on @ A @ B @ C @ F2 @ F @ G2 )
% 0.26/0.46 => ( ( graph_homomorphism @ A @ B @ B @ F2 @ R2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F2 ) ) )
% 0.26/0.46 => ( agree_on @ A @ B @ C @ F2 @ F @ H ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % agree_on_subg_compose
% 0.26/0.46 thf(fact_161_singleton__insert__inj__eq,axiom,
% 0.26/0.46 ! [A: $tType,B3: A,A3: A,A2: set @ A] :
% 0.26/0.46 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( insert @ A @ A3 @ A2 ) )
% 0.26/0.46 = ( ( A3 = B3 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % singleton_insert_inj_eq
% 0.26/0.46 thf(fact_162_singleton__insert__inj__eq_H,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 0.26/0.46 ( ( ( insert @ A @ A3 @ A2 )
% 0.26/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( ( A3 = B3 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % singleton_insert_inj_eq'
% 0.26/0.46 thf(fact_163_equiv__class__subset,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equiv_class_subset
% 0.26/0.46 thf(fact_164_subset__equiv__class,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),B3: A,A3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 => ( ( member @ A @ B3 @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_equiv_class
% 0.26/0.46 thf(fact_165_refines__equiv__class__eq2,axiom,
% 0.26/0.46 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A2: set @ A,A3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
% 0.26/0.46 => ( ( equiv_equiv @ A @ A2 @ R2 )
% 0.26/0.46 => ( ( equiv_equiv @ A @ A2 @ S )
% 0.26/0.46 => ( ( image @ A @ A @ S @ ( image @ A @ A @ R2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 = ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refines_equiv_class_eq2
% 0.26/0.46 thf(fact_166_empty__Collect__eq,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o] :
% 0.26/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.26/0.46 = ( collect @ A @ P ) )
% 0.26/0.46 = ( ! [X7: A] :
% 0.26/0.46 ~ ( P @ X7 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % empty_Collect_eq
% 0.26/0.46 thf(fact_167_Collect__empty__eq,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o] :
% 0.26/0.46 ( ( ( collect @ A @ P )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( ! [X7: A] :
% 0.26/0.46 ~ ( P @ X7 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_empty_eq
% 0.26/0.46 thf(fact_168_all__not__in__conv,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] :
% 0.26/0.46 ( ( ! [X7: A] :
% 0.26/0.46 ~ ( member @ A @ X7 @ A2 ) )
% 0.26/0.46 = ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % all_not_in_conv
% 0.26/0.46 thf(fact_169_empty__iff,axiom,
% 0.26/0.46 ! [A: $tType,C2: A] :
% 0.26/0.46 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % empty_iff
% 0.26/0.46 thf(fact_170_subset__antisym,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.26/0.46 => ( A2 = B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_antisym
% 0.26/0.46 thf(fact_171_subsetI,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ! [X6: A] :
% 0.26/0.46 ( ( member @ A @ X6 @ A2 )
% 0.26/0.46 => ( member @ A @ X6 @ B2 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % subsetI
% 0.26/0.46 thf(fact_172_insert__absorb2,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,A2: set @ A] :
% 0.26/0.46 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A2 ) )
% 0.26/0.46 = ( insert @ A @ X3 @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_absorb2
% 0.26/0.46 thf(fact_173_insert__iff,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.26/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.26/0.46 = ( ( A3 = B3 )
% 0.26/0.46 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_iff
% 0.26/0.46 thf(fact_174_insertCI,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.26/0.46 ( ( ~ ( member @ A @ A3 @ B2 )
% 0.26/0.46 => ( A3 = B3 ) )
% 0.26/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insertCI
% 0.26/0.46 thf(fact_175_empty__subsetI,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.26/0.46
% 0.26/0.46 % empty_subsetI
% 0.26/0.46 thf(fact_176_subset__empty,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_empty
% 0.26/0.46 thf(fact_177_singletonI,axiom,
% 0.26/0.46 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % singletonI
% 0.26/0.46 thf(fact_178_insert__subset,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ B2 )
% 0.26/0.46 = ( ( member @ A @ X3 @ B2 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_subset
% 0.26/0.46 thf(fact_179_relcomp__empty1,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ C @ B )] :
% 0.26/0.46 ( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R2 )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_empty1
% 0.26/0.46 thf(fact_180_relcomp__empty2,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ C )] :
% 0.26/0.46 ( ( relcomp @ A @ C @ B @ R2 @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relcomp_empty2
% 0.26/0.46 thf(fact_181_graph__empty__e,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,V2: set @ B] :
% 0.26/0.46 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V2 )
% 0.26/0.46 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_empty_e
% 0.26/0.46 thf(fact_182_graph__homomorphism__nonempty,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),E4: allegorical_term @ A] :
% 0.26/0.46 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F )
% 0.26/0.46 => ( ( ( semantics @ A @ B @ A2 @ E4 )
% 0.26/0.46 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.26/0.46 => ( ( semantics @ A @ C @ B2 @ E4 )
% 0.26/0.46 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % graph_homomorphism_nonempty
% 0.26/0.46 thf(fact_183_ex__in__conv,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] :
% 0.26/0.46 ( ( ? [X7: A] : ( member @ A @ X7 @ A2 ) )
% 0.26/0.46 = ( A2
% 0.26/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ex_in_conv
% 0.26/0.46 thf(fact_184_equals0I,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] :
% 0.26/0.46 ( ! [Y5: A] :
% 0.26/0.46 ~ ( member @ A @ Y5 @ A2 )
% 0.26/0.46 => ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equals0I
% 0.26/0.46 thf(fact_185_equals0D,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,A3: A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % equals0D
% 0.26/0.46 thf(fact_186_emptyE,axiom,
% 0.26/0.46 ! [A: $tType,A3: A] :
% 0.26/0.46 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % emptyE
% 0.26/0.46 thf(fact_187_Collect__mono__iff,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.26/0.46 = ( ! [X7: A] :
% 0.26/0.46 ( ( P @ X7 )
% 0.26/0.46 => ( Q @ X7 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_mono_iff
% 0.26/0.46 thf(fact_188_set__eq__subset,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ^ [Y6: set @ A,Z4: set @ A] : ( Y6 = Z4 ) )
% 0.26/0.46 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 0.26/0.46 & ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % set_eq_subset
% 0.26/0.46 thf(fact_189_subset__trans,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C7: set @ A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C7 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C7 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_trans
% 0.26/0.46 thf(fact_190_Collect__mono,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.26/0.46 ( ! [X6: A] :
% 0.26/0.46 ( ( P @ X6 )
% 0.26/0.46 => ( Q @ X6 ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_mono
% 0.26/0.46 thf(fact_191_subset__refl,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.26/0.46
% 0.26/0.46 % subset_refl
% 0.26/0.46 thf(fact_192_subset__iff,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.26/0.46 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.26/0.46 ! [T3: A] :
% 0.26/0.46 ( ( member @ A @ T3 @ A8 )
% 0.26/0.46 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_iff
% 0.26/0.46 thf(fact_193_equalityD2,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ( A2 = B2 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % equalityD2
% 0.26/0.46 thf(fact_194_equalityD1,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ( A2 = B2 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % equalityD1
% 0.26/0.46 thf(fact_195_subset__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.26/0.46 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.26/0.46 ! [X7: A] :
% 0.26/0.46 ( ( member @ A @ X7 @ A8 )
% 0.26/0.46 => ( member @ A @ X7 @ B7 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_eq
% 0.26/0.46 thf(fact_196_equalityE,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ( A2 = B2 )
% 0.26/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equalityE
% 0.26/0.46 thf(fact_197_subsetD,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.26/0.46 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subsetD
% 0.26/0.46 thf(fact_198_in__mono,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.46 => ( ( member @ A @ X3 @ A2 )
% 0.26/0.46 => ( member @ A @ X3 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % in_mono
% 0.26/0.46 thf(fact_199_mk__disjoint__insert,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.26/0.46 ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ? [B8: set @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( insert @ A @ A3 @ B8 ) )
% 0.26/0.46 & ~ ( member @ A @ A3 @ B8 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % mk_disjoint_insert
% 0.26/0.46 thf(fact_200_insert__commute,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Y3: A,A2: set @ A] :
% 0.26/0.46 ( ( insert @ A @ X3 @ ( insert @ A @ Y3 @ A2 ) )
% 0.26/0.46 = ( insert @ A @ Y3 @ ( insert @ A @ X3 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_commute
% 0.26/0.46 thf(fact_201_insert__eq__iff,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
% 0.26/0.46 ( ~ ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( ~ ( member @ A @ B3 @ B2 )
% 0.26/0.46 => ( ( ( insert @ A @ A3 @ A2 )
% 0.26/0.46 = ( insert @ A @ B3 @ B2 ) )
% 0.26/0.46 = ( ( ( A3 = B3 )
% 0.26/0.46 => ( A2 = B2 ) )
% 0.26/0.46 & ( ( A3 != B3 )
% 0.26/0.46 => ? [C5: set @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( insert @ A @ B3 @ C5 ) )
% 0.26/0.46 & ~ ( member @ A @ B3 @ C5 )
% 0.26/0.46 & ( B2
% 0.26/0.46 = ( insert @ A @ A3 @ C5 ) )
% 0.26/0.46 & ~ ( member @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_eq_iff
% 0.26/0.46 thf(fact_202_insert__absorb,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.26/0.46 ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( ( insert @ A @ A3 @ A2 )
% 0.26/0.46 = A2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_absorb
% 0.26/0.46 thf(fact_203_insert__ident,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ~ ( member @ A @ X3 @ A2 )
% 0.26/0.46 => ( ~ ( member @ A @ X3 @ B2 )
% 0.26/0.46 => ( ( ( insert @ A @ X3 @ A2 )
% 0.26/0.46 = ( insert @ A @ X3 @ B2 ) )
% 0.26/0.46 = ( A2 = B2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_ident
% 0.26/0.46 thf(fact_204_Set_Oset__insert,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,A2: set @ A] :
% 0.26/0.46 ( ( member @ A @ X3 @ A2 )
% 0.26/0.46 => ~ ! [B8: set @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( insert @ A @ X3 @ B8 ) )
% 0.26/0.46 => ( member @ A @ X3 @ B8 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Set.set_insert
% 0.26/0.46 thf(fact_205_insertI2,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.26/0.46 ( ( member @ A @ A3 @ B2 )
% 0.26/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insertI2
% 0.26/0.46 thf(fact_206_insertI1,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % insertI1
% 0.26/0.46 thf(fact_207_insertE,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.26/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.26/0.46 => ( ( A3 != B3 )
% 0.26/0.46 => ( member @ A @ A3 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insertE
% 0.26/0.46 thf(fact_208_singleton__inject,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B3: A] :
% 0.26/0.46 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 => ( A3 = B3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % singleton_inject
% 0.26/0.46 thf(fact_209_insert__not__empty,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.26/0.46 ( ( insert @ A @ A3 @ A2 )
% 0.26/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_not_empty
% 0.26/0.46 thf(fact_210_doubleton__eq__iff,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,B3: A,C2: A,D3: A] :
% 0.26/0.46 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( insert @ A @ C2 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 = ( ( ( A3 = C2 )
% 0.26/0.46 & ( B3 = D3 ) )
% 0.26/0.46 | ( ( A3 = D3 )
% 0.26/0.46 & ( B3 = C2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % doubleton_eq_iff
% 0.26/0.46 thf(fact_211_singleton__iff,axiom,
% 0.26/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.26/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( B3 = A3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % singleton_iff
% 0.26/0.46 thf(fact_212_singletonD,axiom,
% 0.26/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.26/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 => ( B3 = A3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % singletonD
% 0.26/0.46 thf(fact_213_subset__insertI2,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_insertI2
% 0.26/0.46 thf(fact_214_subset__insertI,axiom,
% 0.26/0.46 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_insertI
% 0.26/0.46 thf(fact_215_subset__insert,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 0.26/0.46 ( ~ ( member @ A @ X3 @ A2 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B2 ) )
% 0.26/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_insert
% 0.26/0.46 thf(fact_216_insert__mono,axiom,
% 0.26/0.46 ! [A: $tType,C7: set @ A,D4: set @ A,A3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ C7 @ D4 )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C7 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_mono
% 0.26/0.46 thf(fact_217_subset__singleton__iff,axiom,
% 0.26/0.46 ! [A: $tType,X5: set @ A,A3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( ( X5
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 | ( X5
% 0.26/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_singleton_iff
% 0.26/0.46 thf(fact_218_subset__singletonD,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,X3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 => ( ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 | ( A2
% 0.26/0.46 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_singletonD
% 0.26/0.46 thf(fact_219_equiv__class__self,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ A @ A3 @ A2 )
% 0.26/0.46 => ( member @ A @ A3 @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equiv_class_self
% 0.26/0.46 thf(fact_220_equiv__class__eq__iff,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 0.26/0.46 = ( ( ( image @ A @ A @ R @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( image @ A @ A @ R @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 & ( member @ A @ X3 @ A2 )
% 0.26/0.46 & ( member @ A @ Y3 @ A2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equiv_class_eq_iff
% 0.26/0.46 thf(fact_221_eq__equiv__class__iff,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ A @ X3 @ A2 )
% 0.26/0.46 => ( ( member @ A @ Y3 @ A2 )
% 0.26/0.46 => ( ( ( image @ A @ A @ R @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( image @ A @ A @ R @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_equiv_class_iff
% 0.26/0.46 thf(fact_222_equiv__class__eq,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
% 0.26/0.46 => ( ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equiv_class_eq
% 0.26/0.46 thf(fact_223_eq__equiv__class,axiom,
% 0.26/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B3: A,A2: set @ A] :
% 0.26/0.46 ( ( ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 => ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( member @ A @ B3 @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_equiv_class
% 0.26/0.46 thf(fact_224_refines__equiv__class__eq,axiom,
% 0.26/0.46 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A2: set @ A,A3: A] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
% 0.26/0.46 => ( ( equiv_equiv @ A @ A2 @ R2 )
% 0.26/0.46 => ( ( equiv_equiv @ A @ A2 @ S )
% 0.26/0.46 => ( ( image @ A @ A @ R2 @ ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 = ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % refines_equiv_class_eq
% 0.26/0.46 thf(fact_225_nonempty__rule,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
% 0.26/0.46 = ( ( labeled_vertices @ A @ B @ G )
% 0.26/0.46 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % nonempty_rule
% 0.26/0.46 thf(fact_226_ma,axiom,
% 0.26/0.46 ! [X4: product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat )] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ X4 @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ ( standa245363227y_rule @ ( standard_Constant @ a ) @ ( standard_S_Idt @ a ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ ( standa2114124375e_rule @ ( standard_Constant @ a ) @ ( standard_S_Idt @ a ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ ( standa1805702094y_rule @ ( standard_Constant @ a ) @ ( standard_S_Idt @ a ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) ) ) ) ) ) )
% 0.26/0.46 => ( maintained @ ( standard_Constant @ a ) @ nat @ b @ X4 @ g ) ) ).
% 0.26/0.46
% 0.26/0.46 % ma
% 0.26/0.46 thf(fact_227_proj__iff,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 0.26/0.46 ( ( equiv_equiv @ A @ A2 @ R )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A2 )
% 0.26/0.46 => ( ( ( equiv_proj @ A @ A @ R @ X3 )
% 0.26/0.46 = ( equiv_proj @ A @ A @ R @ Y3 ) )
% 0.26/0.46 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % proj_iff
% 0.26/0.46 thf(fact_228_proj__def,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType] :
% 0.26/0.46 ( ( equiv_proj @ B @ A )
% 0.26/0.46 = ( ^ [R4: set @ ( product_prod @ B @ A ),X7: B] : ( image @ B @ A @ R4 @ ( insert @ B @ X7 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % proj_def
% 0.26/0.46 thf(fact_229_equivalence,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,I: A] :
% 0.26/0.46 ( ( G
% 0.26/0.46 = ( restrict @ A @ B @ G ) )
% 0.26/0.46 => ( ! [X6: product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat )] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ X6 @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa245363227y_rule @ A @ I ) @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa2114124375e_rule @ A @ I ) @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa1805702094y_rule @ A @ I ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) ) ) ) ) ) )
% 0.26/0.46 => ( maintained @ A @ nat @ B @ X6 @ G ) )
% 0.26/0.46 => ( equiv_equiv @ B @ ( labeled_vertices @ A @ B @ G ) @ ( getRel @ A @ B @ I @ G ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % equivalence
% 0.26/0.46 thf(fact_230_f,axiom,
% 0.26/0.46 ( f
% 0.26/0.46 = ( ^ [X7: b] :
% 0.26/0.46 ( if @ b
% 0.26/0.46 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X7 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ b ) ) )
% 0.26/0.46 @ X7
% 0.26/0.46 @ ( hilbert_Eps @ b @ ( p @ X7 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % f
% 0.26/0.46 thf(fact_231_the__elem__eq,axiom,
% 0.26/0.46 ! [A: $tType,X3: A] :
% 0.26/0.46 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.26/0.46 = X3 ) ).
% 0.26/0.46
% 0.26/0.46 % the_elem_eq
% 0.26/0.46 thf(fact_232_tuple__disj,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Z: A] :
% 0.26/0.46 ( ( collect @ A
% 0.26/0.46 @ ^ [Y7: A] :
% 0.26/0.46 ( ( Y7 = X3 )
% 0.26/0.46 | ( Y7 = Z ) ) )
% 0.26/0.46 = ( insert @ A @ X3 @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % tuple_disj
% 0.26/0.46 thf(fact_233_singleton__conv,axiom,
% 0.26/0.46 ! [A: $tType,A3: A] :
% 0.26/0.46 ( ( collect @ A
% 0.26/0.46 @ ^ [X7: A] : ( X7 = A3 ) )
% 0.26/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % singleton_conv
% 0.26/0.46 thf(fact_234_singleton__conv2,axiom,
% 0.26/0.46 ! [A: $tType,A3: A] :
% 0.26/0.46 ( ( collect @ A
% 0.26/0.46 @ ( ^ [Y6: A,Z4: A] : ( Y6 = Z4 )
% 0.26/0.46 @ A3 ) )
% 0.26/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % singleton_conv2
% 0.26/0.46 thf(fact_235_P,axiom,
% 0.26/0.46 ( p
% 0.26/0.46 = ( ^ [X7: b,Y7: b] : ( member @ b @ Y7 @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X7 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % P
% 0.26/0.46 thf(fact_236_empty__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.26/0.46 = ( collect @ A
% 0.26/0.46 @ ^ [X7: A] : $false ) ) ).
% 0.26/0.46
% 0.26/0.46 % empty_def
% 0.26/0.46 thf(fact_237_insert__compr,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( insert @ A )
% 0.26/0.46 = ( ^ [A6: A,B7: set @ A] :
% 0.26/0.46 ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( X7 = A6 )
% 0.26/0.46 | ( member @ A @ X7 @ B7 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_compr
% 0.26/0.46 thf(fact_238_insert__Collect,axiom,
% 0.26/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.26/0.46 ( ( insert @ A @ A3 @ ( collect @ A @ P ) )
% 0.26/0.46 = ( collect @ A
% 0.26/0.46 @ ^ [U2: A] :
% 0.26/0.46 ( ( U2 != A3 )
% 0.26/0.46 => ( P @ U2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_Collect
% 0.26/0.46 thf(fact_239_Collect__conv__if,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,A3: A] :
% 0.26/0.46 ( ( ( P @ A3 )
% 0.26/0.46 => ( ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( X7 = A3 )
% 0.26/0.46 & ( P @ X7 ) ) )
% 0.26/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 & ( ~ ( P @ A3 )
% 0.26/0.46 => ( ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( X7 = A3 )
% 0.26/0.46 & ( P @ X7 ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_conv_if
% 0.26/0.46 thf(fact_240_Collect__conv__if2,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,A3: A] :
% 0.26/0.46 ( ( ( P @ A3 )
% 0.26/0.46 => ( ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( A3 = X7 )
% 0.26/0.46 & ( P @ X7 ) ) )
% 0.26/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.26/0.46 & ( ~ ( P @ A3 )
% 0.26/0.46 => ( ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( A3 = X7 )
% 0.26/0.46 & ( P @ X7 ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_conv_if2
% 0.26/0.46 thf(fact_241_Image__singleton,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
% 0.26/0.46 ( ( image @ B @ A @ R @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
% 0.26/0.46 = ( collect @ A
% 0.26/0.46 @ ^ [B6: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B6 ) @ R ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Image_singleton
% 0.26/0.46 thf(fact_242_edges__def,axiom,
% 0.26/0.46 ! [V: $tType,L: $tType] :
% 0.26/0.46 ( ( labeled_edges @ L @ V )
% 0.26/0.46 = ( labele1974067554_graph @ L @ V @ ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) )
% 0.26/0.46 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X13 ) ) ).
% 0.26/0.46
% 0.26/0.46 % edges_def
% 0.26/0.46 thf(fact_243_vertices__def,axiom,
% 0.26/0.46 ! [V: $tType,L: $tType] :
% 0.26/0.46 ( ( labeled_vertices @ L @ V )
% 0.26/0.46 = ( labele1974067554_graph @ L @ V @ ( set @ V )
% 0.26/0.46 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X24 ) ) ).
% 0.26/0.46
% 0.26/0.46 % vertices_def
% 0.26/0.46 thf(fact_244_labeled__graph_Ocase__distrib,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,V: $tType,L: $tType,H: A > B,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.26/0.46 ( ( H @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 0.26/0.46 = ( labele1974067554_graph @ L @ V @ B
% 0.26/0.46 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : ( H @ ( F @ X13 @ X24 ) )
% 0.26/0.46 @ Labeled_graph ) ) ).
% 0.26/0.46
% 0.26/0.46 % labeled_graph.case_distrib
% 0.26/0.46 thf(fact_245_pred__equals__eq2,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( ( ^ [X7: A,Y7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X7 @ Y7 ) @ R2 ) )
% 0.26/0.46 = ( ^ [X7: A,Y7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X7 @ Y7 ) @ S ) ) )
% 0.26/0.46 = ( R2 = S ) ) ).
% 0.26/0.46
% 0.26/0.46 % pred_equals_eq2
% 0.26/0.46 thf(fact_246_subset__CollectI,axiom,
% 0.26/0.46 ! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.26/0.46 => ( ! [X6: A] :
% 0.26/0.46 ( ( member @ A @ X6 @ B2 )
% 0.26/0.46 => ( ( Q @ X6 )
% 0.26/0.46 => ( P @ X6 ) ) )
% 0.26/0.46 => ( ord_less_eq @ ( set @ A )
% 0.26/0.46 @ ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( member @ A @ X7 @ B2 )
% 0.26/0.46 & ( Q @ X7 ) ) )
% 0.26/0.46 @ ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( member @ A @ X7 @ A2 )
% 0.26/0.46 & ( P @ X7 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_CollectI
% 0.26/0.46 thf(fact_247_subset__Collect__iff,axiom,
% 0.26/0.46 ! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
% 0.26/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.26/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.26/0.46 @ ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( member @ A @ X7 @ A2 )
% 0.26/0.46 & ( P @ X7 ) ) ) )
% 0.26/0.46 = ( ! [X7: A] :
% 0.26/0.46 ( ( member @ A @ X7 @ B2 )
% 0.26/0.46 => ( P @ X7 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subset_Collect_iff
% 0.26/0.46 thf(fact_248_Collect__subset,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A,P: A > $o] :
% 0.26/0.46 ( ord_less_eq @ ( set @ A )
% 0.26/0.46 @ ( collect @ A
% 0.26/0.46 @ ^ [X7: A] :
% 0.26/0.46 ( ( member @ A @ X7 @ A2 )
% 0.26/0.46 & ( P @ X7 ) ) )
% 0.26/0.46 @ A2 ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_subset
% 0.26/0.46 thf(fact_249_congr_I1_J,axiom,
% 0.26/0.46 ! [X2: b,Y: b,L2: standard_Constant @ a] :
% 0.26/0.46 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X2 @ Y ) @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) )
% 0.26/0.46 => ( equiv_congruent @ b @ ( set @ b ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g )
% 0.26/0.46 @ ^ [V3: b] : ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) @ ( insert @ b @ V3 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % congr(1)
% 0.26/0.46 thf(fact_250_ident,axiom,
% 0.26/0.46 ( ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) )
% 0.26/0.46 = ( image2 @ b @ ( product_prod @ b @ b )
% 0.26/0.46 @ ^ [X7: b] : ( product_Pair @ b @ b @ X7 @ X7 )
% 0.26/0.46 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ident
% 0.26/0.46 thf(fact_251_image__empty,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: B > A] :
% 0.26/0.46 ( ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % image_empty
% 0.26/0.46 thf(fact_252_empty__is__image,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 0.26/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.26/0.46 = ( image2 @ B @ A @ F @ A2 ) )
% 0.26/0.46 = ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % empty_is_image
% 0.26/0.46 thf(fact_253_image__is__empty,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 0.26/0.46 ( ( ( image2 @ B @ A @ F @ A2 )
% 0.26/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.26/0.46 = ( A2
% 0.26/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % image_is_empty
% 0.26/0.46 thf(fact_254_image__insert,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: B > A,A3: B,B2: set @ B] :
% 0.26/0.46 ( ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ B2 ) )
% 0.26/0.46 = ( insert @ A @ ( F @ A3 ) @ ( image2 @ B @ A @ F @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % image_insert
% 0.26/0.46 thf(fact_255_insert__image,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
% 0.26/0.46 ( ( member @ A @ X3 @ A2 )
% 0.26/0.46 => ( ( insert @ B @ ( F @ X3 ) @ ( image2 @ A @ B @ F @ A2 ) )
% 0.26/0.46 = ( image2 @ A @ B @ F @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_image
% 0.26/0.46
% 0.26/0.46 % Helper facts (3)
% 0.26/0.46 thf(help_If_3_1_T,axiom,
% 0.26/0.46 ! [P: $o] :
% 0.26/0.46 ( ( P = $true )
% 0.26/0.46 | ( P = $false ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(help_If_2_1_T,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Y3: A] :
% 0.26/0.46 ( ( if @ A @ $false @ X3 @ Y3 )
% 0.26/0.46 = Y3 ) ).
% 0.26/0.46
% 0.26/0.46 thf(help_If_1_1_T,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Y3: A] :
% 0.26/0.46 ( ( if @ A @ $true @ X3 @ Y3 )
% 0.26/0.46 = X3 ) ).
% 0.26/0.46
% 0.26/0.46 % Conjectures (1)
% 0.26/0.46 thf(conj_0,conjecture,
% 0.26/0.46 member @ b @ x @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ).
% 0.26/0.46
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 ------------------------
% 2.95/1.53 ---- Embedded in THF ---
% 2.95/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.95/1.53 %%% Generated on Tue Feb 24 08:28:52 EST 2026
% 2.95/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.95/1.53 %%% Logic specification used:
% 2.95/1.53 %%% thf(spec, logic, $$dhol).
% 2.95/1.53
% 2.95/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2DTF32619.p
% 2.95/1.53 thf(ty_t_LabeledGraphSemantics_OStandard__Constant, type, standard_Constant: ($tType > $tType)).
% 2.95/1.53 thf(ty_t_LabeledGraphSemantics_Oallegorical__term, type, allegorical_term: ($tType > $tType)).
% 2.95/1.53 thf(ty_t_LabeledGraphs_Olabeled__graph, type, labeled_graph: ($tType > ($tType > $tType))).
% 2.95/1.53 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.95/1.53 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.95/1.53 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.95/1.53 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.95/1.53 thf(ty_tf_b, type, b: $tType).
% 2.95/1.53 thf(ty_tf_a, type, a: $tType).
% 2.95/1.53 thf(sy_c_BNF__Def_OGr, type, bNF_Gr: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (set @ ((product_prod @ A) @ B))))))).
% 2.95/1.53 thf(sy_c_Equiv__Relations_Ocongruent, type, equiv_congruent: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.95/1.53 thf(sy_c_Equiv__Relations_Oequiv, type, equiv_equiv: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.95/1.53 thf(sy_c_Equiv__Relations_Oproj, type, equiv_proj: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > (set @ A)))))).
% 2.95/1.53 thf(sy_c_Hilbert__Choice_OEps, type, hilbert_Eps: (!> [A:$tType]: (((A > $o) > A)))).
% 2.95/1.53 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.95/1.53 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt, type, standard_S_Idt: (!> [V:$tType]: ((standard_Constant @ V)))).
% 2.95/1.53 thf(sy_c_LabeledGraphSemantics_OgetRel, type, getRel: (!> [B:$tType,A:$tType]: ((B > (((labeled_graph @ B) @ A) > (set @ ((product_prod @ A) @ A))))))).
% 2.95/1.53 thf(sy_c_LabeledGraphSemantics_Osemantics, type, semantics: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((allegorical_term @ A) > (set @ ((product_prod @ B) @ B))))))).
% 2.95/1.53 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))) > $o)))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Ograph__homomorphism, type, graph_homomorphism: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ C) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Ograph__union, type, graph_union: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B)))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG, type, labeled_LG: (!> [L:$tType,V:$tType]: (((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > ((labeled_graph @ L) @ V)))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph, type, labele1974067554_graph: (!> [L:$tType,V:$tType,A:$tType]: ((((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)) > (((labeled_graph @ L) @ V) > A))))).
% 2.95/1.53 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))))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices, type, labeled_vertices: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ V))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Omap__graph, type, map_graph: (!> [C:$tType,B:$tType,A:$tType]: (((set @ ((product_prod @ C) @ B)) > (((labeled_graph @ A) @ C) > ((labeled_graph @ A) @ B)))))).
% 2.95/1.53 thf(sy_c_LabeledGraphs_Orestrict, type, restrict: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B))))).
% 2.95/1.53 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.95/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.95/1.53 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.95/1.53 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.95/1.53 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 2.95/1.53 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 2.95/1.53 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.95/1.53 thf(sy_c_Relation_Orelcomp, type, relcomp: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ A) @ C))))))).
% 2.95/1.53 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)) > $o)))))).
% 2.95/1.53 thf(sy_c_RulesAndChains_Oconsequence__graph, type, consequence_graph: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) > (((labeled_graph @ A) @ C) > $o))))).
% 2.95/1.53 thf(sy_c_RulesAndChains_Oextensible, type, extensible: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > ((set @ ((product_prod @ X) @ V)) > $o)))))).
% 2.95/1.53 thf(sy_c_RulesAndChains_Oleast, type, least: (!> [X:$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) > $o))))))).
% 2.95/1.53 thf(sy_c_RulesAndChains_Oleast__consequence__graph, type, least_559130134_graph: (!> [X:$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) > $o))))))).
% 2.95/1.53 thf(sy_c_RulesAndChains_Omaintained, type, maintained: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > $o))))).
% 2.95/1.53 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.95/1.53 thf(sy_c_Set_Oimage, type, image2: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.95/1.53 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.95/1.53 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.95/1.53 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules, type, standa2002409347_rules: (!> [A:$tType]: (((set @ (standard_Constant @ A)) > (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ A)) @ nat)) @ ((labeled_graph @ (standard_Constant @ A)) @ nat))))))).
% 2.95/1.53 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule, type, standa244753842y_rule: (!> [L:$tType]: (((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat))))).
% 2.95/1.53 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule, type, standa245363227y_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 2.95/1.53 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule, type, standa1805702094y_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 2.95/1.53 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule, type, standa2114124375e_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 2.95/1.53 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.95/1.53 thf(sy_v_G, type, g: ((labeled_graph @ (standard_Constant @ a)) @ b)).
% 2.95/1.53 thf(sy_v_L, type, l: (set @ (standard_Constant @ a))).
% 2.95/1.53 thf(sy_v_P____, type, p: (b > (b > $o))).
% 2.95/1.53 thf(sy_v_f____, type, f: (b > b)).
% 2.95/1.53 thf(sy_v_l____, type, l2: (standard_Constant @ a)).
% 2.95/1.53 thf(sy_v_x____, type, x: b).
% 2.95/1.53 thf(sy_v_y____, type, y: b).
% 2.95/1.53 thf(fact_0__092_060open_062_092_060And_062xa_O_A_092_060lbrakk_062f_Axa_A_092_060in_062_Avertices_AG_059_Axa_A_092_060notin_062_Avertices_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom, (! [Xa:b]: (((((member @ b) @ (f @ Xa)) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ b) @ Xa) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)))))).
% 2.95/1.53 thf(fact_1_fv_I2_J, axiom, (((member @ b) @ (f @ y)) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g))).
% 2.95/1.53 thf(fact_2_fv_I1_J, axiom, (((member @ b) @ (f @ x)) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g))).
% 2.95/1.53 thf(fact_3_gu2, axiom, (! [X2:b]: (((((member @ b) @ X2) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ b) @ (f @ X2)) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)))))).
% 2.95/1.53 thf(fact_4_assms_I1_J, axiom, (g = (((restrict @ (standard_Constant @ a)) @ b) @ g))).
% 2.95/1.53 thf(fact_5_r1, axiom, (! [X3:b]: (((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) = ((p @ X3) @ X3))))).
% 2.95/1.53 thf(fact_6_assms_I2_J, axiom, (! [X4:((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))) @ X4) @ ((standa2002409347_rules @ a) @ l)) => (((((maintained @ (standard_Constant @ a)) @ nat) @ b) @ X4) @ g))))).
% 2.95/1.53 thf(fact_7_gr, axiom, (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ (f @ x)) @ (f @ y))) @ ((((getRel @ (standard_Constant @ a)) @ b) @ l2) @ g))).
% 2.95/1.53 thf(fact_8_a, axiom, (((member @ ((product_prod @ (standard_Constant @ a)) @ ((product_prod @ b) @ b))) @ ((((product_Pair @ (standard_Constant @ a)) @ ((product_prod @ b) @ b)) @ l2) @ ((((product_Pair @ b) @ b) @ (f @ x)) @ (f @ y)))) @ (((labeled_edges @ (standard_Constant @ a)) @ b) @ g))).
% 2.95/1.53 thf(fact_9_local_Orefl, axiom, (((refl_on @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g))).
% 2.95/1.53 thf(fact_10_equiv, axiom, (((equiv_equiv @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g))).
% 2.95/1.53 thf(fact_11__092_060open_062graph__union_A_Imap__graph__fn_AG_Af_J_AG_A_061_AG_092_060close_062, axiom, (((((graph_union @ (standard_Constant @ a)) @ b) @ (((((map_graph @ b) @ b) @ (standard_Constant @ a)) @ ((((bNF_Gr @ b) @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ f)) @ g)) @ g) = g)).
% 2.95/1.53 thf(fact_12_gu1, axiom, (! [L2:(standard_Constant @ a),X2:b,Y:b]: (((((member @ ((product_prod @ (standard_Constant @ a)) @ ((product_prod @ b) @ b))) @ ((((product_Pair @ (standard_Constant @ a)) @ ((product_prod @ b) @ b)) @ L2) @ ((((product_Pair @ b) @ b) @ X2) @ Y))) @ (((labeled_edges @ (standard_Constant @ a)) @ b) @ g)) => ((((member @ b) @ X2) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => ((((member @ b) @ Y) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ ((product_prod @ (standard_Constant @ a)) @ ((product_prod @ b) @ b))) @ ((((product_Pair @ (standard_Constant @ a)) @ ((product_prod @ b) @ b)) @ L2) @ ((((product_Pair @ b) @ b) @ (f @ X2)) @ (f @ Y)))) @ (((labeled_edges @ (standard_Constant @ a)) @ b) @ g)))))))).
% 2.95/1.53 thf(fact_13__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062, axiom, (! [X3:b]: (((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X3) @ X3)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)))))).
% 2.95/1.53 thf(fact_14_labeled__graph_Osel_I2_J, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((labeled_vertices @ L) @ V) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = X22)))).
% 2.95/1.53 thf(fact_15_subg, axiom, ((((((graph_homomorphism @ (standard_Constant @ a)) @ b) @ b) @ (((((map_graph @ b) @ b) @ (standard_Constant @ a)) @ ((((bNF_Gr @ b) @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ f)) @ g)) @ g) @ ((id_on @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ (((((map_graph @ b) @ b) @ (standard_Constant @ a)) @ ((((bNF_Gr @ b) @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ f)) @ g))))).
% 2.95/1.53 thf(fact_16_vertices__restrict, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ B) @ A)]: (((((labeled_vertices @ B) @ A) @ (((restrict @ B) @ A) @ G)) = (((labeled_vertices @ B) @ A) @ G))))).
% 2.95/1.53 thf(fact_17_restrict__idemp, axiom, (! [B:$tType,A:$tType,X3:((labeled_graph @ A) @ B)]: (((((restrict @ A) @ B) @ (((restrict @ A) @ B) @ X3)) = (((restrict @ A) @ B) @ X3))))).
% 2.95/1.53 thf(fact_18_labeled__graph_Oinject, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V),Y1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),Y2:(set @ V)]: (((((((labeled_LG @ L) @ V) @ X1) @ X22) = ((((labeled_LG @ L) @ V) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 2.95/1.53 thf(fact_19_graph__union__idemp_I3_J, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A2) @ ((((graph_union @ A) @ B) @ B2) @ A2)) = ((((graph_union @ A) @ B) @ B2) @ A2))))).
% 2.95/1.53 thf(fact_20_graph__union__idemp_I2_J, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A2) @ ((((graph_union @ A) @ B) @ A2) @ B2)) = ((((graph_union @ A) @ B) @ A2) @ B2))))).
% 2.95/1.53 thf(fact_21_graph__union__idemp_I1_J, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A2) @ A2) = A2)))).
% 2.95/1.53 thf(fact_22_map__graph__preserves__restricted, axiom, (! [C:$tType,B:$tType,A:$tType,G:((labeled_graph @ A) @ B),F:(set @ ((product_prod @ B) @ C))]: (((G = (((restrict @ A) @ B) @ G)) => ((((((map_graph @ B) @ C) @ A) @ F) @ G) = (((restrict @ A) @ C) @ (((((map_graph @ B) @ C) @ A) @ F) @ G))))))).
% 2.95/1.53 thf(fact_23_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)))))))).
% 2.95/1.53 thf(fact_24_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)))).
% 2.95/1.53 thf(fact_25_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)))))).
% 2.95/1.53 thf(fact_26_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)))))).
% 2.95/1.53 thf(fact_27_graph__homomorphism__Id, axiom, (! [B:$tType,A:$tType,A3:((labeled_graph @ A) @ B)]: (((((((graph_homomorphism @ A) @ B) @ B) @ (((restrict @ A) @ B) @ A3)) @ (((restrict @ A) @ B) @ A3)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A3)))))).
% 2.95/1.53 thf(fact_28_map__graph__preserves__subgraph, axiom, (! [C:$tType,B:$tType,A:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),F:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A2) @ B2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2))) => ((((((graph_homomorphism @ A) @ C) @ C) @ (((((map_graph @ B) @ C) @ A) @ F) @ A2)) @ (((((map_graph @ B) @ C) @ A) @ F) @ B2)) @ ((id_on @ C) @ (((labeled_vertices @ A) @ C) @ (((((map_graph @ B) @ C) @ A) @ F) @ A2)))))))).
% 2.95/1.53 thf(fact_29_map__graph__fn__id_I2_J, axiom, (! [B:$tType,A:$tType,X5:((labeled_graph @ A) @ B)]: (((((((map_graph @ B) @ B) @ A) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ X5))) @ X5) = (((restrict @ A) @ B) @ X5))))).
% 2.95/1.53 thf(fact_30_map__graph__fn__graphI, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ C),F:(C > B)]: (((((((map_graph @ C) @ B) @ A) @ ((((bNF_Gr @ C) @ B) @ (((labeled_vertices @ A) @ C) @ G)) @ F)) @ G) = (((restrict @ A) @ B) @ (((((map_graph @ C) @ B) @ A) @ ((((bNF_Gr @ C) @ B) @ (((labeled_vertices @ A) @ C) @ G)) @ F)) @ G)))))).
% 2.95/1.53 thf(fact_31_graph__homo, axiom, (! [A:$tType,C:$tType,B:$tType,G:((labeled_graph @ A) @ B),F:(B > C)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((graph_homomorphism @ A) @ B) @ C) @ G) @ (((((map_graph @ B) @ C) @ A) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ G)) @ F)) @ G)) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ G)) @ F)))))).
% 2.95/1.53 thf(fact_32_labeled__graph_Osel_I1_J, axiom, (! [V:$tType,L:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((labeled_edges @ L) @ V) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = X1)))).
% 2.95/1.53 thf(fact_33_subgraph__def, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) = ((G_1 = (((restrict @ A) @ B) @ G_1)) & ((G_2 = (((restrict @ A) @ B) @ G_2)) & (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2))))))).
% 2.95/1.53 thf(fact_34_subgraph__trans, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B),G_3:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) => (((((((graph_homomorphism @ A) @ B) @ B) @ G_2) @ G_3) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_2))) => ((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_3) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1)))))))).
% 2.95/1.53 thf(fact_35_map__graph__fn__eqI, axiom, (! [C:$tType,A:$tType,B:$tType,G:((labeled_graph @ B) @ A),F:(A > C),G2:(A > C)]: (((! [X6:A]: (((((member @ A) @ X6) @ (((labeled_vertices @ B) @ A) @ G)) => ((F @ X6) = (G2 @ X6))))) => ((((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ F)) @ G) = (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ G2)) @ G)))))).
% 2.95/1.53 thf(fact_36_labeled__graph_Oinduct, axiom, (! [V:$tType,L:$tType,P:(((labeled_graph @ L) @ V) > $o),Labeled_graph:((labeled_graph @ L) @ V)]: (((! [X1a:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X2a:(set @ V)]: ((P @ ((((labeled_LG @ L) @ V) @ X1a) @ X2a)))) => (P @ Labeled_graph))))).
% 2.95/1.53 thf(fact_37_labeled__graph_Oexhaust, axiom, (! [L:$tType,V:$tType,Y3:((labeled_graph @ L) @ V)]: ((~ ((! [X12:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X23:(set @ V)]: ((Y3 != ((((labeled_LG @ L) @ V) @ X12) @ X23))))))))).
% 2.95/1.53 thf(fact_38_subgraph__preserves__hom, axiom, (! [A:$tType,B:$tType,C:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),X5:((labeled_graph @ A) @ C),H:(set @ ((product_prod @ C) @ B))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A2) @ B2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2))) => (((((((graph_homomorphism @ A) @ C) @ B) @ X5) @ A2) @ H) => ((((((graph_homomorphism @ A) @ C) @ B) @ X5) @ B2) @ H)))))).
% 2.95/1.53 thf(fact_39_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)))))).
% 2.95/1.53 thf(fact_40_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))))).
% 2.95/1.53 thf(fact_41_getRel__hom, axiom, (! [C:$tType,A:$tType,B:$tType,Y3:A,Z:A,L3:B,G:((labeled_graph @ B) @ A),F:(A > C)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z)) @ ((((getRel @ B) @ A) @ L3) @ G)) => ((((member @ A) @ Y3) @ (((labeled_vertices @ B) @ A) @ G)) => ((((member @ A) @ Z) @ (((labeled_vertices @ B) @ A) @ G)) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ (F @ Y3)) @ (F @ Z))) @ ((((getRel @ B) @ C) @ L3) @ (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ F)) @ G))))))))).
% 2.95/1.53 thf(fact_42_getRel__map__fn, axiom, (! [C:$tType,A:$tType,B:$tType,A22:A,G:((labeled_graph @ B) @ A),B22:A,L3:B,F:(A > C),A3:C,B3: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) @ L3) @ G)) => (((F @ A22) = A3) => (((F @ B22) = B3) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ A3) @ B3)) @ ((((getRel @ B) @ C) @ L3) @ (((((map_graph @ A) @ C) @ B) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ F)) @ G))))))))))).
% 2.95/1.53 thf(fact_43_idt__eq, axiom, (! [X2:b,Y:b]: (((((member @ ((product_prod @ (standard_Constant @ a)) @ ((product_prod @ b) @ b))) @ ((((product_Pair @ (standard_Constant @ a)) @ ((product_prod @ b) @ b)) @ (standard_S_Idt @ a)) @ ((((product_Pair @ b) @ b) @ X2) @ Y))) @ (((labeled_edges @ (standard_Constant @ a)) @ b) @ g)) => (((hilbert_Eps @ b) @ (p @ X2)) = ((hilbert_Eps @ b) @ (p @ Y))))))).
% 2.95/1.53 thf(fact_44_vert__P, axiom, (! [X3:b]: (((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X3) @ ((hilbert_Eps @ b) @ (p @ X3)))) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)))))).
% 2.95/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.95/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X7:A]: ((((member @ A) @ X7) @ A2)))) = A2)))).
% 2.95/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X6:A]: (((P @ X6) = (Q @ X6)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.95/1.53 thf(fact_48_getRel__subgraph, axiom, (! [A:$tType,B:$tType,Y3:A,Z:A,L3:B,G:((labeled_graph @ B) @ A),G3:((labeled_graph @ B) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z)) @ ((((getRel @ B) @ A) @ L3) @ G)) => (((((((graph_homomorphism @ B) @ A) @ A) @ G) @ G3) @ ((id_on @ A) @ (((labeled_vertices @ B) @ A) @ G))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z)) @ ((((getRel @ B) @ A) @ L3) @ G3))))))).
% 2.95/1.53 thf(fact_49_in__Gr, axiom, (! [A:$tType,B:$tType,X3:A,Y3:B,A2:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((((bNF_Gr @ A) @ B) @ A2) @ F)) = ((((member @ A) @ X3) @ A2) & ((F @ X3) = Y3)))))).
% 2.95/1.53 thf(fact_50_Gr__not__in, axiom, (! [B:$tType,A:$tType,X3:A,F2:(set @ A),F:(A > B),Y3:B]: ((((~ ((((member @ A) @ X3) @ F2))) | ((F @ X3) != Y3)) => (~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((((bNF_Gr @ A) @ B) @ F2) @ F)))))))).
% 2.95/1.53 thf(fact_51_Id__onI, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ ((id_on @ A) @ A2)))))).
% 2.95/1.53 thf(fact_52_getRel__dom_I2_J, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),A3:B,B3:B,L3:A]: (((G = (((restrict @ A) @ B) @ G)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((getRel @ A) @ B) @ L3) @ G)) => (((member @ B) @ B3) @ (((labeled_vertices @ A) @ B) @ G))))))).
% 2.95/1.53 thf(fact_53_getRel__dom_I1_J, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),A3:B,B3:B,L3:A]: (((G = (((restrict @ A) @ B) @ G)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((getRel @ A) @ B) @ L3) @ G)) => (((member @ B) @ A3) @ (((labeled_vertices @ A) @ B) @ G))))))).
% 2.95/1.53 thf(fact_54_Id__on__iff, axiom, (! [A:$tType,X3:A,Y3:A,A2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ ((id_on @ A) @ A2)) = ((X3 = Y3) & (((member @ A) @ X3) @ A2)))))).
% 2.95/1.53 thf(fact_55_Id__on__eqI, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((A3 = B3) => ((((member @ A) @ A3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ ((id_on @ A) @ A2))))))).
% 2.95/1.53 thf(fact_56_Id__onE, axiom, (! [A:$tType,C2:((product_prod @ A) @ A),A2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C2) @ ((id_on @ A) @ A2)) => (~ ((! [X6:A]: (((((member @ A) @ X6) @ A2) => (C2 != ((((product_Pair @ A) @ A) @ X6) @ X6))))))))))).
% 2.95/1.53 thf(fact_57_refl__onD2, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),X3:A,Y3:A]: (((((refl_on @ A) @ A2) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R) => (((member @ A) @ Y3) @ A2)))))).
% 2.95/1.53 thf(fact_58_refl__onD1, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),X3:A,Y3:A]: (((((refl_on @ A) @ A2) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R) => (((member @ A) @ X3) @ A2)))))).
% 2.95/1.53 thf(fact_59_refl__onD, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((refl_on @ A) @ A2) @ R) => ((((member @ A) @ A3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ R)))))).
% 2.95/1.53 thf(fact_60_refl__on__Id__on, axiom, (! [A:$tType,A2:(set @ A)]: ((((refl_on @ A) @ A2) @ ((id_on @ A) @ A2))))).
% 2.95/1.53 thf(fact_61_getRel__homR, axiom, (! [C:$tType,A:$tType,B:$tType,Y3:A,Z:A,L3:B,G:((labeled_graph @ B) @ A),U:C,F:(set @ ((product_prod @ A) @ C)),V2:C]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z)) @ ((((getRel @ B) @ A) @ L3) @ G)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ Y3) @ U)) @ F) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ Z) @ V2)) @ F) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ U) @ V2)) @ ((((getRel @ B) @ C) @ L3) @ (((((map_graph @ A) @ C) @ B) @ F) @ G))))))))).
% 2.95/1.53 thf(fact_62_maintained__refl, axiom, (! [B:$tType,C:$tType,A:$tType,R2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C)]: ((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R2) @ R2)) @ G)))).
% 2.95/1.53 thf(fact_63_edge__preserving__on__graphI, axiom, (! [B:$tType,C:$tType,A:$tType,X5:((labeled_graph @ A) @ B),F:(B > C),Y4:(set @ ((product_prod @ A) @ ((product_prod @ C) @ C)))]: (((! [L4:A,X6:B,Y5:B]: (((((member @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ L4) @ ((((product_Pair @ B) @ B) @ X6) @ Y5))) @ (((labeled_edges @ A) @ B) @ X5)) => ((((member @ B) @ X6) @ (((labeled_vertices @ A) @ B) @ X5)) => ((((member @ B) @ Y5) @ (((labeled_vertices @ A) @ B) @ X5)) => (((member @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ ((((product_Pair @ A) @ ((product_prod @ C) @ C)) @ L4) @ ((((product_Pair @ C) @ C) @ (F @ X6)) @ (F @ Y5)))) @ Y4)))))) => ((((((edge_preserving @ B) @ C) @ A) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ X5)) @ F)) @ (((labeled_edges @ A) @ B) @ X5)) @ Y4))))).
% 2.95/1.53 thf(fact_64_reflexivity__rule, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),L3:A]: (((G = (((restrict @ A) @ B) @ G)) => ((((((maintained @ A) @ nat) @ B) @ ((standa245363227y_rule @ A) @ L3)) @ G) => (((refl_on @ B) @ (((labeled_vertices @ A) @ B) @ G)) @ ((((getRel @ A) @ B) @ L3) @ G))))))).
% 2.95/1.53 thf(fact_65_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((A3 = A4) & (B3 = B4)))))).
% 2.95/1.53 thf(fact_66_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)))))).
% 2.95/1.53 thf(fact_67_refl__on__domain, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B3:A]: (((((refl_on @ A) @ A2) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R) => ((((member @ A) @ A3) @ A2) & (((member @ A) @ B3) @ A2))))))).
% 2.95/1.53 thf(fact_68_GrD2, axiom, (! [A:$tType,B:$tType,X3:A,Fx:B,A2:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A2) @ F)) => ((F @ X3) = Fx))))).
% 2.95/1.53 thf(fact_69_GrD1, axiom, (! [B:$tType,A:$tType,X3:A,Fx:B,A2:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A2) @ F)) => (((member @ A) @ X3) @ A2))))).
% 2.95/1.53 thf(fact_70_edge__preserving__Id, axiom, (! [A:$tType,B:$tType,Y3:(set @ A),X3:(set @ ((product_prod @ B) @ ((product_prod @ A) @ A)))]: (((((((edge_preserving @ A) @ A) @ B) @ ((id_on @ A) @ Y3)) @ X3) @ X3)))).
% 2.95/1.53 thf(fact_71_map__graph__edge__preserving, axiom, (! [B:$tType,A:$tType,C:$tType,F:(set @ ((product_prod @ A) @ B)),G:((labeled_graph @ C) @ A)]: (((((((edge_preserving @ A) @ B) @ C) @ F) @ (((labeled_edges @ C) @ A) @ G)) @ (((labeled_edges @ C) @ B) @ (((((map_graph @ A) @ B) @ C) @ F) @ G)))))).
% 2.95/1.53 thf(fact_72_edge__preserving__atomic, axiom, (! [A:$tType,B:$tType,C:$tType,H1:(set @ ((product_prod @ A) @ B)),E1:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A))),E2:(set @ ((product_prod @ C) @ ((product_prod @ B) @ B))),V1:A,V12:B,V22:A,V23:B,K:C]: ((((((((edge_preserving @ A) @ B) @ C) @ H1) @ E1) @ E2) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ V1) @ V12)) @ H1) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ V22) @ V23)) @ H1) => ((((member @ ((product_prod @ C) @ ((product_prod @ A) @ A))) @ ((((product_Pair @ C) @ ((product_prod @ A) @ A)) @ K) @ ((((product_Pair @ A) @ A) @ V1) @ V22))) @ E1) => (((member @ ((product_prod @ C) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ C) @ ((product_prod @ B) @ B)) @ K) @ ((((product_Pair @ B) @ B) @ V12) @ V23))) @ E2)))))))).
% 2.95/1.53 thf(fact_73_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X6:A,Y5:B]: ((P2 = ((((product_Pair @ A) @ B) @ X6) @ Y5))))))).
% 2.95/1.53 thf(fact_74_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ P2))))).
% 2.95/1.53 thf(fact_75_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) => (~ (((A3 = A4) => (B3 != B4)))))))).
% 2.95/1.53 thf(fact_76_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 2.95/1.53 thf(fact_77_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ Prod))))).
% 2.95/1.53 thf(fact_78_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C3:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))).
% 2.95/1.53 thf(fact_79_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 2.95/1.53 thf(fact_80_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E3:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E3)))))))))))).
% 2.95/1.53 thf(fact_81_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E3:E,F4:F3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E3) @ F4))))))))))))).
% 2.95/1.53 thf(fact_82_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,G4:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4))))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E3:E,F4:F3,G5:G4]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G4)) @ E3) @ ((((product_Pair @ F3) @ G4) @ F4) @ G5)))))))))))))).
% 2.95/1.53 thf(fact_83_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X3))))).
% 2.95/1.53 thf(fact_84_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X3))))).
% 2.95/1.53 thf(fact_85_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B5:B,C3:C,D2:D,E3:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E3))))))) => (P @ X3))))).
% 2.95/1.53 thf(fact_86_prod__induct6, axiom, (! [F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: (((! [A5:A,B5:B,C3:C,D2:D,E3:E,F4:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E3) @ F4)))))))) => (P @ X3))))).
% 2.95/1.53 thf(fact_87_prod__induct7, axiom, (! [G4:$tType,F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4))))))]: (((! [A5:A,B5:B,C3:C,D2:D,E3:E,F4:F3,G5:G4]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G4))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G4)) @ E3) @ ((((product_Pair @ F3) @ G4) @ F4) @ G5))))))))) => (P @ X3))))).
% 2.95/1.53 thf(fact_88_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 2.95/1.53 thf(fact_89_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))) => ((! [C4:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs) @ C4) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S) @ G)) @ C4)))) => ((((((((least_559130134_graph @ X) @ L) @ A) @ C) @ T2) @ Rs) @ S) @ G))))))).
% 2.95/1.53 thf(fact_90_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))) => ((! [C4:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs) @ C4) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S) @ G)) @ C4)))) => ((((((((least @ X) @ L) @ A) @ C) @ T2) @ Rs) @ S) @ G)))))).
% 2.95/1.53 thf(fact_91_map__graph__in, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ B),A3:B,B3:B,E4:(allegorical_term @ A),F:(B > C)]: (((G = (((restrict @ A) @ B) @ G)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((semantics @ A) @ B) @ G) @ E4)) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ (F @ A3)) @ (F @ B3))) @ ((((semantics @ A) @ C) @ (((((map_graph @ B) @ C) @ A) @ ((((bNF_Gr @ B) @ C) @ (((labeled_vertices @ A) @ B) @ G)) @ F)) @ G)) @ E4))))))).
% 2.95/1.53 thf(fact_92_maintainedI, axiom, (! [A:$tType,B:$tType,C:$tType,A2:((labeled_graph @ C) @ A),G:((labeled_graph @ C) @ B),B2:((labeled_graph @ C) @ A)]: (((! [F4:(set @ ((product_prod @ A) @ B))]: ((((((((graph_homomorphism @ C) @ A) @ B) @ A2) @ G) @ F4) => ((((((extensible @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A2) @ B2)) @ G) @ F4)))) => (((((maintained @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A2) @ B2)) @ G))))).
% 2.95/1.53 thf(fact_93_extensible__refl__concr, axiom, (! [A:$tType,C:$tType,B:$tType,E_1:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B))),V2:(set @ B),G:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C)),E_2:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_1) @ V2)) @ G) @ F) => (((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ ((((labeled_LG @ A) @ B) @ E_1) @ V2)) @ ((((labeled_LG @ A) @ B) @ E_2) @ V2))) @ G) @ F) = ((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_2) @ V2)) @ G) @ F)))))).
% 2.95/1.53 thf(fact_94_labeled__graph_Osplit__sel, axiom, (! [A:$tType,V:$tType,L:$tType,P:(A > $o),F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P @ (((((labele1974067554_graph @ L) @ V) @ A) @ F) @ Labeled_graph)) = ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) => (P @ ((F @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))).
% 2.95/1.53 thf(fact_95_extensible__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R2) @ G) @ F) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R2) @ R2)) @ G) @ F))))).
% 2.95/1.53 thf(fact_96_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),G6:((labeled_graph @ L) @ C)]: (((((((consequence_graph @ L) @ V) @ C) @ Rs2) @ G6) & ((((((((least @ X) @ L) @ V) @ C) @ T3) @ Rs2) @ S2) @ G6)))))))).
% 2.95/1.53 thf(fact_97_consequence__graphD_I3_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C)]: (((((((consequence_graph @ A) @ B) @ C) @ Rs) @ G) => (G = (((restrict @ A) @ C) @ G)))))).
% 2.95/1.53 thf(fact_98_graph__homomorphism__semantics, axiom, (! [B:$tType,C:$tType,A:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C)),A3:B,B3:B,E4:(allegorical_term @ A),A4:C,B4:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ B2) @ F) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((semantics @ A) @ B) @ A2) @ E4)) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ A3) @ A4)) @ F) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B3) @ B4)) @ F) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ A4) @ B4)) @ ((((semantics @ A) @ C) @ B2) @ E4))))))))).
% 2.95/1.53 thf(fact_99_consequence__graphD_I1_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G:((labeled_graph @ A) @ C),R2:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((((consequence_graph @ A) @ B) @ C) @ Rs) @ G) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R2) @ Rs) => (((((maintained @ A) @ B) @ C) @ R2) @ G)))))).
% 2.95/1.53 thf(fact_100_labeled__graph_Ocase, axiom, (! [L:$tType,A:$tType,V:$tType,F:((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) @ F) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 2.95/1.53 thf(fact_101_semantics__in__vertices_I1_J, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B),A3:B,B3:B,E4:(allegorical_term @ A)]: (((A2 = (((restrict @ A) @ B) @ A2)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((semantics @ A) @ B) @ A2) @ E4)) => (((member @ B) @ A3) @ (((labeled_vertices @ A) @ B) @ A2))))))).
% 2.95/1.53 thf(fact_102_semantics__in__vertices_I2_J, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B),A3:B,B3:B,E4:(allegorical_term @ A)]: (((A2 = (((restrict @ A) @ B) @ A2)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((semantics @ A) @ B) @ A2) @ E4)) => (((member @ B) @ B3) @ (((labeled_vertices @ A) @ B) @ A2))))))).
% 2.95/1.53 thf(fact_103_labeled__graph_Ocase__eq__if, axiom, (! [A:$tType,V:$tType,L:$tType]: (((((labele1974067554_graph @ L) @ V) @ A) = (^ [F5:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph3:((labeled_graph @ L) @ V)]: (((F5 @ (((labeled_edges @ L) @ V) @ Labeled_graph3)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph3)))))))).
% 2.95/1.53 thf(fact_104_subgraph__semantics, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),A3:B,B3:B,E4:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A2) @ B2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2))) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((semantics @ A) @ B) @ A2) @ E4)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A3) @ B3)) @ ((((semantics @ A) @ B) @ B2) @ E4))))))).
% 2.95/1.53 thf(fact_105_maintainedD, axiom, (! [A:$tType,C:$tType,B:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A2) @ B2)) @ G) => (((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ G) @ F) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A2) @ B2)) @ G) @ F)))))).
% 2.95/1.53 thf(fact_106_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),G6:((labeled_graph @ L) @ C)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S2) @ G6) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S2))) & (! [C5:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ V) @ X) @ Rs2) @ C5) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S2) @ G6)) @ C5))))))))))).
% 2.95/1.53 thf(fact_107_labeled__graph_Osplit__sel__asm, axiom, (! [A:$tType,V:$tType,L:$tType,P:(A > $o),F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P @ (((((labele1974067554_graph @ L) @ V) @ A) @ F) @ Labeled_graph)) = (~ (((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) & (~ ((P @ ((F @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))))))).
% 2.95/1.53 thf(fact_108_extensibleI, axiom, (! [A:$tType,C:$tType,B:$tType,R22:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),G2:(set @ ((product_prod @ B) @ C)),R1:((labeled_graph @ A) @ B),F:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R22) @ G) @ G2) => (((((((agree_on @ A) @ B) @ C) @ R1) @ F) @ G2) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R1) @ R22)) @ G) @ F)))))).
% 2.95/1.53 thf(fact_109_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)))))).
% 2.95/1.53 thf(fact_110_graph__homo__union__id_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((graph_union @ A) @ B) @ A2) @ B2)) @ G) @ F) => ((A2 = (((restrict @ A) @ B) @ A2)) => ((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ G) @ (((((relcomp @ B) @ B) @ C) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2))) @ F))))))).
% 2.95/1.53 thf(fact_111_graph__homo__union__id_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((graph_union @ A) @ B) @ A2) @ B2)) @ G) @ F) => ((B2 = (((restrict @ A) @ B) @ B2)) => ((((((graph_homomorphism @ A) @ B) @ C) @ B2) @ G) @ (((((relcomp @ B) @ B) @ C) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ B2))) @ F))))))).
% 2.95/1.53 thf(fact_112__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062, axiom, (f = (^ [X7:b]: (((((if @ b) @ (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X7) @ (bot_bot @ (set @ b)))) = (bot_bot @ (set @ b)))) @ X7) @ ((hilbert_Eps @ b) @ (p @ X7))))))).
% 2.95/1.53 thf(fact_113_agree__on__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R2:((labeled_graph @ A) @ B),F:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ R2) @ F) @ F)))).
% 2.95/1.53 thf(fact_114_agree__on__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((agree_on @ A) @ B) @ C) = (^ [X8:((labeled_graph @ A) @ B),F5:(set @ ((product_prod @ B) @ C)),G7:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ X8) @ G7) @ F5))))))).
% 2.95/1.53 thf(fact_115_ImageI, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R:(set @ ((product_prod @ A) @ B)),A2:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R) => ((((member @ A) @ A3) @ A2) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ R) @ A2))))))).
% 2.95/1.53 thf(fact_116_Image__empty2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ B) @ A))]: ((((((image @ B) @ A) @ R2) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.95/1.53 thf(fact_117_Image__empty1, axiom, (! [B:$tType,A:$tType,X5:(set @ B)]: ((((((image @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) @ X5) = (bot_bot @ (set @ A)))))).
% 2.95/1.53 thf(fact_118_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.95/1.53 thf(fact_119_Gr__empty, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((bNF_Gr @ A) @ B) @ (bot_bot @ (set @ A))) @ F) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.95/1.53 thf(fact_120_graph__homomorphism__composes, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A3:((labeled_graph @ A) @ B),B3:((labeled_graph @ A) @ C),X3:(set @ ((product_prod @ B) @ C)),C2:((labeled_graph @ A) @ D),Y3:(set @ ((product_prod @ C) @ D))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A3) @ B3) @ X3) => (((((((graph_homomorphism @ A) @ C) @ D) @ B3) @ C2) @ Y3) => ((((((graph_homomorphism @ A) @ B) @ D) @ A3) @ C2) @ (((((relcomp @ B) @ C) @ D) @ X3) @ Y3))))))).
% 2.95/1.53 thf(fact_121_P__eq, axiom, (! [X2:b]: (((((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X2) @ (bot_bot @ (set @ b)))) != (bot_bot @ (set @ b))) => (((hilbert_Eps @ b) @ (p @ ((hilbert_Eps @ b) @ (p @ X2)))) = ((hilbert_Eps @ b) @ (p @ X2))))))).
% 2.95/1.53 thf(fact_122_Gr__insert, axiom, (! [B:$tType,A:$tType,X3:A,F2:(set @ A),F:(A > B)]: ((((((bNF_Gr @ A) @ B) @ (((insert @ A) @ X3) @ F2)) @ F) = (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ (F @ X3))) @ ((((bNF_Gr @ A) @ B) @ F2) @ F)))))).
% 2.95/1.53 thf(fact_123_map__graph__selectors_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,F:(set @ ((product_prod @ C) @ A)),G:((labeled_graph @ B) @ C)]: (((((labeled_vertices @ B) @ A) @ (((((map_graph @ C) @ A) @ B) @ F) @ G)) = ((((image @ C) @ A) @ F) @ (((labeled_vertices @ B) @ C) @ G)))))).
% 2.95/1.53 thf(fact_124_agree__on__empty, axiom, (! [A:$tType,C:$tType,B:$tType,F:(set @ ((product_prod @ B) @ C)),G2:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ (bot_bot @ (set @ B)))) @ F) @ G2)))).
% 2.95/1.53 thf(fact_125_Image__singleton__iff, axiom, (! [A:$tType,B:$tType,B3:A,R:(set @ ((product_prod @ B) @ A)),A3:B]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ R) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B))))) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A3) @ B3)) @ R))))).
% 2.95/1.53 thf(fact_126_in__on__graph, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,X3:A,G:((labeled_graph @ B) @ A),A3:(A > C),Y3:D,B3:(set @ ((product_prod @ C) @ D))]: (((((member @ A) @ X3) @ (((labeled_vertices @ B) @ A) @ G)) => ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (A3 @ X3)) @ Y3)) @ B3) => (((member @ ((product_prod @ A) @ D)) @ ((((product_Pair @ A) @ D) @ X3) @ Y3)) @ (((((relcomp @ A) @ C) @ D) @ ((((bNF_Gr @ A) @ C) @ (((labeled_vertices @ B) @ A) @ G)) @ A3)) @ B3))))))).
% 2.95/1.53 thf(fact_127_graph__homomorphism__empty, axiom, (! [B:$tType,C:$tType,A:$tType,G:((labeled_graph @ A) @ C),F:(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) @ F) = ((F = (bot_bot @ (set @ ((product_prod @ B) @ C)))) & (G = (((restrict @ A) @ C) @ G))))))).
% 2.95/1.53 thf(fact_128_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)))))).
% 2.95/1.53 thf(fact_129_graph__single, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,C2:B]: ((((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A3) @ ((((product_Pair @ B) @ B) @ B3) @ C2))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B3) @ (((insert @ B) @ C2) @ (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)) @ A3) @ ((((product_Pair @ B) @ B) @ B3) @ C2))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B3) @ (((insert @ B) @ C2) @ (bot_bot @ (set @ B)))))))))).
% 2.95/1.53 thf(fact_130_r2, axiom, (! [X3:b]: (((((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X3) @ (bot_bot @ (set @ b)))) = (bot_bot @ (set @ b))) = (~ ((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)))))))).
% 2.95/1.53 thf(fact_131_refl__on__singleton, axiom, (! [A:$tType,X3:A]: ((((refl_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.95/1.53 thf(fact_132_rev__ImageI, axiom, (! [B:$tType,A:$tType,A3:A,A2:(set @ A),B3:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ A2) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ R) @ A2))))))).
% 2.95/1.53 thf(fact_133_Image__iff, axiom, (! [A:$tType,B:$tType,B3:A,R:(set @ ((product_prod @ B) @ A)),A2:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ R) @ A2)) = (? [X7:B]: (((((member @ B) @ X7) @ A2) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X7) @ B3)) @ R)))))))).
% 2.95/1.53 thf(fact_134_ImageE, axiom, (! [A:$tType,B:$tType,B3:A,R:(set @ ((product_prod @ B) @ A)),A2:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ R) @ A2)) => (~ ((! [X6:B]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X6) @ B3)) @ R) => (~ ((((member @ B) @ X6) @ A2)))))))))))).
% 2.95/1.53 thf(fact_135_relcomp_OrelcompI, axiom, (! [A:$tType,C:$tType,B:$tType,A3:A,B3:B,R:(set @ ((product_prod @ A) @ B)),C2:C,S3:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B3) @ C2)) @ S3) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A3) @ C2)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S3))))))).
% 2.95/1.53 thf(fact_136_relcomp_Oinducts, axiom, (! [B:$tType,A:$tType,C:$tType,X1:A,X22:C,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ B) @ C)),P:(A > (C > $o))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X1) @ X22)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S3)) => ((! [A5:A,B5:B,C3:C]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B5) @ C3)) @ S3) => ((P @ A5) @ C3))))) => ((P @ X1) @ X22)))))).
% 2.95/1.53 thf(fact_137_relcomp_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,A1:A,A22:C,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A1) @ A22)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S3)) = (? [A6:A,B6:B,C6:C]: (((A1 = A6) & ((A22 = C6) & ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B6)) @ R) & (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B6) @ C6)) @ S3)))))))))).
% 2.95/1.53 thf(fact_138_relcomp_Ocases, axiom, (! [A:$tType,C:$tType,B:$tType,A1:A,A22:C,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A1) @ A22)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S3)) => (~ ((! [B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A1) @ B5)) @ R) => (~ ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B5) @ A22)) @ S3)))))))))))).
% 2.95/1.53 thf(fact_139_relcompEpair, axiom, (! [A:$tType,B:$tType,C:$tType,A3:A,C2:B,R:(set @ ((product_prod @ A) @ C)),S3:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ C2)) @ (((((relcomp @ A) @ C) @ B) @ R) @ S3)) => (~ ((! [B5:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A3) @ B5)) @ R) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ B5) @ C2)) @ S3)))))))))))).
% 2.95/1.53 thf(fact_140_relcompE, axiom, (! [A:$tType,B:$tType,C:$tType,Xz:((product_prod @ A) @ B),R:(set @ ((product_prod @ A) @ C)),S3:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ Xz) @ (((((relcomp @ A) @ C) @ B) @ R) @ S3)) => (~ ((! [X6:A,Y5:C,Z2:B]: (((Xz = ((((product_Pair @ A) @ B) @ X6) @ Z2)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X6) @ Y5)) @ R) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ Y5) @ Z2)) @ S3))))))))))))).
% 2.95/1.53 thf(fact_141_relcomp__Image, axiom, (! [A:$tType,C:$tType,B:$tType,X5:(set @ ((product_prod @ B) @ C)),Y4:(set @ ((product_prod @ C) @ A)),Z3:(set @ B)]: ((((((image @ B) @ A) @ (((((relcomp @ B) @ C) @ A) @ X5) @ Y4)) @ Z3) = ((((image @ C) @ A) @ Y4) @ ((((image @ B) @ C) @ X5) @ Z3)))))).
% 2.95/1.53 thf(fact_142_relcomp__mono, axiom, (! [A:$tType,C:$tType,B:$tType,R3:(set @ ((product_prod @ A) @ B)),R:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ B) @ C)),S3:(set @ ((product_prod @ B) @ C))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ R) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ C))) @ S4) @ S3) => (((ord_less_eq @ (set @ ((product_prod @ A) @ C))) @ (((((relcomp @ A) @ B) @ C) @ R3) @ S4)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S3))))))).
% 2.95/1.53 thf(fact_143_Image__mono, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),R:(set @ ((product_prod @ A) @ B)),A7:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ R) => ((((ord_less_eq @ (set @ A)) @ A7) @ A2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R3) @ A7)) @ ((((image @ A) @ B) @ R) @ A2))))))).
% 2.95/1.53 thf(fact_144_O__assoc, axiom, (! [A:$tType,D:$tType,B:$tType,C:$tType,R2:(set @ ((product_prod @ A) @ D)),S:(set @ ((product_prod @ D) @ C)),T4:(set @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ (((((relcomp @ A) @ D) @ C) @ R2) @ S)) @ T4) = (((((relcomp @ A) @ D) @ B) @ R2) @ (((((relcomp @ D) @ C) @ B) @ S) @ T4)))))).
% 2.95/1.53 thf(fact_145_agree__on__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((agree_on @ A) @ B) @ C) = (^ [G6:((labeled_graph @ A) @ B),F_1:(set @ ((product_prod @ B) @ C)),F_2:(set @ ((product_prod @ B) @ C))]: ((! [X7:B]: (((((member @ B) @ X7) @ (((labeled_vertices @ A) @ B) @ G6)) => (((((image @ B) @ C) @ F_1) @ (((insert @ B) @ X7) @ (bot_bot @ (set @ B)))) = ((((image @ B) @ C) @ F_2) @ (((insert @ B) @ X7) @ (bot_bot @ (set @ B)))))))))))))).
% 2.95/1.53 thf(fact_146_edge__preserving__subset, axiom, (! [A:$tType,B:$tType,C:$tType,R_1:(set @ ((product_prod @ A) @ B)),R_2:(set @ ((product_prod @ A) @ B)),E_12:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A))),E_22:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A))),G:(set @ ((product_prod @ C) @ ((product_prod @ B) @ B)))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R_1) @ R_2) => ((((ord_less_eq @ (set @ ((product_prod @ C) @ ((product_prod @ A) @ A)))) @ E_12) @ E_22) => (((((((edge_preserving @ A) @ B) @ C) @ R_2) @ E_22) @ G) => ((((((edge_preserving @ A) @ B) @ C) @ R_1) @ E_12) @ G))))))).
% 2.95/1.53 thf(fact_147_agree__on__trans, axiom, (! [A:$tType,C:$tType,B:$tType,X5:((labeled_graph @ A) @ B),F:(set @ ((product_prod @ B) @ C)),G2:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ X5) @ F) @ G2) => (((((((agree_on @ A) @ B) @ C) @ X5) @ G2) @ H) => ((((((agree_on @ A) @ B) @ C) @ X5) @ F) @ H)))))).
% 2.95/1.53 thf(fact_148_agree__on__ext, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,G:((labeled_graph @ A) @ B),F_12:(set @ ((product_prod @ B) @ C)),F_22:(set @ ((product_prod @ B) @ C)),G2:(set @ ((product_prod @ C) @ D))]: ((((((((agree_on @ A) @ B) @ C) @ G) @ F_12) @ F_22) => ((((((agree_on @ A) @ B) @ D) @ G) @ (((((relcomp @ B) @ C) @ D) @ F_12) @ G2)) @ (((((relcomp @ B) @ C) @ D) @ F_22) @ G2)))))).
% 2.95/1.53 thf(fact_149_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.95/1.53 thf(fact_150_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((! [X6:A,Y5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Y5)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X6) @ Y5)) @ S3)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S3))))).
% 2.95/1.53 thf(fact_151_compose__preserves__edge__preserving, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,H1:(set @ ((product_prod @ A) @ B)),E1:(set @ ((product_prod @ C) @ ((product_prod @ A) @ A))),E2:(set @ ((product_prod @ C) @ ((product_prod @ B) @ B))),H2:(set @ ((product_prod @ B) @ D)),E32:(set @ ((product_prod @ C) @ ((product_prod @ D) @ D)))]: ((((((((edge_preserving @ A) @ B) @ C) @ H1) @ E1) @ E2) => (((((((edge_preserving @ B) @ D) @ C) @ H2) @ E2) @ E32) => ((((((edge_preserving @ A) @ D) @ C) @ (((((relcomp @ A) @ B) @ D) @ H1) @ H2)) @ E1) @ E32)))))).
% 2.95/1.53 thf(fact_152_graph__union__iff, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2) = ((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2)) & (((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2))))))).
% 2.95/1.53 thf(fact_153_restrict__subsD, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G)) @ (((labeled_edges @ A) @ B) @ (((restrict @ A) @ B) @ G))) => (G = (((restrict @ A) @ B) @ G)))))).
% 2.95/1.53 thf(fact_154_subgraph__def2, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((G_1 = (((restrict @ A) @ B) @ G_1)) => ((G_2 = (((restrict @ A) @ B) @ G_2)) => (((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) = ((((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)) & (((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2))))))))).
% 2.95/1.53 thf(fact_155_subgraph__subset_I1_J, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) => (((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)))))).
% 2.95/1.53 thf(fact_156_maintainedD2, axiom, (! [A:$tType,C:$tType,B:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ B),G:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A2) @ B2)) @ G) => (((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ G) @ F) => (~ ((! [G5:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ B2) @ G) @ G5) => (~ ((((ord_less_eq @ (set @ ((product_prod @ B) @ C))) @ F) @ G5))))))))))))).
% 2.95/1.53 thf(fact_157_Id__on__vertices__identity_I2_J, axiom, (! [B:$tType,C:$tType,A:$tType,A3:((labeled_graph @ A) @ B),B3:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C)),Aa:B,Ba:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A3) @ B3) @ F) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa) @ Ba)) @ F) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa) @ Ba)) @ (((((relcomp @ B) @ C) @ C) @ F) @ ((id_on @ C) @ (((labeled_vertices @ A) @ C) @ B3))))))))).
% 2.95/1.53 thf(fact_158_Id__on__vertices__identity_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,A3:((labeled_graph @ A) @ B),B3:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C)),Aa:B,Ba:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A3) @ B3) @ F) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa) @ Ba)) @ F) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ Aa) @ Ba)) @ (((((relcomp @ B) @ B) @ C) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A3))) @ F))))))).
% 2.95/1.53 thf(fact_159_graph__homomorphism__on__graph, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ C),R2:(set @ ((product_prod @ B) @ C)),F:(C > D)]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ B2) @ R2) => ((((((graph_homomorphism @ A) @ B) @ D) @ A2) @ (((((map_graph @ C) @ D) @ A) @ ((((bNF_Gr @ C) @ D) @ (((labeled_vertices @ A) @ C) @ B2)) @ F)) @ B2)) @ (((((relcomp @ B) @ C) @ D) @ R2) @ ((((bNF_Gr @ C) @ D) @ (((labeled_vertices @ A) @ C) @ B2)) @ F))))))).
% 2.95/1.53 thf(fact_160_agree__on__subg__compose, axiom, (! [A:$tType,C:$tType,B:$tType,R2:((labeled_graph @ A) @ B),G2:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C)),F2:((labeled_graph @ A) @ B),F:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ R2) @ G2) @ H) => (((((((agree_on @ A) @ B) @ C) @ F2) @ F) @ G2) => (((((((graph_homomorphism @ A) @ B) @ B) @ F2) @ R2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ F2))) => ((((((agree_on @ A) @ B) @ C) @ F2) @ F) @ H))))))).
% 2.95/1.53 thf(fact_161_singleton__insert__inj__eq, axiom, (! [A:$tType,B3:A,A3:A,A2:(set @ A)]: ((((((insert @ A) @ B3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A2)) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.95/1.53 thf(fact_162_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A]: ((((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.95/1.53 thf(fact_163_equiv__class__subset, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))))))))).
% 2.95/1.53 thf(fact_164_subset__equiv__class, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),B3:A,A3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) => ((((member @ A) @ B3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R))))))).
% 2.95/1.53 thf(fact_165_refines__equiv__class__eq2, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A)),A2:(set @ A),A3:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ S) => ((((equiv_equiv @ A) @ A2) @ R2) => ((((equiv_equiv @ A) @ A2) @ S) => (((((image @ A) @ A) @ S) @ ((((image @ A) @ A) @ R2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = ((((image @ A) @ A) @ S) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))))).
% 2.95/1.53 thf(fact_166_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X7:A]: ((~ ((P @ X7))))))))).
% 2.95/1.53 thf(fact_167_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X7:A]: ((~ ((P @ X7))))))))).
% 2.95/1.53 thf(fact_168_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X7:A]: ((~ ((((member @ A) @ X7) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_169_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.95/1.53 thf(fact_170_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.95/1.53 thf(fact_171_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X6:A]: (((((member @ A) @ X6) @ A2) => (((member @ A) @ X6) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.95/1.53 thf(fact_172_insert__absorb2, axiom, (! [A:$tType,X3:A,A2:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ X3) @ A2)) = (((insert @ A) @ X3) @ A2))))).
% 2.95/1.53 thf(fact_173_insert__iff, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) = ((A3 = B3) | (((member @ A) @ A3) @ A2)))))).
% 2.95/1.53 thf(fact_174_insertCI, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A3) @ B2))) => (A3 = B3)) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.95/1.53 thf(fact_175_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 2.95/1.53 thf(fact_176_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_177_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_178_insert__subset, axiom, (! [A:$tType,X3:A,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A2)) @ B2) = ((((member @ A) @ X3) @ B2) & (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.95/1.53 thf(fact_179_relcomp__empty1, axiom, (! [C:$tType,B:$tType,A:$tType,R2:(set @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ C)))) @ R2) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.95/1.53 thf(fact_180_relcomp__empty2, axiom, (! [C:$tType,B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ C))]: (((((((relcomp @ A) @ C) @ B) @ R2) @ (bot_bot @ (set @ ((product_prod @ C) @ B)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.95/1.53 thf(fact_181_graph__empty__e, axiom, (! [A:$tType,B:$tType,V2:(set @ B)]: ((((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V2) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V2)))))).
% 2.95/1.53 thf(fact_182_graph__homomorphism__nonempty, axiom, (! [B:$tType,A:$tType,C:$tType,A2:((labeled_graph @ A) @ B),B2:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C)),E4:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A2) @ B2) @ F) => ((((((semantics @ A) @ B) @ A2) @ E4) != (bot_bot @ (set @ ((product_prod @ B) @ B)))) => (((((semantics @ A) @ C) @ B2) @ E4) != (bot_bot @ (set @ ((product_prod @ C) @ C))))))))).
% 2.95/1.53 thf(fact_183_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X7:A]: ((((member @ A) @ X7) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_184_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y5:A]: ((~ ((((member @ A) @ Y5) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_185_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 2.95/1.53 thf(fact_186_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.95/1.53 thf(fact_187_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X7:A]: (((P @ X7) => (Q @ X7)))))))).
% 2.95/1.53 thf(fact_188_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y6:(set @ A),Z4:(set @ A)]: ((Y6 = Z4))) = (^ [A8:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A8)))))))).
% 2.95/1.53 thf(fact_189_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C7) => (((ord_less_eq @ (set @ A)) @ A2) @ C7)))))).
% 2.95/1.53 thf(fact_190_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X6:A]: (((P @ X6) => (Q @ X6)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.95/1.53 thf(fact_191_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.95/1.53 thf(fact_192_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A8) => (((member @ A) @ T3) @ B7)))))))))).
% 2.95/1.53 thf(fact_193_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.95/1.53 thf(fact_194_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.95/1.53 thf(fact_195_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: ((! [X7:A]: (((((member @ A) @ X7) @ A8) => (((member @ A) @ X7) @ B7)))))))))).
% 2.95/1.53 thf(fact_196_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.95/1.53 thf(fact_197_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.95/1.53 thf(fact_198_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))))).
% 2.95/1.53 thf(fact_199_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (? [B8:(set @ A)]: (((A2 = (((insert @ A) @ A3) @ B8)) & (~ ((((member @ A) @ A3) @ B8)))))))))).
% 2.95/1.53 thf(fact_200_insert__commute, axiom, (! [A:$tType,X3:A,Y3:A,A2:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ Y3) @ A2)) = (((insert @ A) @ Y3) @ (((insert @ A) @ X3) @ A2)))))).
% 2.95/1.53 thf(fact_201_insert__eq__iff, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A,B2:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((~ ((((member @ A) @ B3) @ B2))) => (((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ B2)) = (((A3 = B3) => (A2 = B2)) & ((A3 != B3) => (? [C5:(set @ A)]: (((A2 = (((insert @ A) @ B3) @ C5)) & ((~ ((((member @ A) @ B3) @ C5))) & ((B2 = (((insert @ A) @ A3) @ C5)) & (~ ((((member @ A) @ A3) @ C5)))))))))))))))).
% 2.95/1.53 thf(fact_202_insert__absorb, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ A2) = A2))))).
% 2.95/1.53 thf(fact_203_insert__ident, axiom, (! [A:$tType,X3:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X3) @ A2))) => ((~ ((((member @ A) @ X3) @ B2))) => (((((insert @ A) @ X3) @ A2) = (((insert @ A) @ X3) @ B2)) = (A2 = B2))))))).
% 2.95/1.53 thf(fact_204_Set_Oset__insert, axiom, (! [A:$tType,X3:A,A2:(set @ A)]: (((((member @ A) @ X3) @ A2) => (~ ((! [B8:(set @ A)]: (((A2 = (((insert @ A) @ X3) @ B8)) => (((member @ A) @ X3) @ B8)))))))))).
% 2.95/1.53 thf(fact_205_insertI2, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: (((((member @ A) @ A3) @ B2) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.95/1.53 thf(fact_206_insertI1, axiom, (! [A:$tType,A3:A,B2:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B2))))).
% 2.95/1.53 thf(fact_207_insertE, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) => ((A3 != B3) => (((member @ A) @ A3) @ A2)))))).
% 2.95/1.53 thf(fact_208_singleton__inject, axiom, (! [A:$tType,A3:A,B3:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A3 = B3))))).
% 2.95/1.53 thf(fact_209_insert__not__empty, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((insert @ A) @ A3) @ A2) != (bot_bot @ (set @ A)))))).
% 2.95/1.53 thf(fact_210_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B3:A,C2:A,D3:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C2) @ (((insert @ A) @ D3) @ (bot_bot @ (set @ A))))) = (((A3 = C2) & (B3 = D3)) | ((A3 = D3) & (B3 = C2))))))).
% 2.95/1.53 thf(fact_211_singleton__iff, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B3 = A3))))).
% 2.95/1.53 thf(fact_212_singletonD, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B3 = A3))))).
% 2.95/1.53 thf(fact_213_subset__insertI2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),B3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 2.95/1.53 thf(fact_214_subset__insertI, axiom, (! [A:$tType,B2:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((insert @ A) @ A3) @ B2))))).
% 2.95/1.53 thf(fact_215_subset__insert, axiom, (! [A:$tType,X3:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X3) @ A2))) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X3) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.95/1.53 thf(fact_216_insert__mono, axiom, (! [A:$tType,C7:(set @ A),D4:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C7) @ D4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C7)) @ (((insert @ A) @ A3) @ D4)))))).
% 2.95/1.53 thf(fact_217_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 2.95/1.53 thf(fact_218_subset__singletonD, axiom, (! [A:$tType,A2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => ((A2 = (bot_bot @ (set @ A))) | (A2 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.95/1.53 thf(fact_219_equiv__class__self, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((member @ A) @ A3) @ A2) => (((member @ A) @ A3) @ ((((image @ A) @ A) @ R) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))))).
% 2.95/1.53 thf(fact_220_equiv__class__eq__iff, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),X3:A,Y3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R) = ((((((image @ A) @ A) @ R) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert @ A) @ Y3) @ (bot_bot @ (set @ A))))) & ((((member @ A) @ X3) @ A2) & (((member @ A) @ Y3) @ A2)))))))).
% 2.95/1.53 thf(fact_221_eq__equiv__class__iff, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),X3:A,Y3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y3) @ A2) => ((((((image @ A) @ A) @ R) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert @ A) @ Y3) @ (bot_bot @ (set @ A))))) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))))))).
% 2.95/1.53 thf(fact_222_equiv__class__eq, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R) => (((((image @ A) @ A) @ R) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))))))))).
% 2.95/1.53 thf(fact_223_eq__equiv__class, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B3:A,A2:(set @ A)]: (((((((image @ A) @ A) @ R) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))) => ((((equiv_equiv @ A) @ A2) @ R) => ((((member @ A) @ B3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R))))))).
% 2.95/1.53 thf(fact_224_refines__equiv__class__eq, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A)),A2:(set @ A),A3:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ S) => ((((equiv_equiv @ A) @ A2) @ R2) => ((((equiv_equiv @ A) @ A2) @ S) => (((((image @ A) @ A) @ R2) @ ((((image @ A) @ A) @ S) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = ((((image @ A) @ A) @ S) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))))).
% 2.95/1.53 thf(fact_225_nonempty__rule, axiom, (! [A:$tType,B:$tType,G:((labeled_graph @ A) @ B)]: (((G = (((restrict @ A) @ B) @ G)) => ((((((maintained @ A) @ nat) @ B) @ (standa244753842y_rule @ A)) @ G) = ((((labeled_vertices @ A) @ B) @ G) != (bot_bot @ (set @ B)))))))).
% 2.95/1.53 thf(fact_226_ma, axiom, (! [X4:((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))) @ X4) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))) @ ((standa245363227y_rule @ (standard_Constant @ a)) @ (standard_S_Idt @ a))) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))) @ ((standa2114124375e_rule @ (standard_Constant @ a)) @ (standard_S_Idt @ a))) @ (((insert @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat))) @ ((standa1805702094y_rule @ (standard_Constant @ a)) @ (standard_S_Idt @ a))) @ (bot_bot @ (set @ ((product_prod @ ((labeled_graph @ (standard_Constant @ a)) @ nat)) @ ((labeled_graph @ (standard_Constant @ a)) @ nat)))))))) => (((((maintained @ (standard_Constant @ a)) @ nat) @ b) @ X4) @ g))))).
% 2.95/1.53 thf(fact_227_proj__iff, axiom, (! [A:$tType,A2:(set @ A),R:(set @ ((product_prod @ A) @ A)),X3:A,Y3:A]: (((((equiv_equiv @ A) @ A2) @ R) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y3) @ (bot_bot @ (set @ A))))) @ A2) => ((((((equiv_proj @ A) @ A) @ R) @ X3) = ((((equiv_proj @ A) @ A) @ R) @ Y3)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R))))))).
% 2.95/1.53 thf(fact_228_proj__def, axiom, (! [A:$tType,B:$tType]: ((((equiv_proj @ B) @ A) = (^ [R4:(set @ ((product_prod @ B) @ A)),X7:B]: (((((image @ B) @ A) @ R4) @ (((insert @ B) @ X7) @ (bot_bot @ (set @ B)))))))))).
% 2.95/1.53 thf(fact_229_equivalence, axiom, (! [B:$tType,A:$tType,G:((labeled_graph @ A) @ B),I:A]: (((G = (((restrict @ A) @ B) @ G)) => ((! [X6:((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))) @ X6) @ (((insert @ ((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))) @ ((standa245363227y_rule @ A) @ I)) @ (((insert @ ((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))) @ ((standa2114124375e_rule @ A) @ I)) @ (((insert @ ((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))) @ ((standa1805702094y_rule @ A) @ I)) @ (bot_bot @ (set @ ((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat)))))))) => (((((maintained @ A) @ nat) @ B) @ X6) @ G)))) => (((equiv_equiv @ B) @ (((labeled_vertices @ A) @ B) @ G)) @ ((((getRel @ A) @ B) @ I) @ G))))))).
% 2.95/1.53 thf(fact_230_f, axiom, (f = (^ [X7:b]: (((((if @ b) @ (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X7) @ (bot_bot @ (set @ b)))) = (bot_bot @ (set @ b)))) @ X7) @ ((hilbert_Eps @ b) @ (p @ X7))))))).
% 2.95/1.53 thf(fact_231_the__elem__eq, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3)))).
% 2.95/1.53 thf(fact_232_tuple__disj, axiom, (! [A:$tType,X3:A,Z:A]: ((((collect @ A) @ (^ [Y7:A]: (((Y7 = X3) | (Y7 = Z))))) = (((insert @ A) @ X3) @ (((insert @ A) @ Z) @ (bot_bot @ (set @ A)))))))).
% 2.95/1.53 thf(fact_233_singleton__conv, axiom, (! [A:$tType,A3:A]: ((((collect @ A) @ (^ [X7:A]: ((X7 = A3)))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_234_singleton__conv2, axiom, (! [A:$tType,A3:A]: ((((collect @ A) @ ((^ [Y6:A,Z4:A]: ((Y6 = Z4))) @ A3)) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.95/1.53 thf(fact_235_P, axiom, (p = (^ [X7:b,Y7:b]: ((((member @ b) @ Y7) @ ((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X7) @ (bot_bot @ (set @ b))))))))).
% 2.95/1.53 thf(fact_236_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X7:A]: ($false))))))).
% 2.95/1.53 thf(fact_237_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A6:A,B7:(set @ A)]: (((collect @ A) @ (^ [X7:A]: (((X7 = A6) | (((member @ A) @ X7) @ B7))))))))))).
% 2.95/1.53 thf(fact_238_insert__Collect, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((insert @ A) @ A3) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U2:A]: (((U2 != A3) => (P @ U2))))))))).
% 2.95/1.53 thf(fact_239_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A3:A]: ((((P @ A3) => (((collect @ A) @ (^ [X7:A]: (((X7 = A3) & (P @ X7))))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A3))) => (((collect @ A) @ (^ [X7:A]: (((X7 = A3) & (P @ X7))))) = (bot_bot @ (set @ A)))))))).
% 2.95/1.53 thf(fact_240_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A3:A]: ((((P @ A3) => (((collect @ A) @ (^ [X7:A]: (((A3 = X7) & (P @ X7))))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A3))) => (((collect @ A) @ (^ [X7:A]: (((A3 = X7) & (P @ X7))))) = (bot_bot @ (set @ A)))))))).
% 2.95/1.53 thf(fact_241_Image__singleton, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A)),A3:B]: ((((((image @ B) @ A) @ R) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B)))) = ((collect @ A) @ (^ [B6:A]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A3) @ B6)) @ R)))))))).
% 2.95/1.53 thf(fact_242_edges__def, axiom, (! [V:$tType,L:$tType]: ((((labeled_edges @ L) @ V) = ((((labele1974067554_graph @ L) @ V) @ (set @ ((product_prod @ L) @ ((product_prod @ V) @ V)))) @ (^ [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X24:(set @ V)]: (X13))))))).
% 2.95/1.53 thf(fact_243_vertices__def, axiom, (! [V:$tType,L:$tType]: ((((labeled_vertices @ L) @ V) = ((((labele1974067554_graph @ L) @ V) @ (set @ V)) @ (^ [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X24:(set @ V)]: (X24))))))).
% 2.95/1.53 thf(fact_244_labeled__graph_Ocase__distrib, axiom, (! [A:$tType,B:$tType,V:$tType,L:$tType,H:(A > B),F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((H @ (((((labele1974067554_graph @ L) @ V) @ A) @ F) @ Labeled_graph)) = (((((labele1974067554_graph @ L) @ V) @ B) @ (^ [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X24:(set @ V)]: ((H @ ((F @ X13) @ X24))))) @ Labeled_graph))))).
% 2.95/1.53 thf(fact_245_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X7:A,Y7:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X7) @ Y7)) @ R2))) = (^ [X7:A,Y7:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X7) @ Y7)) @ S)))) = (R2 = S))))).
% 2.95/1.53 thf(fact_246_subset__CollectI, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((! [X6:A]: (((((member @ A) @ X6) @ B2) => ((Q @ X6) => (P @ X6))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X7:A]: (((((member @ A) @ X7) @ B2) & (Q @ X7)))))) @ ((collect @ A) @ (^ [X7:A]: (((((member @ A) @ X7) @ A2) & (P @ X7))))))))))).
% 2.95/1.53 thf(fact_247_subset__Collect__iff, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((ord_less_eq @ (set @ A)) @ B2) @ ((collect @ A) @ (^ [X7:A]: (((((member @ A) @ X7) @ A2) & (P @ X7)))))) = (! [X7:A]: (((((member @ A) @ X7) @ B2) => (P @ X7))))))))).
% 2.95/1.53 thf(fact_248_Collect__subset, axiom, (! [A:$tType,A2:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X7:A]: (((((member @ A) @ X7) @ A2) & (P @ X7)))))) @ A2)))).
% 2.95/1.53 thf(fact_249_congr_I1_J, axiom, (! [X2:b,Y:b,L2:(standard_Constant @ a)]: (((((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X2) @ Y)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ L2) @ g)) => ((((equiv_congruent @ b) @ (set @ b)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (^ [V3:b]: (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ L2) @ g)) @ (((insert @ b) @ V3) @ (bot_bot @ (set @ b))))))))))).
% 2.95/1.53 thf(fact_250_ident, axiom, (((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ (((((map_graph @ b) @ b) @ (standard_Constant @ a)) @ ((((bNF_Gr @ b) @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ f)) @ g)) = ((((image2 @ b) @ ((product_prod @ b) @ b)) @ (^ [X7:b]: (((((product_Pair @ b) @ b) @ X7) @ X7)))) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ (((((map_graph @ b) @ b) @ (standard_Constant @ a)) @ ((((bNF_Gr @ b) @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ f)) @ g))))).
% 2.95/1.53 thf(fact_251_image__empty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((image2 @ B) @ A) @ F) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.95/1.53 thf(fact_252_empty__is__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image2 @ B) @ A) @ F) @ A2)) = (A2 = (bot_bot @ (set @ B))))))).
% 2.95/1.53 thf(fact_253_image__is__empty, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B)]: (((((((image2 @ B) @ A) @ F) @ A2) = (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ B))))))).
% 2.95/1.53 thf(fact_254_image__insert, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:B,B2:(set @ B)]: ((((((image2 @ B) @ A) @ F) @ (((insert @ B) @ A3) @ B2)) = (((insert @ A) @ (F @ A3)) @ ((((image2 @ B) @ A) @ F) @ B2)))))).
% 2.95/1.53 thf(fact_255_insert__image, axiom, (! [B:$tType,A:$tType,X3:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X3) @ A2) => ((((insert @ B) @ (F @ X3)) @ ((((image2 @ A) @ B) @ F) @ A2)) = ((((image2 @ A) @ B) @ F) @ A2)))))).
% 2.95/1.53 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.95/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y3:A]: ((((((if @ A) @ $false) @ X3) @ Y3) = Y3)))).
% 2.95/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y3:A]: ((((((if @ A) @ $true) @ X3) @ Y3) = X3)))).
% 2.95/1.53 thf(conj_0, conjecture, (((member @ b) @ x) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g))).
% 2.95/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2DTF32619.p
% 2.95/1.53 ------------------------
% 2.95/1.59 ---- Cleaned THF ---
% 2.95/1.59 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 2.95/1.59 standard_Constant: $tType > $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 2.95/1.59 allegorical_term: $tType > $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 2.95/1.59 labeled_graph: $tType > $tType > $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_t_Product__Type_Oprod,type,
% 2.95/1.59 product_prod: $tType > $tType > $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_t_Set_Oset,type,
% 2.95/1.59 set: $tType > $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_t_Nat_Onat,type,
% 2.95/1.59 nat: $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_t_itself,type,
% 2.95/1.59 itself: $tType > $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_tf_b,type,
% 2.95/1.59 b: $tType ).
% 2.95/1.59
% 2.95/1.59 thf(ty_tf_a,type,
% 2.95/1.59 a: $tType ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_BNF__Def_OGr,type,
% 2.95/1.59 bNF_Gr:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 2.95/1.59 equiv_congruent:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Equiv__Relations_Oequiv,type,
% 2.95/1.59 equiv_equiv:
% 2.95/1.59 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Equiv__Relations_Oproj,type,
% 2.95/1.59 equiv_proj:
% 2.95/1.59 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Hilbert__Choice_OEps,type,
% 2.95/1.59 hilbert_Eps:
% 2.95/1.59 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_If,type,
% 2.95/1.59 if:
% 2.95/1.59 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt,type,
% 2.95/1.59 standard_S_Idt:
% 2.95/1.59 !>[V: $tType] : ( standard_Constant @ V ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 2.95/1.59 getRel:
% 2.95/1.59 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 2.95/1.59 semantics:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Oedge__preserving,type,
% 2.95/1.59 edge_preserving:
% 2.95/1.59 !>[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 ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 2.95/1.59 graph_homomorphism:
% 2.95/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 2.95/1.59 graph_union:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 2.95/1.59 labeled_LG:
% 2.95/1.59 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 2.95/1.59 labele1974067554_graph:
% 2.95/1.59 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 2.95/1.59 labeled_edges:
% 2.95/1.59 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 2.95/1.59 labeled_vertices:
% 2.95/1.59 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Omap__graph,type,
% 2.95/1.59 map_graph:
% 2.95/1.59 !>[C: $tType,B: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ B ) ) > ( labeled_graph @ A @ C ) > ( labeled_graph @ A @ B ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_LabeledGraphs_Orestrict,type,
% 2.95/1.59 restrict:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.95/1.59 bot_bot:
% 2.95/1.59 !>[A: $tType] : A ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.95/1.59 ord_less_eq:
% 2.95/1.59 !>[A: $tType] : ( A > A > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Product__Type_OPair,type,
% 2.95/1.59 product_Pair:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.95/1.59 product_rec_prod:
% 2.95/1.59 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Relation_OId__on,type,
% 2.95/1.59 id_on:
% 2.95/1.59 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Relation_OImage,type,
% 2.95/1.59 image:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Relation_Orefl__on,type,
% 2.95/1.59 refl_on:
% 2.95/1.59 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Relation_Orelcomp,type,
% 2.95/1.59 relcomp:
% 2.95/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_RulesAndChains_Oagree__on,type,
% 2.95/1.59 agree_on:
% 2.95/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 2.95/1.59 consequence_graph:
% 2.95/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_RulesAndChains_Oextensible,type,
% 2.95/1.59 extensible:
% 2.95/1.59 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ X @ V ) ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_RulesAndChains_Oleast,type,
% 2.95/1.59 least:
% 2.95/1.59 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 2.95/1.59 least_559130134_graph:
% 2.95/1.59 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_RulesAndChains_Omaintained,type,
% 2.95/1.59 maintained:
% 2.95/1.59 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Set_OCollect,type,
% 2.95/1.59 collect:
% 2.95/1.59 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Set_Oimage,type,
% 2.95/1.59 image2:
% 2.95/1.59 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Set_Oinsert,type,
% 2.95/1.59 insert:
% 2.95/1.59 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_Set_Othe__elem,type,
% 2.95/1.59 the_elem:
% 2.95/1.59 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oidentity__rules,type,
% 2.95/1.59 standa2002409347_rules:
% 2.95/1.59 !>[A: $tType] : ( ( set @ ( standard_Constant @ A ) ) > ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ A ) @ nat ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 2.95/1.59 standa244753842y_rule:
% 2.95/1.59 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule,type,
% 2.95/1.59 standa245363227y_rule:
% 2.95/1.59 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule,type,
% 2.95/1.59 standa1805702094y_rule:
% 2.95/1.59 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule,type,
% 2.95/1.59 standa2114124375e_rule:
% 2.95/1.59 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_c_member,type,
% 2.95/1.59 member:
% 2.95/1.59 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_G,type,
% 2.95/1.59 g: labeled_graph @ ( standard_Constant @ a ) @ b ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_L,type,
% 2.95/1.59 l: set @ ( standard_Constant @ a ) ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_P____,type,
% 2.95/1.59 p: b > b > $o ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_f____,type,
% 2.95/1.59 f: b > b ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_l____,type,
% 2.95/1.59 l2: standard_Constant @ a ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_x____,type,
% 2.95/1.59 x: b ).
% 2.95/1.59
% 2.95/1.59 thf(sy_v_y____,type,
% 2.95/1.59 y: b ).
% 2.95/1.59
% 2.95/1.59 thf(fact_0__092_060open_062_092_060And_062xa_O_A_092_060lbrakk_062f_Axa_A_092_060in_062_Avertices_AG_059_Axa_A_092_060notin_062_Avertices_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
% 2.95/1.59 ! [Xa: b] :
% 2.95/1.59 ( ( member @ b @ ( f @ Xa ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( member @ b @ Xa @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_1_fv_I2_J,axiom,
% 2.95/1.59 member @ b @ ( f @ y ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_2_fv_I1_J,axiom,
% 2.95/1.59 member @ b @ ( f @ x ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_3_gu2,axiom,
% 2.95/1.59 ! [X2: b] :
% 2.95/1.59 ( ( member @ b @ X2 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( member @ b @ ( f @ X2 ) @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_4_assms_I1_J,axiom,
% 2.95/1.59 ( g
% 2.95/1.59 = ( restrict @ ( standard_Constant @ a ) @ b @ g ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_5_r1,axiom,
% 2.95/1.59 ! [X3: b] :
% 2.95/1.59 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 = ( p @ X3 @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_6_assms_I2_J,axiom,
% 2.95/1.59 ! [X4: product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat )] :
% 2.95/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ X4 @ ( standa2002409347_rules @ a @ l ) )
% 2.95/1.59 => ( maintained @ ( standard_Constant @ a ) @ nat @ b @ X4 @ g ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_7_gr,axiom,
% 2.95/1.59 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ ( f @ x ) @ ( f @ y ) ) @ ( getRel @ ( standard_Constant @ a ) @ b @ l2 @ g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_8_a,axiom,
% 2.95/1.59 member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ l2 @ ( product_Pair @ b @ b @ ( f @ x ) @ ( f @ y ) ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_9_local_Orefl,axiom,
% 2.95/1.59 refl_on @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_10_equiv,axiom,
% 2.95/1.59 equiv_equiv @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_11__092_060open_062graph__union_A_Imap__graph__fn_AG_Af_J_AG_A_061_AG_092_060close_062,axiom,
% 2.95/1.59 ( ( graph_union @ ( standard_Constant @ a ) @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) @ g )
% 2.95/1.59 = g ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_12_gu1,axiom,
% 2.95/1.59 ! [L2: standard_Constant @ a,X2: b,Y: b] :
% 2.95/1.59 ( ( member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ L2 @ ( product_Pair @ b @ b @ X2 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( ( member @ b @ X2 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( ( member @ b @ Y @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ L2 @ ( product_Pair @ b @ b @ ( f @ X2 ) @ ( f @ Y ) ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_13__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 2.95/1.59 ! [X3: b] :
% 2.95/1.59 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ X3 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_14_labeled__graph_Osel_I2_J,axiom,
% 2.95/1.59 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.95/1.59 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.95/1.59 = X22 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_15_subg,axiom,
% 2.95/1.59 graph_homomorphism @ ( standard_Constant @ a ) @ b @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) @ g @ ( id_on @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_16_vertices__restrict,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,G: labeled_graph @ B @ A] :
% 2.95/1.59 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G ) )
% 2.95/1.59 = ( labeled_vertices @ B @ A @ G ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_17_restrict__idemp,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X3: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X3 ) )
% 2.95/1.59 = ( restrict @ A @ B @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_18_labeled__graph_Oinject,axiom,
% 2.95/1.59 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V,Y1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),Y2: set @ V] :
% 2.95/1.59 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 2.95/1.59 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 2.95/1.59 = ( ( X1 = Y1 )
% 2.95/1.59 & ( X22 = Y2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_19_graph__union__idemp_I3_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_union @ A @ B @ A2 @ ( graph_union @ A @ B @ B2 @ A2 ) )
% 2.95/1.59 = ( graph_union @ A @ B @ B2 @ A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_20_graph__union__idemp_I2_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_union @ A @ B @ A2 @ ( graph_union @ A @ B @ A2 @ B2 ) )
% 2.95/1.59 = ( graph_union @ A @ B @ A2 @ B2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_21_graph__union__idemp_I1_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_union @ A @ B @ A2 @ A2 )
% 2.95/1.59 = A2 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_22_map__graph__preserves__restricted,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType,G: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ( map_graph @ B @ C @ A @ F @ G )
% 2.95/1.59 = ( restrict @ A @ C @ ( map_graph @ B @ C @ A @ F @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_23_graph__union__preserves__restrict,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( G_1
% 2.95/1.59 = ( restrict @ A @ B @ G_1 ) )
% 2.95/1.59 => ( ( G_2
% 2.95/1.59 = ( restrict @ A @ B @ G_2 ) )
% 2.95/1.59 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.95/1.59 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_24_labeled__graph_Ocollapse,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.95/1.59 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 2.95/1.59 = Labeled_graph ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_25_subgraph__refl,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ G @ G @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 2.95/1.59 = ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_26_subgraph__restrict,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ G @ ( restrict @ A @ B @ G ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G ) ) )
% 2.95/1.59 = ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_27_graph__homomorphism__Id,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A3 ) @ ( restrict @ A @ B @ A3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A3 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_28_map__graph__preserves__subgraph,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ A2 @ B2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) )
% 2.95/1.59 => ( graph_homomorphism @ A @ C @ C @ ( map_graph @ B @ C @ A @ F @ A2 ) @ ( map_graph @ B @ C @ A @ F @ B2 ) @ ( id_on @ C @ ( labeled_vertices @ A @ C @ ( map_graph @ B @ C @ A @ F @ A2 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_29_map__graph__fn__id_I2_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X5: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( map_graph @ B @ B @ A @ ( id_on @ B @ ( labeled_vertices @ A @ B @ X5 ) ) @ X5 )
% 2.95/1.59 = ( restrict @ A @ B @ X5 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_30_map__graph__fn__graphI,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F: C > B] :
% 2.95/1.59 ( ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F ) @ G )
% 2.95/1.59 = ( restrict @ A @ B @ ( map_graph @ C @ B @ A @ ( bNF_Gr @ C @ B @ ( labeled_vertices @ A @ C @ G ) @ F ) @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_31_graph__homo,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,G: labeled_graph @ A @ B,F: B > C] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( graph_homomorphism @ A @ B @ C @ G @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F ) @ G ) @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_32_labeled__graph_Osel_I1_J,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.95/1.59 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.95/1.59 = X1 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_33_subgraph__def,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.95/1.59 = ( ( G_1
% 2.95/1.59 = ( restrict @ A @ B @ G_1 ) )
% 2.95/1.59 & ( G_2
% 2.95/1.59 = ( restrict @ A @ B @ G_2 ) )
% 2.95/1.59 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.95/1.59 = G_2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_34_subgraph__trans,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 2.95/1.59 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_35_map__graph__fn__eqI,axiom,
% 2.95/1.59 ! [C: $tType,A: $tType,B: $tType,G: labeled_graph @ B @ A,F: A > C,G2: A > C] :
% 2.95/1.59 ( ! [X6: A] :
% 2.95/1.59 ( ( member @ A @ X6 @ ( labeled_vertices @ B @ A @ G ) )
% 2.95/1.59 => ( ( F @ X6 )
% 2.95/1.59 = ( G2 @ X6 ) ) )
% 2.95/1.59 => ( ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F ) @ G )
% 2.95/1.59 = ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ G2 ) @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_36_labeled__graph_Oinduct,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType,P: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 2.95/1.59 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 2.95/1.59 => ( P @ Labeled_graph ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_37_labeled__graph_Oexhaust,axiom,
% 2.95/1.59 ! [L: $tType,V: $tType,Y3: labeled_graph @ L @ V] :
% 2.95/1.59 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_38_subgraph__preserves__hom,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,X5: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ A2 @ B2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ C @ B @ X5 @ A2 @ H )
% 2.95/1.59 => ( graph_homomorphism @ A @ C @ B @ X5 @ B2 @ H ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_39_labeled__graph_Oexhaust__sel,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.95/1.59 ( Labeled_graph
% 2.95/1.59 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_40_labeled__graph_Oexpand,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 2.95/1.59 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 2.95/1.59 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 2.95/1.59 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 2.95/1.59 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 2.95/1.59 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_41_getRel__hom,axiom,
% 2.95/1.59 ! [C: $tType,A: $tType,B: $tType,Y3: A,Z: A,L3: B,G: labeled_graph @ B @ A,F: A > C] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G ) )
% 2.95/1.59 => ( ( member @ A @ Y3 @ ( labeled_vertices @ B @ A @ G ) )
% 2.95/1.59 => ( ( member @ A @ Z @ ( labeled_vertices @ B @ A @ G ) )
% 2.95/1.59 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F @ Y3 ) @ ( F @ Z ) ) @ ( getRel @ B @ C @ L3 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F ) @ G ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_42_getRel__map__fn,axiom,
% 2.95/1.59 ! [C: $tType,A: $tType,B: $tType,A22: A,G: labeled_graph @ B @ A,B22: A,L3: B,F: A > C,A3: C,B3: C] :
% 2.95/1.59 ( ( member @ A @ A22 @ ( labeled_vertices @ B @ A @ G ) )
% 2.95/1.59 => ( ( member @ A @ B22 @ ( labeled_vertices @ B @ A @ G ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B22 ) @ ( getRel @ B @ A @ L3 @ G ) )
% 2.95/1.59 => ( ( ( F @ A22 )
% 2.95/1.59 = A3 )
% 2.95/1.59 => ( ( ( F @ B22 )
% 2.95/1.59 = B3 )
% 2.95/1.59 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( getRel @ B @ C @ L3 @ ( map_graph @ A @ C @ B @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ F ) @ G ) ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_43_idt__eq,axiom,
% 2.95/1.59 ! [X2: b,Y: b] :
% 2.95/1.59 ( ( member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ ( standard_S_Idt @ a ) @ ( product_Pair @ b @ b @ X2 @ Y ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( ( hilbert_Eps @ b @ ( p @ X2 ) )
% 2.95/1.59 = ( hilbert_Eps @ b @ ( p @ Y ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_44_vert__P,axiom,
% 2.95/1.59 ! [X3: b] :
% 2.95/1.59 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.95/1.59 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ ( hilbert_Eps @ b @ ( p @ X3 ) ) ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,P: A > $o] :
% 2.95/1.59 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.95/1.59 = ( P @ A3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] :
% 2.95/1.59 ( ( collect @ A
% 2.95/1.59 @ ^ [X7: A] : ( member @ A @ X7 @ A2 ) )
% 2.95/1.59 = A2 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_47_Collect__cong,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.95/1.59 ( ! [X6: A] :
% 2.95/1.59 ( ( P @ X6 )
% 2.95/1.59 = ( Q @ X6 ) )
% 2.95/1.59 => ( ( collect @ A @ P )
% 2.95/1.59 = ( collect @ A @ Q ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_48_getRel__subgraph,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,Y3: A,Z: A,L3: B,G: labeled_graph @ B @ A,G3: labeled_graph @ B @ A] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G ) )
% 2.95/1.59 => ( ( graph_homomorphism @ B @ A @ A @ G @ G3 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G ) ) )
% 2.95/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_49_in__Gr,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,X3: A,Y3: B,A2: set @ A,F: A > B] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bNF_Gr @ A @ B @ A2 @ F ) )
% 2.95/1.59 = ( ( member @ A @ X3 @ A2 )
% 2.95/1.59 & ( ( F @ X3 )
% 2.95/1.59 = Y3 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_50_Gr__not__in,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X3: A,F2: set @ A,F: A > B,Y3: B] :
% 2.95/1.59 ( ( ~ ( member @ A @ X3 @ F2 )
% 2.95/1.59 | ( ( F @ X3 )
% 2.95/1.59 != Y3 ) )
% 2.95/1.59 => ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bNF_Gr @ A @ B @ F2 @ F ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_51_Id__onI,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_52_getRel__dom_I2_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A3: B,B3: B,L3: A] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( getRel @ A @ B @ L3 @ G ) )
% 2.95/1.59 => ( member @ B @ B3 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_53_getRel__dom_I1_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,A3: B,B3: B,L3: A] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( getRel @ A @ B @ L3 @ G ) )
% 2.95/1.59 => ( member @ B @ A3 @ ( labeled_vertices @ A @ B @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_54_Id__on__iff,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,Y3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( id_on @ A @ A2 ) )
% 2.95/1.59 = ( ( X3 = Y3 )
% 2.95/1.59 & ( member @ A @ X3 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_55_Id__on__eqI,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.95/1.59 ( ( A3 = B3 )
% 2.95/1.59 => ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id_on @ A @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_56_Id__onE,axiom,
% 2.95/1.59 ! [A: $tType,C2: product_prod @ A @ A,A2: set @ A] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A2 ) )
% 2.95/1.59 => ~ ! [X6: A] :
% 2.95/1.59 ( ( member @ A @ X6 @ A2 )
% 2.95/1.59 => ( C2
% 2.95/1.59 != ( product_Pair @ A @ A @ X6 @ X6 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_57_refl__onD2,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 2.95/1.59 ( ( refl_on @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 2.95/1.59 => ( member @ A @ Y3 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_58_refl__onD1,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 2.95/1.59 ( ( refl_on @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 2.95/1.59 => ( member @ A @ X3 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_59_refl__onD,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 2.95/1.59 ( ( refl_on @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_60_refl__on__Id__on,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] : ( refl_on @ A @ A2 @ ( id_on @ A @ A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_61_getRel__homR,axiom,
% 2.95/1.59 ! [C: $tType,A: $tType,B: $tType,Y3: A,Z: A,L3: B,G: labeled_graph @ B @ A,U: C,F: set @ ( product_prod @ A @ C ),V2: C] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( getRel @ B @ A @ L3 @ G ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Y3 @ U ) @ F )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ Z @ V2 ) @ F )
% 2.95/1.59 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ U @ V2 ) @ ( getRel @ B @ C @ L3 @ ( map_graph @ A @ C @ B @ F @ G ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_62_maintained__refl,axiom,
% 2.95/1.59 ! [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 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_63_edge__preserving__on__graphI,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,X5: labeled_graph @ A @ B,F: B > C,Y4: set @ ( product_prod @ A @ ( product_prod @ C @ C ) )] :
% 2.95/1.59 ( ! [L4: A,X6: B,Y5: B] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L4 @ ( product_Pair @ B @ B @ X6 @ Y5 ) ) @ ( labeled_edges @ A @ B @ X5 ) )
% 2.95/1.59 => ( ( member @ B @ X6 @ ( labeled_vertices @ A @ B @ X5 ) )
% 2.95/1.59 => ( ( member @ B @ Y5 @ ( labeled_vertices @ A @ B @ X5 ) )
% 2.95/1.59 => ( member @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L4 @ ( product_Pair @ C @ C @ ( F @ X6 ) @ ( F @ Y5 ) ) ) @ Y4 ) ) ) )
% 2.95/1.59 => ( edge_preserving @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ X5 ) @ F ) @ ( labeled_edges @ A @ B @ X5 ) @ Y4 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_64_reflexivity__rule,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,L3: A] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ( maintained @ A @ nat @ B @ ( standa245363227y_rule @ A @ L3 ) @ G )
% 2.95/1.59 => ( refl_on @ B @ ( labeled_vertices @ A @ B @ G ) @ ( getRel @ A @ B @ L3 @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_65_old_Oprod_Oinject,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 2.95/1.59 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.95/1.59 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.95/1.59 = ( ( A3 = A4 )
% 2.95/1.59 & ( B3 = B4 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_66_prod_Oinject,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.95/1.59 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.95/1.59 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.95/1.59 = ( ( X1 = Y1 )
% 2.95/1.59 & ( X22 = Y2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_67_refl__on__domain,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 2.95/1.59 ( ( refl_on @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
% 2.95/1.59 => ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 & ( member @ A @ B3 @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_68_GrD2,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,X3: A,Fx: B,A2: set @ A,F: A > B] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A2 @ F ) )
% 2.95/1.59 => ( ( F @ X3 )
% 2.95/1.59 = Fx ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_69_GrD1,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X3: A,Fx: B,A2: set @ A,F: A > B] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A2 @ F ) )
% 2.95/1.59 => ( member @ A @ X3 @ A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_70_edge__preserving__Id,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,Y3: set @ A,X3: set @ ( product_prod @ B @ ( product_prod @ A @ A ) )] : ( edge_preserving @ A @ A @ B @ ( id_on @ A @ Y3 ) @ X3 @ X3 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_71_map__graph__edge__preserving,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,C: $tType,F: set @ ( product_prod @ A @ B ),G: labeled_graph @ C @ A] : ( edge_preserving @ A @ B @ C @ F @ ( labeled_edges @ C @ A @ G ) @ ( labeled_edges @ C @ B @ ( map_graph @ A @ B @ C @ F @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_72_edge__preserving__atomic,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E2: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),V1: A,V12: B,V22: A,V23: B,K: C] :
% 2.95/1.59 ( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E2 )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V1 @ V12 ) @ H1 )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ V22 @ V23 ) @ H1 )
% 2.95/1.59 => ( ( member @ ( product_prod @ C @ ( product_prod @ A @ A ) ) @ ( product_Pair @ C @ ( product_prod @ A @ A ) @ K @ ( product_Pair @ A @ A @ V1 @ V22 ) ) @ E1 )
% 2.95/1.59 => ( member @ ( product_prod @ C @ ( product_prod @ B @ B ) ) @ ( product_Pair @ C @ ( product_prod @ B @ B ) @ K @ ( product_Pair @ B @ B @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_73_surj__pair,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.95/1.59 ? [X6: A,Y5: B] :
% 2.95/1.59 ( P2
% 2.95/1.59 = ( product_Pair @ A @ B @ X6 @ Y5 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_74_prod__cases,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.95/1.59 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.95/1.59 => ( P @ P2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_75_Pair__inject,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 2.95/1.59 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.95/1.59 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.95/1.59 => ~ ( ( A3 = A4 )
% 2.95/1.59 => ( B3 != B4 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_76_old_Oprod_Oexhaust,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.95/1.59 ~ ! [A5: A,B5: B] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_77_old_Oprod_Oinducts,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.95/1.59 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.95/1.59 => ( P @ Prod ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_78_prod__cases3,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.95/1.59 ~ ! [A5: A,B5: B,C3: C] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_79_prod__cases4,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.95/1.59 ~ ! [A5: A,B5: B,C3: C,D2: D] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_80_prod__cases5,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.95/1.59 ~ ! [A5: A,B5: B,C3: C,D2: D,E3: E] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_81_prod__cases6,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 2.95/1.59 ~ ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_82_prod__cases7,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G4: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) )] :
% 2.95/1.59 ~ ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3,G5: G4] :
% 2.95/1.59 ( Y3
% 2.95/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G4 ) @ E3 @ ( product_Pair @ F3 @ G4 @ F4 @ G5 ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_83_prod__induct3,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 2.95/1.59 => ( P @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_84_prod__induct4,axiom,
% 2.95/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.95/1.59 ( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 2.95/1.59 => ( P @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_85_prod__induct5,axiom,
% 2.95/1.59 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.95/1.59 ( ! [A5: A,B5: B,C3: C,D2: D,E3: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) )
% 2.95/1.59 => ( P @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_86_prod__induct6,axiom,
% 2.95/1.59 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 2.95/1.59 ( ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E3 @ F4 ) ) ) ) ) )
% 2.95/1.59 => ( P @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_87_prod__induct7,axiom,
% 2.95/1.59 ! [G4: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) )] :
% 2.95/1.59 ( ! [A5: A,B5: B,C3: C,D2: D,E3: E,F4: F3,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G4 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G4 ) @ E3 @ ( product_Pair @ F3 @ G4 @ F4 @ G5 ) ) ) ) ) ) )
% 2.95/1.59 => ( P @ X3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_88_old_Oprod_Orec,axiom,
% 2.95/1.59 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 2.95/1.59 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.95/1.59 = ( F1 @ A3 @ B3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_89_least__consequence__graphI,axiom,
% 2.95/1.59 ! [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] :
% 2.95/1.59 ( ( consequence_graph @ L @ A @ C @ Rs @ G )
% 2.95/1.59 => ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 2.95/1.59 => ( ! [C4: labeled_graph @ L @ X] :
% 2.95/1.59 ( ( consequence_graph @ L @ A @ X @ Rs @ C4 )
% 2.95/1.59 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C4 ) )
% 2.95/1.59 => ( least_559130134_graph @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_90_leastI,axiom,
% 2.95/1.59 ! [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] :
% 2.95/1.59 ( ( graph_homomorphism @ L @ C @ C @ S @ G @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S ) ) )
% 2.95/1.59 => ( ! [C4: labeled_graph @ L @ X] :
% 2.95/1.59 ( ( consequence_graph @ L @ A @ X @ Rs @ C4 )
% 2.95/1.59 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S @ G ) @ C4 ) )
% 2.95/1.59 => ( least @ X @ L @ A @ C @ T2 @ Rs @ S @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_91_map__graph__in,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A,F: B > C] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ G @ E4 ) )
% 2.95/1.59 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ ( F @ A3 ) @ ( F @ B3 ) ) @ ( semantics @ A @ C @ ( map_graph @ B @ C @ A @ ( bNF_Gr @ B @ C @ ( labeled_vertices @ A @ B @ G ) @ F ) @ G ) @ E4 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_92_maintainedI,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,A2: labeled_graph @ C @ A,G: labeled_graph @ C @ B,B2: labeled_graph @ C @ A] :
% 2.95/1.59 ( ! [F4: set @ ( product_prod @ A @ B )] :
% 2.95/1.59 ( ( graph_homomorphism @ C @ A @ B @ A2 @ G @ F4 )
% 2.95/1.59 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A2 @ B2 ) @ G @ F4 ) )
% 2.95/1.59 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A2 @ B2 ) @ G ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_93_extensible__refl__concr,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V2: set @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V2 ) @ G @ F )
% 2.95/1.59 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V2 ) @ ( labeled_LG @ A @ B @ E_2 @ V2 ) ) @ G @ F )
% 2.95/1.59 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V2 ) @ G @ F ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_94_labeled__graph_Osplit__sel,axiom,
% 2.95/1.59 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.95/1.59 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 2.95/1.59 = ( ( Labeled_graph
% 2.95/1.59 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.95/1.59 => ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_95_extensible__refl,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ R2 @ G @ F )
% 2.95/1.59 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R2 @ R2 ) @ G @ F ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_96_least__consequence__graph__def,axiom,
% 2.95/1.59 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 2.95/1.59 ( ( least_559130134_graph @ X @ L @ V @ C )
% 2.95/1.59 = ( ^ [T3: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G6: labeled_graph @ L @ C] :
% 2.95/1.59 ( ( consequence_graph @ L @ V @ C @ Rs2 @ G6 )
% 2.95/1.59 & ( least @ X @ L @ V @ C @ T3 @ Rs2 @ S2 @ G6 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_97_consequence__graphD_I3_J,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G: labeled_graph @ A @ C] :
% 2.95/1.59 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 2.95/1.59 => ( G
% 2.95/1.59 = ( restrict @ A @ C @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_98_graph__homomorphism__semantics,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),A3: B,B3: B,E4: allegorical_term @ A,A4: C,B4: C] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A3 @ A4 ) @ F )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ B4 ) @ F )
% 2.95/1.59 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A4 @ B4 ) @ ( semantics @ A @ C @ B2 @ E4 ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_99_consequence__graphD_I1_J,axiom,
% 2.95/1.59 ! [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 )] :
% 2.95/1.59 ( ( consequence_graph @ A @ B @ C @ Rs @ G )
% 2.95/1.59 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R2 @ Rs )
% 2.95/1.59 => ( maintained @ A @ B @ C @ R2 @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_100_labeled__graph_Ocase,axiom,
% 2.95/1.59 ! [L: $tType,A: $tType,V: $tType,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.95/1.59 ( ( labele1974067554_graph @ L @ V @ A @ F @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.95/1.59 = ( F @ X1 @ X22 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_101_semantics__in__vertices_I1_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A] :
% 2.95/1.59 ( ( A2
% 2.95/1.59 = ( restrict @ A @ B @ A2 ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 2.95/1.59 => ( member @ B @ A3 @ ( labeled_vertices @ A @ B @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_102_semantics__in__vertices_I2_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A] :
% 2.95/1.59 ( ( A2
% 2.95/1.59 = ( restrict @ A @ B @ A2 ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 2.95/1.59 => ( member @ B @ B3 @ ( labeled_vertices @ A @ B @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_103_labeled__graph_Ocase__eq__if,axiom,
% 2.95/1.59 ! [A: $tType,V: $tType,L: $tType] :
% 2.95/1.59 ( ( labele1974067554_graph @ L @ V @ A )
% 2.95/1.59 = ( ^ [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 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_104_subgraph__semantics,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,A3: B,B3: B,E4: allegorical_term @ A] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ A2 @ B2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ A2 @ E4 ) )
% 2.95/1.59 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A3 @ B3 ) @ ( semantics @ A @ B @ B2 @ E4 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_105_maintainedD,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A2 @ B2 ) @ G )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ B @ C @ A2 @ G @ F )
% 2.95/1.59 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A2 @ B2 ) @ G @ F ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_106_least__def,axiom,
% 2.95/1.59 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 2.95/1.59 ( ( least @ X @ L @ V @ C )
% 2.95/1.59 = ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S2: labeled_graph @ L @ C,G6: labeled_graph @ L @ C] :
% 2.95/1.59 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G6 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 2.95/1.59 & ! [C5: labeled_graph @ L @ X] :
% 2.95/1.59 ( ( consequence_graph @ L @ V @ X @ Rs2 @ C5 )
% 2.95/1.59 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G6 ) @ C5 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_107_labeled__graph_Osplit__sel__asm,axiom,
% 2.95/1.59 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.95/1.59 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 2.95/1.59 = ( ~ ( ( Labeled_graph
% 2.95/1.59 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.95/1.59 & ~ ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_108_extensibleI,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G: labeled_graph @ A @ C,G2: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G @ G2 )
% 2.95/1.59 => ( ( agree_on @ A @ B @ C @ R1 @ F @ G2 )
% 2.95/1.59 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G @ F ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_109_subgraph__subset_I2_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.95/1.59 => ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G_1 ) ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_110_graph__homo__union__id_I1_J,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A2 @ B2 ) @ G @ F )
% 2.95/1.59 => ( ( A2
% 2.95/1.59 = ( restrict @ A @ B @ A2 ) )
% 2.95/1.59 => ( graph_homomorphism @ A @ B @ C @ A2 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) @ F ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_111_graph__homo__union__id_I2_J,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ ( graph_union @ A @ B @ A2 @ B2 ) @ G @ F )
% 2.95/1.59 => ( ( B2
% 2.95/1.59 = ( restrict @ A @ B @ B2 ) )
% 2.95/1.59 => ( graph_homomorphism @ A @ B @ C @ B2 @ G @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ B2 ) ) @ F ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_112__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
% 2.95/1.59 ( f
% 2.95/1.59 = ( ^ [X7: b] :
% 2.95/1.59 ( if @ b
% 2.95/1.59 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X7 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ b ) ) )
% 2.95/1.59 @ X7
% 2.95/1.59 @ ( hilbert_Eps @ b @ ( p @ X7 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_113_agree__on__refl,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R2 @ F @ F ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_114_agree__on__comm,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.95/1.59 ( ( agree_on @ A @ B @ C )
% 2.95/1.59 = ( ^ [X8: labeled_graph @ A @ B,F5: set @ ( product_prod @ B @ C ),G7: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X8 @ G7 @ F5 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_115_ImageI,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),A2: set @ A] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 2.95/1.59 => ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( member @ B @ B3 @ ( image @ A @ B @ R @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_116_Image__empty2,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
% 2.95/1.59 ( ( image @ B @ A @ R2 @ ( bot_bot @ ( set @ B ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_117_Image__empty1,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X5: set @ B] :
% 2.95/1.59 ( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X5 )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_118_Id__on__empty,axiom,
% 2.95/1.59 ! [A: $tType] :
% 2.95/1.59 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_119_Gr__empty,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,F: A > B] :
% 2.95/1.59 ( ( bNF_Gr @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F )
% 2.95/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_120_graph__homomorphism__composes,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,D: $tType,C: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ C,X3: set @ ( product_prod @ B @ C ),C2: labeled_graph @ A @ D,Y3: set @ ( product_prod @ C @ D )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ A3 @ B3 @ X3 )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ C @ D @ B3 @ C2 @ Y3 )
% 2.95/1.59 => ( graph_homomorphism @ A @ B @ D @ A3 @ C2 @ ( relcomp @ B @ C @ D @ X3 @ Y3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_121_P__eq,axiom,
% 2.95/1.59 ! [X2: b] :
% 2.95/1.59 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.95/1.59 != ( bot_bot @ ( set @ b ) ) )
% 2.95/1.59 => ( ( hilbert_Eps @ b @ ( p @ ( hilbert_Eps @ b @ ( p @ X2 ) ) ) )
% 2.95/1.59 = ( hilbert_Eps @ b @ ( p @ X2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_122_Gr__insert,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X3: A,F2: set @ A,F: A > B] :
% 2.95/1.59 ( ( bNF_Gr @ A @ B @ ( insert @ A @ X3 @ F2 ) @ F )
% 2.95/1.59 = ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ ( F @ X3 ) ) @ ( bNF_Gr @ A @ B @ F2 @ F ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_123_map__graph__selectors_I1_J,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,F: set @ ( product_prod @ C @ A ),G: labeled_graph @ B @ C] :
% 2.95/1.59 ( ( labeled_vertices @ B @ A @ ( map_graph @ C @ A @ B @ F @ G ) )
% 2.95/1.59 = ( image @ C @ A @ F @ ( labeled_vertices @ B @ C @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_124_agree__on__empty,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,F: set @ ( product_prod @ B @ C ),G2: 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 ) ) ) @ F @ G2 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_125_Image__singleton__iff,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A ),A3: B] :
% 2.95/1.59 ( ( member @ A @ B3 @ ( image @ B @ A @ R @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.95/1.59 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B3 ) @ R ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_126_in__on__graph,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,D: $tType,C: $tType,X3: A,G: labeled_graph @ B @ A,A3: A > C,Y3: D,B3: set @ ( product_prod @ C @ D )] :
% 2.95/1.59 ( ( member @ A @ X3 @ ( labeled_vertices @ B @ A @ G ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( A3 @ X3 ) @ Y3 ) @ B3 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ D ) @ ( product_Pair @ A @ D @ X3 @ Y3 ) @ ( relcomp @ A @ C @ D @ ( bNF_Gr @ A @ C @ ( labeled_vertices @ B @ A @ G ) @ A3 ) @ B3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_127_graph__homomorphism__empty,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G @ F )
% 2.95/1.59 = ( ( F
% 2.95/1.59 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 2.95/1.59 & ( G
% 2.95/1.59 = ( restrict @ A @ C @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_128_graph__unionI,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 2.95/1.59 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.95/1.59 = G_2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_129_graph__single,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,A3: A,B3: B,C2: B] :
% 2.95/1.59 ( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A3 @ ( product_Pair @ B @ B @ B3 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B3 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.95/1.59 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A3 @ ( product_Pair @ B @ B @ B3 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B3 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_130_r2,axiom,
% 2.95/1.59 ! [X3: b] :
% 2.95/1.59 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X3 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ b ) ) )
% 2.95/1.59 = ( ~ ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_131_refl__on__singleton,axiom,
% 2.95/1.59 ! [A: $tType,X3: A] : ( refl_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_132_rev__ImageI,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 2.95/1.59 ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 2.95/1.59 => ( member @ B @ B3 @ ( image @ A @ B @ R @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_133_Image__iff,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A ),A2: set @ B] :
% 2.95/1.59 ( ( member @ A @ B3 @ ( image @ B @ A @ R @ A2 ) )
% 2.95/1.59 = ( ? [X7: B] :
% 2.95/1.59 ( ( member @ B @ X7 @ A2 )
% 2.95/1.59 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X7 @ B3 ) @ R ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_134_ImageE,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A ),A2: set @ B] :
% 2.95/1.59 ( ( member @ A @ B3 @ ( image @ B @ A @ R @ A2 ) )
% 2.95/1.59 => ~ ! [X6: B] :
% 2.95/1.59 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X6 @ B3 ) @ R )
% 2.95/1.59 => ~ ( member @ B @ X6 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_135_relcomp_OrelcompI,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),C2: C,S3: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C2 ) @ S3 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_136_relcomp_Oinducts,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,C: $tType,X1: A,X22: C,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C ),P: A > C > $o] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X1 @ X22 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) )
% 2.95/1.59 => ( ! [A5: A,B5: B,C3: C] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C3 ) @ S3 )
% 2.95/1.59 => ( P @ A5 @ C3 ) ) )
% 2.95/1.59 => ( P @ X1 @ X22 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_137_relcomp_Osimps,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) )
% 2.95/1.59 = ( ? [A6: A,B6: B,C6: C] :
% 2.95/1.59 ( ( A1 = A6 )
% 2.95/1.59 & ( A22 = C6 )
% 2.95/1.59 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B6 ) @ R )
% 2.95/1.59 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B6 @ C6 ) @ S3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_138_relcomp_Ocases,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) )
% 2.95/1.59 => ~ ! [B5: B] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B5 ) @ R )
% 2.95/1.59 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A22 ) @ S3 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_139_relcompEpair,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R @ S3 ) )
% 2.95/1.59 => ~ ! [B5: C] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B5 ) @ R )
% 2.95/1.59 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S3 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_140_relcompE,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S3 ) )
% 2.95/1.59 => ~ ! [X6: A,Y5: C,Z2: B] :
% 2.95/1.59 ( ( Xz
% 2.95/1.59 = ( product_Pair @ A @ B @ X6 @ Z2 ) )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X6 @ Y5 ) @ R )
% 2.95/1.59 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y5 @ Z2 ) @ S3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_141_relcomp__Image,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,X5: set @ ( product_prod @ B @ C ),Y4: set @ ( product_prod @ C @ A ),Z3: set @ B] :
% 2.95/1.59 ( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X5 @ Y4 ) @ Z3 )
% 2.95/1.59 = ( image @ C @ A @ Y4 @ ( image @ B @ C @ X5 @ Z3 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_142_relcomp__mono,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C ),S3: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ R )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S4 @ S3 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R3 @ S4 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_143_Image__mono,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),A7: set @ A,A2: set @ A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ R )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R3 @ A7 ) @ ( image @ A @ B @ R @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_144_O__assoc,axiom,
% 2.95/1.59 ! [A: $tType,D: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ A @ D ),S: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
% 2.95/1.59 ( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R2 @ S ) @ T4 )
% 2.95/1.59 = ( relcomp @ A @ D @ B @ R2 @ ( relcomp @ D @ C @ B @ S @ T4 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_145_agree__on__def,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.95/1.59 ( ( agree_on @ A @ B @ C )
% 2.95/1.59 = ( ^ [G6: labeled_graph @ A @ B,F_1: set @ ( product_prod @ B @ C ),F_2: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ! [X7: B] :
% 2.95/1.59 ( ( member @ B @ X7 @ ( labeled_vertices @ A @ B @ G6 ) )
% 2.95/1.59 => ( ( image @ B @ C @ F_1 @ ( insert @ B @ X7 @ ( bot_bot @ ( set @ B ) ) ) )
% 2.95/1.59 = ( image @ B @ C @ F_2 @ ( insert @ B @ X7 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_146_edge__preserving__subset,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,C: $tType,R_1: set @ ( product_prod @ A @ B ),R_2: set @ ( product_prod @ A @ B ),E_12: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E_22: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),G: set @ ( product_prod @ C @ ( product_prod @ B @ B ) )] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R_1 @ R_2 )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ ( product_prod @ A @ A ) ) ) @ E_12 @ E_22 )
% 2.95/1.59 => ( ( edge_preserving @ A @ B @ C @ R_2 @ E_22 @ G )
% 2.95/1.59 => ( edge_preserving @ A @ B @ C @ R_1 @ E_12 @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_147_agree__on__trans,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,X5: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C ),G2: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( agree_on @ A @ B @ C @ X5 @ F @ G2 )
% 2.95/1.59 => ( ( agree_on @ A @ B @ C @ X5 @ G2 @ H )
% 2.95/1.59 => ( agree_on @ A @ B @ C @ X5 @ F @ H ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_148_agree__on__ext,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,D: $tType,C: $tType,G: labeled_graph @ A @ B,F_12: set @ ( product_prod @ B @ C ),F_22: set @ ( product_prod @ B @ C ),G2: set @ ( product_prod @ C @ D )] :
% 2.95/1.59 ( ( agree_on @ A @ B @ C @ G @ F_12 @ F_22 )
% 2.95/1.59 => ( agree_on @ A @ B @ D @ G @ ( relcomp @ B @ C @ D @ F_12 @ G2 ) @ ( relcomp @ B @ C @ D @ F_22 @ G2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_149_refl__on__empty,axiom,
% 2.95/1.59 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_150_subrelI,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.95/1.59 ( ! [X6: A,Y5: B] :
% 2.95/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y5 ) @ R )
% 2.95/1.59 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X6 @ Y5 ) @ S3 ) )
% 2.95/1.59 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_151_compose__preserves__edge__preserving,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,D: $tType,C: $tType,H1: set @ ( product_prod @ A @ B ),E1: set @ ( product_prod @ C @ ( product_prod @ A @ A ) ),E2: set @ ( product_prod @ C @ ( product_prod @ B @ B ) ),H2: set @ ( product_prod @ B @ D ),E32: set @ ( product_prod @ C @ ( product_prod @ D @ D ) )] :
% 2.95/1.59 ( ( edge_preserving @ A @ B @ C @ H1 @ E1 @ E2 )
% 2.95/1.59 => ( ( edge_preserving @ B @ D @ C @ H2 @ E2 @ E32 )
% 2.95/1.59 => ( edge_preserving @ A @ D @ C @ ( relcomp @ A @ B @ D @ H1 @ H2 ) @ E1 @ E32 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_152_graph__union__iff,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.95/1.59 = G_2 )
% 2.95/1.59 = ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 2.95/1.59 & ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_153_restrict__subsD,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G ) ) )
% 2.95/1.59 => ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_154_subgraph__def2,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( G_1
% 2.95/1.59 = ( restrict @ A @ B @ G_1 ) )
% 2.95/1.59 => ( ( G_2
% 2.95/1.59 = ( restrict @ A @ B @ G_2 ) )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.95/1.59 = ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 2.95/1.59 & ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_155_subgraph__subset_I1_J,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 2.95/1.59 => ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_156_maintainedD2,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ B,G: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A2 @ B2 ) @ G )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ B @ C @ A2 @ G @ F )
% 2.95/1.59 => ~ ! [G5: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ B2 @ G @ G5 )
% 2.95/1.59 => ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F @ G5 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_157_Id__on__vertices__identity_I2_J,axiom,
% 2.95/1.59 ! [B: $tType,C: $tType,A: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),Aa: B,Ba: C] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ A3 @ B3 @ F )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ F )
% 2.95/1.59 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ ( relcomp @ B @ C @ C @ F @ ( id_on @ C @ ( labeled_vertices @ A @ C @ B3 ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_158_Id__on__vertices__identity_I1_J,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),Aa: B,Ba: C] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ A3 @ B3 @ F )
% 2.95/1.59 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ F )
% 2.95/1.59 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ Aa @ Ba ) @ ( relcomp @ B @ B @ C @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A3 ) ) @ F ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_159_graph__homomorphism__on__graph,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,R2: set @ ( product_prod @ B @ C ),F: C > D] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ R2 )
% 2.95/1.59 => ( graph_homomorphism @ A @ B @ D @ A2 @ ( map_graph @ C @ D @ A @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ B2 ) @ F ) @ B2 ) @ ( relcomp @ B @ C @ D @ R2 @ ( bNF_Gr @ C @ D @ ( labeled_vertices @ A @ C @ B2 ) @ F ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_160_agree__on__subg__compose,axiom,
% 2.95/1.59 ! [A: $tType,C: $tType,B: $tType,R2: labeled_graph @ A @ B,G2: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F2: labeled_graph @ A @ B,F: set @ ( product_prod @ B @ C )] :
% 2.95/1.59 ( ( agree_on @ A @ B @ C @ R2 @ G2 @ H )
% 2.95/1.59 => ( ( agree_on @ A @ B @ C @ F2 @ F @ G2 )
% 2.95/1.59 => ( ( graph_homomorphism @ A @ B @ B @ F2 @ R2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F2 ) ) )
% 2.95/1.59 => ( agree_on @ A @ B @ C @ F2 @ F @ H ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_161_singleton__insert__inj__eq,axiom,
% 2.95/1.59 ! [A: $tType,B3: A,A3: A,A2: set @ A] :
% 2.95/1.59 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 = ( insert @ A @ A3 @ A2 ) )
% 2.95/1.59 = ( ( A3 = B3 )
% 2.95/1.59 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_162_singleton__insert__inj__eq_H,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 2.95/1.59 ( ( ( insert @ A @ A3 @ A2 )
% 2.95/1.59 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( ( A3 = B3 )
% 2.95/1.59 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_163_equiv__class__subset,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_164_subset__equiv__class,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),B3: A,A3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 => ( ( member @ A @ B3 @ A2 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_165_refines__equiv__class__eq2,axiom,
% 2.95/1.59 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A2: set @ A,A3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
% 2.95/1.59 => ( ( equiv_equiv @ A @ A2 @ R2 )
% 2.95/1.59 => ( ( equiv_equiv @ A @ A2 @ S )
% 2.95/1.59 => ( ( image @ A @ A @ S @ ( image @ A @ A @ R2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 = ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_166_empty__Collect__eq,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o] :
% 2.95/1.59 ( ( ( bot_bot @ ( set @ A ) )
% 2.95/1.59 = ( collect @ A @ P ) )
% 2.95/1.59 = ( ! [X7: A] :
% 2.95/1.59 ~ ( P @ X7 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_167_Collect__empty__eq,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o] :
% 2.95/1.59 ( ( ( collect @ A @ P )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 = ( ! [X7: A] :
% 2.95/1.59 ~ ( P @ X7 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_168_all__not__in__conv,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] :
% 2.95/1.59 ( ( ! [X7: A] :
% 2.95/1.59 ~ ( member @ A @ X7 @ A2 ) )
% 2.95/1.59 = ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_169_empty__iff,axiom,
% 2.95/1.59 ! [A: $tType,C2: A] :
% 2.95/1.59 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_170_subset__antisym,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.95/1.59 => ( A2 = B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_171_subsetI,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ! [X6: A] :
% 2.95/1.59 ( ( member @ A @ X6 @ A2 )
% 2.95/1.59 => ( member @ A @ X6 @ B2 ) )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_172_insert__absorb2,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,A2: set @ A] :
% 2.95/1.59 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A2 ) )
% 2.95/1.59 = ( insert @ A @ X3 @ A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_173_insert__iff,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.95/1.59 = ( ( A3 = B3 )
% 2.95/1.59 | ( member @ A @ A3 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_174_insertCI,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.95/1.59 ( ( ~ ( member @ A @ A3 @ B2 )
% 2.95/1.59 => ( A3 = B3 ) )
% 2.95/1.59 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_175_empty__subsetI,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_176_subset__empty,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 = ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_177_singletonI,axiom,
% 2.95/1.59 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_178_insert__subset,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A2 ) @ B2 )
% 2.95/1.59 = ( ( member @ A @ X3 @ B2 )
% 2.95/1.59 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_179_relcomp__empty1,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ C @ B )] :
% 2.95/1.59 ( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R2 )
% 2.95/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_180_relcomp__empty2,axiom,
% 2.95/1.59 ! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ C )] :
% 2.95/1.59 ( ( relcomp @ A @ C @ B @ R2 @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_181_graph__empty__e,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,V2: set @ B] :
% 2.95/1.59 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V2 )
% 2.95/1.59 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_182_graph__homomorphism__nonempty,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,C: $tType,A2: labeled_graph @ A @ B,B2: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C ),E4: allegorical_term @ A] :
% 2.95/1.59 ( ( graph_homomorphism @ A @ B @ C @ A2 @ B2 @ F )
% 2.95/1.59 => ( ( ( semantics @ A @ B @ A2 @ E4 )
% 2.95/1.59 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 2.95/1.59 => ( ( semantics @ A @ C @ B2 @ E4 )
% 2.95/1.59 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_183_ex__in__conv,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] :
% 2.95/1.59 ( ( ? [X7: A] : ( member @ A @ X7 @ A2 ) )
% 2.95/1.59 = ( A2
% 2.95/1.59 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_184_equals0I,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] :
% 2.95/1.59 ( ! [Y5: A] :
% 2.95/1.59 ~ ( member @ A @ Y5 @ A2 )
% 2.95/1.59 => ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_185_equals0D,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,A3: A] :
% 2.95/1.59 ( ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_186_emptyE,axiom,
% 2.95/1.59 ! [A: $tType,A3: A] :
% 2.95/1.59 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_187_Collect__mono__iff,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.95/1.59 = ( ! [X7: A] :
% 2.95/1.59 ( ( P @ X7 )
% 2.95/1.59 => ( Q @ X7 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_188_set__eq__subset,axiom,
% 2.95/1.59 ! [A: $tType] :
% 2.95/1.59 ( ( ^ [Y6: set @ A,Z4: set @ A] : ( Y6 = Z4 ) )
% 2.95/1.59 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 2.95/1.59 & ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_189_subset__trans,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A,C7: set @ A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C7 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ A2 @ C7 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_190_Collect__mono,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.95/1.59 ( ! [X6: A] :
% 2.95/1.59 ( ( P @ X6 )
% 2.95/1.59 => ( Q @ X6 ) )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_191_subset__refl,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_192_subset__iff,axiom,
% 2.95/1.59 ! [A: $tType] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) )
% 2.95/1.59 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.95/1.59 ! [T3: A] :
% 2.95/1.59 ( ( member @ A @ T3 @ A8 )
% 2.95/1.59 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_193_equalityD2,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ( A2 = B2 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_194_equalityD1,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ( A2 = B2 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_195_subset__eq,axiom,
% 2.95/1.59 ! [A: $tType] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) )
% 2.95/1.59 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.95/1.59 ! [X7: A] :
% 2.95/1.59 ( ( member @ A @ X7 @ A8 )
% 2.95/1.59 => ( member @ A @ X7 @ B7 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_196_equalityE,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ( A2 = B2 )
% 2.95/1.59 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.95/1.59 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_197_subsetD,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.95/1.59 => ( ( member @ A @ C2 @ A2 )
% 2.95/1.59 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_198_in__mono,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.95/1.59 => ( ( member @ A @ X3 @ A2 )
% 2.95/1.59 => ( member @ A @ X3 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_199_mk__disjoint__insert,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ? [B8: set @ A] :
% 2.95/1.59 ( ( A2
% 2.95/1.59 = ( insert @ A @ A3 @ B8 ) )
% 2.95/1.59 & ~ ( member @ A @ A3 @ B8 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_200_insert__commute,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,Y3: A,A2: set @ A] :
% 2.95/1.59 ( ( insert @ A @ X3 @ ( insert @ A @ Y3 @ A2 ) )
% 2.95/1.59 = ( insert @ A @ Y3 @ ( insert @ A @ X3 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_201_insert__eq__iff,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
% 2.95/1.59 ( ~ ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( ~ ( member @ A @ B3 @ B2 )
% 2.95/1.59 => ( ( ( insert @ A @ A3 @ A2 )
% 2.95/1.59 = ( insert @ A @ B3 @ B2 ) )
% 2.95/1.59 = ( ( ( A3 = B3 )
% 2.95/1.59 => ( A2 = B2 ) )
% 2.95/1.59 & ( ( A3 != B3 )
% 2.95/1.59 => ? [C5: set @ A] :
% 2.95/1.59 ( ( A2
% 2.95/1.59 = ( insert @ A @ B3 @ C5 ) )
% 2.95/1.59 & ~ ( member @ A @ B3 @ C5 )
% 2.95/1.59 & ( B2
% 2.95/1.59 = ( insert @ A @ A3 @ C5 ) )
% 2.95/1.59 & ~ ( member @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_202_insert__absorb,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( ( insert @ A @ A3 @ A2 )
% 2.95/1.59 = A2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_203_insert__ident,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ~ ( member @ A @ X3 @ A2 )
% 2.95/1.59 => ( ~ ( member @ A @ X3 @ B2 )
% 2.95/1.59 => ( ( ( insert @ A @ X3 @ A2 )
% 2.95/1.59 = ( insert @ A @ X3 @ B2 ) )
% 2.95/1.59 = ( A2 = B2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_204_Set_Oset__insert,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ A @ X3 @ A2 )
% 2.95/1.59 => ~ ! [B8: set @ A] :
% 2.95/1.59 ( ( A2
% 2.95/1.59 = ( insert @ A @ X3 @ B8 ) )
% 2.95/1.59 => ( member @ A @ X3 @ B8 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_205_insertI2,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.95/1.59 ( ( member @ A @ A3 @ B2 )
% 2.95/1.59 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_206_insertI1,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_207_insertE,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.95/1.59 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.95/1.59 => ( ( A3 != B3 )
% 2.95/1.59 => ( member @ A @ A3 @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_208_singleton__inject,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B3: A] :
% 2.95/1.59 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 => ( A3 = B3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_209_insert__not__empty,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,A2: set @ A] :
% 2.95/1.59 ( ( insert @ A @ A3 @ A2 )
% 2.95/1.59 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_210_doubleton__eq__iff,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,B3: A,C2: A,D3: A] :
% 2.95/1.59 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( insert @ A @ C2 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 = ( ( ( A3 = C2 )
% 2.95/1.59 & ( B3 = D3 ) )
% 2.95/1.59 | ( ( A3 = D3 )
% 2.95/1.59 & ( B3 = C2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_211_singleton__iff,axiom,
% 2.95/1.59 ! [A: $tType,B3: A,A3: A] :
% 2.95/1.59 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( B3 = A3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_212_singletonD,axiom,
% 2.95/1.59 ! [A: $tType,B3: A,A3: A] :
% 2.95/1.59 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 => ( B3 = A3 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_213_subset__insertI2,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_214_subset__insertI,axiom,
% 2.95/1.59 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_215_subset__insert,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 2.95/1.59 ( ~ ( member @ A @ X3 @ A2 )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ B2 ) )
% 2.95/1.59 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_216_insert__mono,axiom,
% 2.95/1.59 ! [A: $tType,C7: set @ A,D4: set @ A,A3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ C7 @ D4 )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C7 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_217_subset__singleton__iff,axiom,
% 2.95/1.59 ! [A: $tType,X5: set @ A,A3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( ( X5
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 | ( X5
% 2.95/1.59 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_218_subset__singletonD,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,X3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 => ( ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 | ( A2
% 2.95/1.59 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_219_equiv__class__self,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ A @ A3 @ A2 )
% 2.95/1.59 => ( member @ A @ A3 @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_220_equiv__class__eq__iff,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 2.95/1.59 = ( ( ( image @ A @ A @ R @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( image @ A @ A @ R @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 & ( member @ A @ X3 @ A2 )
% 2.95/1.59 & ( member @ A @ Y3 @ A2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_221_eq__equiv__class__iff,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ A @ X3 @ A2 )
% 2.95/1.59 => ( ( member @ A @ Y3 @ A2 )
% 2.95/1.59 => ( ( ( image @ A @ A @ R @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( image @ A @ A @ R @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_222_equiv__class__eq,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R )
% 2.95/1.59 => ( ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_223_eq__equiv__class,axiom,
% 2.95/1.59 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B3: A,A2: set @ A] :
% 2.95/1.59 ( ( ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = ( image @ A @ A @ R @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 => ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( member @ A @ B3 @ A2 )
% 2.95/1.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_224_refines__equiv__class__eq,axiom,
% 2.95/1.59 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A2: set @ A,A3: A] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
% 2.95/1.59 => ( ( equiv_equiv @ A @ A2 @ R2 )
% 2.95/1.59 => ( ( equiv_equiv @ A @ A2 @ S )
% 2.95/1.59 => ( ( image @ A @ A @ R2 @ ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 = ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_225_nonempty__rule,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,G: labeled_graph @ A @ B] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G )
% 2.95/1.59 = ( ( labeled_vertices @ A @ B @ G )
% 2.95/1.59 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_226_ma,axiom,
% 2.95/1.59 ! [X4: product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat )] :
% 2.95/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ X4 @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ ( standa245363227y_rule @ ( standard_Constant @ a ) @ ( standard_S_Idt @ a ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ ( standa2114124375e_rule @ ( standard_Constant @ a ) @ ( standard_S_Idt @ a ) ) @ ( insert @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) @ ( standa1805702094y_rule @ ( standard_Constant @ a ) @ ( standard_S_Idt @ a ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) @ ( labeled_graph @ ( standard_Constant @ a ) @ nat ) ) ) ) ) ) ) )
% 2.95/1.59 => ( maintained @ ( standard_Constant @ a ) @ nat @ b @ X4 @ g ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_227_proj__iff,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,R: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
% 2.95/1.59 ( ( equiv_equiv @ A @ A2 @ R )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A2 )
% 2.95/1.59 => ( ( ( equiv_proj @ A @ A @ R @ X3 )
% 2.95/1.59 = ( equiv_proj @ A @ A @ R @ Y3 ) )
% 2.95/1.59 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_228_proj__def,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType] :
% 2.95/1.59 ( ( equiv_proj @ B @ A )
% 2.95/1.59 = ( ^ [R4: set @ ( product_prod @ B @ A ),X7: B] : ( image @ B @ A @ R4 @ ( insert @ B @ X7 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_229_equivalence,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,G: labeled_graph @ A @ B,I: A] :
% 2.95/1.59 ( ( G
% 2.95/1.59 = ( restrict @ A @ B @ G ) )
% 2.95/1.59 => ( ! [X6: product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat )] :
% 2.95/1.59 ( ( member @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ X6 @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa245363227y_rule @ A @ I ) @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa2114124375e_rule @ A @ I ) @ ( insert @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ ( standa1805702094y_rule @ A @ I ) @ ( bot_bot @ ( set @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) ) ) ) ) ) )
% 2.95/1.59 => ( maintained @ A @ nat @ B @ X6 @ G ) )
% 2.95/1.59 => ( equiv_equiv @ B @ ( labeled_vertices @ A @ B @ G ) @ ( getRel @ A @ B @ I @ G ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_230_f,axiom,
% 2.95/1.59 ( f
% 2.95/1.59 = ( ^ [X7: b] :
% 2.95/1.59 ( if @ b
% 2.95/1.59 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X7 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ b ) ) )
% 2.95/1.59 @ X7
% 2.95/1.59 @ ( hilbert_Eps @ b @ ( p @ X7 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_231_the__elem__eq,axiom,
% 2.95/1.59 ! [A: $tType,X3: A] :
% 2.95/1.59 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.95/1.59 = X3 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_232_tuple__disj,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,Z: A] :
% 2.95/1.59 ( ( collect @ A
% 2.95/1.59 @ ^ [Y7: A] :
% 2.95/1.59 ( ( Y7 = X3 )
% 2.95/1.59 | ( Y7 = Z ) ) )
% 2.95/1.59 = ( insert @ A @ X3 @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_233_singleton__conv,axiom,
% 2.95/1.59 ! [A: $tType,A3: A] :
% 2.95/1.59 ( ( collect @ A
% 2.95/1.59 @ ^ [X7: A] : ( X7 = A3 ) )
% 2.95/1.59 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_234_singleton__conv2,axiom,
% 2.95/1.59 ! [A: $tType,A3: A] :
% 2.95/1.59 ( ( collect @ A
% 2.95/1.59 @ ( ^ [Y6: A,Z4: A] : ( Y6 = Z4 )
% 2.95/1.59 @ A3 ) )
% 2.95/1.59 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_235_P,axiom,
% 2.95/1.59 ( p
% 2.95/1.59 = ( ^ [X7: b,Y7: b] : ( member @ b @ Y7 @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X7 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_236_empty__def,axiom,
% 2.95/1.59 ! [A: $tType] :
% 2.95/1.59 ( ( bot_bot @ ( set @ A ) )
% 2.95/1.59 = ( collect @ A
% 2.95/1.59 @ ^ [X7: A] : $false ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_237_insert__compr,axiom,
% 2.95/1.59 ! [A: $tType] :
% 2.95/1.59 ( ( insert @ A )
% 2.95/1.59 = ( ^ [A6: A,B7: set @ A] :
% 2.95/1.59 ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( X7 = A6 )
% 2.95/1.59 | ( member @ A @ X7 @ B7 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_238_insert__Collect,axiom,
% 2.95/1.59 ! [A: $tType,A3: A,P: A > $o] :
% 2.95/1.59 ( ( insert @ A @ A3 @ ( collect @ A @ P ) )
% 2.95/1.59 = ( collect @ A
% 2.95/1.59 @ ^ [U2: A] :
% 2.95/1.59 ( ( U2 != A3 )
% 2.95/1.59 => ( P @ U2 ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_239_Collect__conv__if,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o,A3: A] :
% 2.95/1.59 ( ( ( P @ A3 )
% 2.95/1.59 => ( ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( X7 = A3 )
% 2.95/1.59 & ( P @ X7 ) ) )
% 2.95/1.59 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 & ( ~ ( P @ A3 )
% 2.95/1.59 => ( ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( X7 = A3 )
% 2.95/1.59 & ( P @ X7 ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_240_Collect__conv__if2,axiom,
% 2.95/1.59 ! [A: $tType,P: A > $o,A3: A] :
% 2.95/1.59 ( ( ( P @ A3 )
% 2.95/1.59 => ( ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( A3 = X7 )
% 2.95/1.59 & ( P @ X7 ) ) )
% 2.95/1.59 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.95/1.59 & ( ~ ( P @ A3 )
% 2.95/1.59 => ( ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( A3 = X7 )
% 2.95/1.59 & ( P @ X7 ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_241_Image__singleton,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
% 2.95/1.59 ( ( image @ B @ A @ R @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
% 2.95/1.59 = ( collect @ A
% 2.95/1.59 @ ^ [B6: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B6 ) @ R ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_242_edges__def,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType] :
% 2.95/1.59 ( ( labeled_edges @ L @ V )
% 2.95/1.59 = ( labele1974067554_graph @ L @ V @ ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) )
% 2.95/1.59 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X13 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_243_vertices__def,axiom,
% 2.95/1.59 ! [V: $tType,L: $tType] :
% 2.95/1.59 ( ( labeled_vertices @ L @ V )
% 2.95/1.59 = ( labele1974067554_graph @ L @ V @ ( set @ V )
% 2.95/1.59 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X24 ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_244_labeled__graph_Ocase__distrib,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,V: $tType,L: $tType,H: A > B,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.95/1.59 ( ( H @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 2.95/1.59 = ( labele1974067554_graph @ L @ V @ B
% 2.95/1.59 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : ( H @ ( F @ X13 @ X24 ) )
% 2.95/1.59 @ Labeled_graph ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_245_pred__equals__eq2,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.95/1.59 ( ( ( ^ [X7: A,Y7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X7 @ Y7 ) @ R2 ) )
% 2.95/1.59 = ( ^ [X7: A,Y7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X7 @ Y7 ) @ S ) ) )
% 2.95/1.59 = ( R2 = S ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_246_subset__CollectI,axiom,
% 2.95/1.59 ! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.95/1.59 => ( ! [X6: A] :
% 2.95/1.59 ( ( member @ A @ X6 @ B2 )
% 2.95/1.59 => ( ( Q @ X6 )
% 2.95/1.59 => ( P @ X6 ) ) )
% 2.95/1.59 => ( ord_less_eq @ ( set @ A )
% 2.95/1.59 @ ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( member @ A @ X7 @ B2 )
% 2.95/1.59 & ( Q @ X7 ) ) )
% 2.95/1.59 @ ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( member @ A @ X7 @ A2 )
% 2.95/1.59 & ( P @ X7 ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_247_subset__Collect__iff,axiom,
% 2.95/1.59 ! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
% 2.95/1.59 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.95/1.59 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 2.95/1.59 @ ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( member @ A @ X7 @ A2 )
% 2.95/1.59 & ( P @ X7 ) ) ) )
% 2.95/1.59 = ( ! [X7: A] :
% 2.95/1.59 ( ( member @ A @ X7 @ B2 )
% 2.95/1.59 => ( P @ X7 ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_248_Collect__subset,axiom,
% 2.95/1.59 ! [A: $tType,A2: set @ A,P: A > $o] :
% 2.95/1.59 ( ord_less_eq @ ( set @ A )
% 2.95/1.59 @ ( collect @ A
% 2.95/1.59 @ ^ [X7: A] :
% 2.95/1.59 ( ( member @ A @ X7 @ A2 )
% 2.95/1.59 & ( P @ X7 ) ) )
% 2.95/1.59 @ A2 ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_249_congr_I1_J,axiom,
% 2.95/1.59 ! [X2: b,Y: b,L2: standard_Constant @ a] :
% 2.95/1.59 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X2 @ Y ) @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) )
% 2.95/1.59 => ( equiv_congruent @ b @ ( set @ b ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g )
% 2.95/1.59 @ ^ [V3: b] : ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) @ ( insert @ b @ V3 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_250_ident,axiom,
% 2.95/1.59 ( ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) )
% 2.95/1.59 = ( image2 @ b @ ( product_prod @ b @ b )
% 2.95/1.59 @ ^ [X7: b] : ( product_Pair @ b @ b @ X7 @ X7 )
% 2.95/1.59 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ ( map_graph @ b @ b @ ( standard_Constant @ a ) @ ( bNF_Gr @ b @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ f ) @ g ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_251_image__empty,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,F: B > A] :
% 2.95/1.59 ( ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_252_empty__is__image,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 2.95/1.59 ( ( ( bot_bot @ ( set @ A ) )
% 2.95/1.59 = ( image2 @ B @ A @ F @ A2 ) )
% 2.95/1.59 = ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_253_image__is__empty,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 2.95/1.59 ( ( ( image2 @ B @ A @ F @ A2 )
% 2.95/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.95/1.59 = ( A2
% 2.95/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_254_image__insert,axiom,
% 2.95/1.59 ! [A: $tType,B: $tType,F: B > A,A3: B,B2: set @ B] :
% 2.95/1.59 ( ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ B2 ) )
% 2.95/1.59 = ( insert @ A @ ( F @ A3 ) @ ( image2 @ B @ A @ F @ B2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(fact_255_insert__image,axiom,
% 2.95/1.59 ! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
% 2.95/1.59 ( ( member @ A @ X3 @ A2 )
% 2.95/1.59 => ( ( insert @ B @ ( F @ X3 ) @ ( image2 @ A @ B @ F @ A2 ) )
% 2.95/1.59 = ( image2 @ A @ B @ F @ A2 ) ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(help_If_3_1_T,axiom,
% 2.95/1.59 ! [P: $o] :
% 2.95/1.59 ( ( P = $true )
% 2.95/1.59 | ( P = $false ) ) ).
% 2.95/1.59
% 2.95/1.59 thf(help_If_2_1_T,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,Y3: A] :
% 2.95/1.59 ( ( if @ A @ $false @ X3 @ Y3 )
% 2.95/1.59 = Y3 ) ).
% 2.95/1.59
% 2.95/1.59 thf(help_If_1_1_T,axiom,
% 2.95/1.59 ! [A: $tType,X3: A,Y3: A] :
% 2.95/1.59 ( ( if @ A @ $true @ X3 @ Y3 )
% 2.95/1.59 = X3 ) ).
% 2.95/1.59
% 2.95/1.59 thf(conj_0,conjecture,
% 2.95/1.59 member @ b @ x @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ).
% 2.95/1.59 ------------------------
% 3.05/1.62 % (351)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/2Mi)
% 3.05/1.62 % (352)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_32619 for (2999ds/2Mi)
% 3.05/1.62 % (353)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_32619 for (2999ds/275Mi)
% 3.05/1.62 % (352)Instruction limit reached!
% 3.05/1.62 % (352)------------------------------
% 3.05/1.62 % (352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.62 % (352)Termination reason: Unknown
% 3.05/1.62 % (350)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_32619 for (2999ds/27Mi)
% 3.05/1.62 % (352)Termination phase: shuffling
% 3.05/1.62
% 3.05/1.62 % (352)Memory used [KB]: 1407
% 3.05/1.62 % (352)Time elapsed: 0.002 s
% 3.05/1.62 % (352)Instructions burned: 2 (million)
% 3.05/1.62 % (352)------------------------------
% 3.05/1.62 % (352)------------------------------
% 3.05/1.62 % (348)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/183Mi)
% 3.05/1.62 % (354)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/18Mi)
% 3.05/1.62 % (351)Instruction limit reached!
% 3.05/1.62 % (351)------------------------------
% 3.05/1.62 % (351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.62 % (351)Termination reason: Unknown
% 3.05/1.62 % (351)Termination phase: shuffling
% 3.05/1.62
% 3.05/1.62 % (351)Memory used [KB]: 1535
% 3.05/1.62 % (351)Time elapsed: 0.003 s
% 3.05/1.62 % (351)Instructions burned: 4 (million)
% 3.05/1.62 % (351)------------------------------
% 3.05/1.62 % (351)------------------------------
% 3.05/1.62 % (349)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_32619 for (2999ds/4Mi)
% 3.05/1.62 % (349)Instruction limit reached!
% 3.05/1.62 % (349)------------------------------
% 3.05/1.62 % (349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.62 % (349)Termination reason: Unknown
% 3.05/1.62 % (349)Termination phase: shuffling
% 3.05/1.62
% 3.05/1.62 % (349)Memory used [KB]: 1535
% 3.05/1.62 % (349)Time elapsed: 0.003 s
% 3.05/1.62 % (349)Instructions burned: 5 (million)
% 3.05/1.62 % (349)------------------------------
% 3.05/1.62 % (349)------------------------------
% 3.05/1.63 % (354)Instruction limit reached!
% 3.05/1.63 % (354)------------------------------
% 3.05/1.63 % (354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.63 % (354)Termination reason: Unknown
% 3.05/1.63 % (354)Termination phase: shuffling
% 3.05/1.63
% 3.05/1.63 % (354)Memory used [KB]: 1791
% 3.05/1.63 % (354)Time elapsed: 0.007 s
% 3.05/1.63 % (354)Instructions burned: 18 (million)
% 3.05/1.63 % (354)------------------------------
% 3.05/1.63 % (354)------------------------------
% 3.05/1.63 % (355)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/3Mi)
% 3.05/1.63 % (350)Instruction limit reached!
% 3.05/1.63 % (350)------------------------------
% 3.05/1.63 % (350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.63 % (350)Termination reason: Unknown
% 3.05/1.63 % (350)Termination phase: shuffling
% 3.05/1.63
% 3.05/1.63 % (350)Memory used [KB]: 1918
% 3.05/1.63 % (350)Time elapsed: 0.011 s
% 3.05/1.63 % (350)Instructions burned: 28 (million)
% 3.05/1.63 % (350)------------------------------
% 3.05/1.63 % (350)------------------------------
% 3.05/1.63 % (356)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_32619 for (2999ds/37Mi)
% 3.05/1.63 % (355)Instruction limit reached!
% 3.05/1.63 % (355)------------------------------
% 3.05/1.63 % (355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.63 % (355)Termination reason: Unknown
% 3.05/1.63 % (355)Termination phase: shuffling
% 3.05/1.63
% 3.05/1.63 % (355)Memory used [KB]: 1407
% 3.05/1.63 % (355)Time elapsed: 0.002 s
% 3.05/1.63 % (355)Instructions burned: 3 (million)
% 3.05/1.63 % (355)------------------------------
% 3.05/1.63 % (355)------------------------------
% 3.05/1.63 % (357)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_32619 for (2999ds/15Mi)
% 3.05/1.64 % (358)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_32619 for (2999ds/3Mi)
% 3.05/1.64 % (358)Instruction limit reached!
% 3.05/1.64 % (358)------------------------------
% 3.05/1.64 % (358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (358)Termination reason: Unknown
% 3.05/1.64 % (358)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (358)Memory used [KB]: 1407
% 3.05/1.64 % (358)Time elapsed: 0.002 s
% 3.05/1.64 % (358)Instructions burned: 3 (million)
% 3.05/1.64 % (358)------------------------------
% 3.05/1.64 % (358)------------------------------
% 3.05/1.64 % (357)Instruction limit reached!
% 3.05/1.64 % (357)------------------------------
% 3.05/1.64 % (357)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (357)Termination reason: Unknown
% 3.05/1.64 % (357)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (357)Memory used [KB]: 1791
% 3.05/1.64 % (357)Time elapsed: 0.006 s
% 3.05/1.64 % (357)Instructions burned: 15 (million)
% 3.05/1.64 % (357)------------------------------
% 3.05/1.64 % (357)------------------------------
% 3.05/1.64 % (353)Refutation not found, incomplete strategy
% 3.05/1.64 % (353)------------------------------
% 3.05/1.64 % (353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (353)Termination reason: Refutation not found, incomplete strategy
% 3.05/1.64
% 3.05/1.64
% 3.05/1.64 % (353)Memory used [KB]: 6652
% 3.05/1.64 % (353)Time elapsed: 0.020 s
% 3.05/1.64 % (353)Instructions burned: 59 (million)
% 3.05/1.64 % (353)------------------------------
% 3.05/1.64 % (353)------------------------------
% 3.05/1.64 % (360)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/7Mi)
% 3.05/1.64 % (359)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_32619 for (2999ds/1041Mi)
% 3.05/1.64 % (356)Instruction limit reached!
% 3.05/1.64 % (356)------------------------------
% 3.05/1.64 % (356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (356)Termination reason: Unknown
% 3.05/1.64 % (356)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (356)Memory used [KB]: 2174
% 3.05/1.64 % (356)Time elapsed: 0.011 s
% 3.05/1.64 % (356)Instructions burned: 37 (million)
% 3.05/1.64 % (356)------------------------------
% 3.05/1.64 % (356)------------------------------
% 3.05/1.64 % (360)Instruction limit reached!
% 3.05/1.64 % (360)------------------------------
% 3.05/1.64 % (360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (360)Termination reason: Unknown
% 3.05/1.64 % (360)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (360)Memory used [KB]: 1535
% 3.05/1.64 % (360)Time elapsed: 0.003 s
% 3.05/1.64 % (360)Instructions burned: 7 (million)
% 3.05/1.64 % (360)------------------------------
% 3.05/1.64 % (360)------------------------------
% 3.05/1.65 % (361)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/16Mi)
% 3.05/1.65 % (362)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/3Mi)
% 3.05/1.65 % (362)Instruction limit reached!
% 3.05/1.65 % (362)------------------------------
% 3.05/1.65 % (362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (362)Termination reason: Unknown
% 3.05/1.65 % (362)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (362)Memory used [KB]: 1407
% 3.05/1.65 % (362)Time elapsed: 0.002 s
% 3.05/1.65 % (362)Instructions burned: 3 (million)
% 3.05/1.65 % (362)------------------------------
% 3.05/1.65 % (362)------------------------------
% 3.05/1.65 % (363)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/3Mi)
% 3.05/1.65 % (364)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/7Mi)
% 3.05/1.65 % (363)Instruction limit reached!
% 3.05/1.65 % (363)------------------------------
% 3.05/1.65 % (363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (363)Termination reason: Unknown
% 3.05/1.65 % (363)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (363)Memory used [KB]: 1407
% 3.05/1.65 % (363)Time elapsed: 0.003 s
% 3.05/1.65 % (363)Instructions burned: 3 (million)
% 3.05/1.65 % (363)------------------------------
% 3.05/1.65 % (363)------------------------------
% 3.05/1.65 % (361)Instruction limit reached!
% 3.05/1.65 % (361)------------------------------
% 3.05/1.65 % (361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (361)Termination reason: Unknown
% 3.05/1.65 % (361)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (361)Memory used [KB]: 1791
% 3.05/1.65 % (361)Time elapsed: 0.006 s
% 3.05/1.65 % (361)Instructions burned: 16 (million)
% 3.05/1.65 % (361)------------------------------
% 3.05/1.65 % (361)------------------------------
% 3.05/1.65 % (365)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/3Mi)
% 3.05/1.65 % (364)Instruction limit reached!
% 3.05/1.65 % (364)------------------------------
% 3.05/1.65 % (364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (364)Termination reason: Unknown
% 3.05/1.65 % (364)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (364)Memory used [KB]: 1535
% 3.05/1.65 % (364)Time elapsed: 0.003 s
% 3.05/1.65 % (364)Instructions burned: 7 (million)
% 3.05/1.65 % (364)------------------------------
% 3.05/1.65 % (364)------------------------------
% 3.05/1.65 % (365)Instruction limit reached!
% 3.05/1.65 % (365)------------------------------
% 3.05/1.65 % (365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (365)Termination reason: Unknown
% 3.05/1.65 % (365)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (365)Memory used [KB]: 1535
% 3.05/1.65 % (365)Time elapsed: 0.002 s
% 3.05/1.65 % (365)Instructions burned: 3 (million)
% 3.05/1.65 % (365)------------------------------
% 3.05/1.65 % (365)------------------------------
% 3.05/1.66 % (367)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/4Mi)
% 3.05/1.66 % (367)Instruction limit reached!
% 3.05/1.66 % (367)------------------------------
% 3.05/1.66 % (367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (367)Termination reason: Unknown
% 3.05/1.66 % (367)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (367)Memory used [KB]: 1535
% 3.05/1.66 % (367)Time elapsed: 0.003 s
% 3.05/1.66 % (367)Instructions burned: 6 (million)
% 3.05/1.66 % (367)------------------------------
% 3.05/1.66 % (367)------------------------------
% 3.05/1.66 % (369)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/18Mi)
% 3.05/1.66 % (370)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/710Mi)
% 3.05/1.66 % (371)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/6Mi)
% 3.05/1.66 % (372)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/902Mi)
% 3.05/1.66 % (371)Instruction limit reached!
% 3.05/1.66 % (371)------------------------------
% 3.05/1.66 % (371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (371)Termination reason: Unknown
% 3.05/1.66 % (371)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (371)Memory used [KB]: 1535
% 3.05/1.66 % (371)Time elapsed: 0.003 s
% 3.05/1.66 % (371)Instructions burned: 6 (million)
% 3.05/1.66 % (371)------------------------------
% 3.05/1.66 % (371)------------------------------
% 3.05/1.67 % (369)Instruction limit reached!
% 3.05/1.67 % (369)------------------------------
% 3.05/1.67 % (369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.67 % (369)Termination reason: Unknown
% 3.05/1.67 % (369)Termination phase: shuffling
% 3.05/1.67
% 3.05/1.67 % (369)Memory used [KB]: 1791
% 3.05/1.67 % (369)Time elapsed: 0.007 s
% 3.05/1.67 % (369)Instructions burned: 18 (million)
% 3.05/1.67 % (369)------------------------------
% 3.05/1.67 % (369)------------------------------
% 3.05/1.67 % (373)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/21Mi)
% 3.05/1.68 % (373)Instruction limit reached!
% 3.05/1.68 % (373)------------------------------
% 3.05/1.68 % (373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (373)Termination reason: Unknown
% 3.05/1.68 % (373)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (373)Memory used [KB]: 1791
% 3.05/1.68 % (373)Time elapsed: 0.008 s
% 3.05/1.68 % (373)Instructions burned: 22 (million)
% 3.05/1.68 % (373)------------------------------
% 3.05/1.68 % (373)------------------------------
% 3.05/1.68 % (374)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/5Mi)
% 3.05/1.68 % (374)Instruction limit reached!
% 3.05/1.68 % (374)------------------------------
% 3.05/1.68 % (374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (374)Termination reason: Unknown
% 3.05/1.68 % (374)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (374)Memory used [KB]: 1535
% 3.05/1.68 % (374)Time elapsed: 0.005 s
% 3.05/1.68 % (374)Instructions burned: 6 (million)
% 3.05/1.68 % (374)------------------------------
% 3.05/1.68 % (374)------------------------------
% 3.05/1.68 % (375)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/6Mi)
% 3.05/1.68 % (348)Instruction limit reached!
% 3.05/1.68 % (348)------------------------------
% 3.05/1.68 % (348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (348)Termination reason: Unknown
% 3.05/1.68 % (348)Termination phase: Saturation
% 3.05/1.68
% 3.05/1.68 % (348)Memory used [KB]: 7291
% 3.05/1.68 % (348)Time elapsed: 0.064 s
% 3.05/1.68 % (348)Instructions burned: 183 (million)
% 3.05/1.68 % (348)------------------------------
% 3.05/1.68 % (348)------------------------------
% 3.05/1.68 % (375)Instruction limit reached!
% 3.05/1.68 % (375)------------------------------
% 3.05/1.68 % (375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (375)Termination reason: Unknown
% 3.05/1.68 % (375)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (375)Memory used [KB]: 1535
% 3.05/1.68 % (375)Time elapsed: 0.004 s
% 3.05/1.68 % (375)Instructions burned: 6 (million)
% 3.05/1.68 % (375)------------------------------
% 3.05/1.68 % (375)------------------------------
% 3.05/1.69 % (384)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/377Mi)
% 3.05/1.69 % (370)Refutation not found, incomplete strategy
% 3.05/1.69 % (370)------------------------------
% 3.05/1.69 % (370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.69 % (370)Termination reason: Refutation not found, incomplete strategy
% 3.05/1.69
% 3.05/1.69
% 3.05/1.69 % (370)Memory used [KB]: 6652
% 3.05/1.69 % (370)Time elapsed: 0.025 s
% 3.05/1.69 % (370)Instructions burned: 57 (million)
% 3.05/1.69 % (370)------------------------------
% 3.05/1.69 % (370)------------------------------
% 3.05/1.70 % (385)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/779Mi)
% 3.05/1.70 % (386)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/19Mi)
% 3.05/1.70 % (387)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/879Mi)
% 3.05/1.70 % (390)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/17Mi)
% 3.05/1.71 % (386)Instruction limit reached!
% 3.05/1.71 % (386)------------------------------
% 3.05/1.71 % (386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.71 % (386)Termination reason: Unknown
% 3.05/1.71 % (386)Termination phase: shuffling
% 3.05/1.71
% 3.05/1.71 % (386)Memory used [KB]: 1918
% 3.05/1.71 % (386)Time elapsed: 0.009 s
% 3.05/1.71 % (386)Instructions burned: 21 (million)
% 3.05/1.71 % (386)------------------------------
% 3.05/1.71 % (386)------------------------------
% 3.05/1.71 % (390)Instruction limit reached!
% 3.05/1.71 % (390)------------------------------
% 3.05/1.71 % (390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.71 % (390)Termination reason: Unknown
% 3.05/1.71 % (390)Termination phase: shuffling
% 3.05/1.71
% 3.05/1.71 % (390)Memory used [KB]: 1791
% 3.05/1.71 % (390)Time elapsed: 0.010 s
% 3.05/1.71 % (390)Instructions burned: 18 (million)
% 3.05/1.71 % (390)------------------------------
% 3.05/1.71 % (390)------------------------------
% 3.05/1.72 % (395)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32619 for (2999ds/3Mi)
% 3.05/1.73 % (395)Instruction limit reached!
% 3.05/1.73 % (395)------------------------------
% 3.05/1.73 % (395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.73 % (395)Termination reason: Unknown
% 3.05/1.73 % (395)Termination phase: shuffling
% 3.05/1.73
% 3.05/1.73 % (395)Memory used [KB]: 1407
% 3.05/1.73 % (395)Time elapsed: 0.003 s
% 3.05/1.73 % (395)Instructions burned: 3 (million)
% 3.05/1.73 % (395)------------------------------
% 3.05/1.73 % (395)------------------------------
% 3.05/1.73 % (372)Refutation not found, incomplete strategy
% 3.05/1.73 % (372)------------------------------
% 3.05/1.73 % (372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.73 % (372)Termination reason: Refutation not found, incomplete strategy
% 3.05/1.73
% 3.05/1.73
% 3.05/1.73 % (372)Memory used [KB]: 7675
% 3.05/1.73 % (372)Time elapsed: 0.071 s
% 3.05/1.73 % (372)Instructions burned: 167 (million)
% 3.05/1.73 % (372)------------------------------
% 3.05/1.73 % (372)------------------------------
% 3.05/1.74 % (398)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_32619 for (2998ds/30Mi)
% 3.05/1.74 % (401)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_32619 for (2998ds/127Mi)
% 3.05/1.74 % (398)Instruction limit reached!
% 3.05/1.74 % (398)------------------------------
% 3.05/1.74 % (398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.74 % (398)Termination reason: Unknown
% 3.05/1.74 % (398)Termination phase: shuffling
% 3.05/1.74
% 3.05/1.74 % (398)Memory used [KB]: 2046
% 3.05/1.74 % (398)Time elapsed: 0.011 s
% 3.05/1.74 % (398)Instructions burned: 30 (million)
% 3.05/1.74 % (398)------------------------------
% 3.05/1.74 % (398)------------------------------
% 3.05/1.75 % (387)First to succeed.
% 3.05/1.75 % (387)Refutation found. Thanks to Tanya!
% 3.05/1.75 % SZS status Theorem for DTF2THF_32619
% 3.05/1.75 % SZS output start Proof for DTF2THF_32619
% 3.05/1.75 thf(type_def_5, type, set: ($tType) > $tType).
% 3.05/1.75 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 3.05/1.75 thf(type_def_8, type, standard_Constant: ($tType) > $tType).
% 3.05/1.75 thf(type_def_9, type, labeled_graph: ($tType * $tType) > $tType).
% 3.05/1.75 thf(type_def_10, type, allegorical_term: ($tType) > $tType).
% 3.05/1.75 thf(type_def_11, type, itself: ($tType) > $tType).
% 3.05/1.75 thf(type_def_12, type, nat: $tType).
% 3.05/1.75 thf(type_def_13, type, a: $tType).
% 3.05/1.75 thf(type_def_14, type, b: $tType).
% 3.05/1.75 thf(func_def_0, type, standard_Constant: ($tType) > $tType).
% 3.05/1.75 thf(func_def_1, type, allegorical_term: ($tType) > $tType).
% 3.05/1.75 thf(func_def_2, type, labeled_graph: ($tType * $tType) > $tType).
% 3.05/1.75 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 3.05/1.75 thf(func_def_4, type, set: ($tType) > $tType).
% 3.05/1.75 thf(func_def_5, type, nat: $tType).
% 3.05/1.75 thf(func_def_6, type, itself: ($tType) > $tType).
% 3.05/1.75 thf(func_def_7, type, b: $tType).
% 3.05/1.75 thf(func_def_8, type, a: $tType).
% 3.05/1.75 thf(func_def_9, type, bNF_Gr: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > set @ product_prod @ X0 @ X1)).
% 3.05/1.75 thf(func_def_10, type, equiv_congruent: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.05/1.75 thf(func_def_11, type, equiv_equiv: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.05/1.75 thf(func_def_12, type, equiv_proj: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > set @ X1)).
% 3.05/1.75 thf(func_def_13, type, hilbert_Eps: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.05/1.75 thf(func_def_14, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.05/1.75 thf(func_def_15, type, standard_S_Idt: !>[X0: $tType]:(standard_Constant @ X0)).
% 3.05/1.75 thf(func_def_16, type, getRel: !>[X0: $tType, X1: $tType]:(X0 > labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X1)).
% 3.05/1.75 thf(func_def_17, type, semantics: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > allegorical_term @ X0 > set @ product_prod @ X1 @ X1)).
% 3.05/1.75 thf(func_def_18, 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)).
% 3.05/1.75 thf(func_def_19, type, graph_homomorphism: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.05/1.75 thf(func_def_20, type, graph_union: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.05/1.75 thf(func_def_21, type, labeled_LG: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > labeled_graph @ X0 @ X1)).
% 3.05/1.75 thf(func_def_22, type, labele1974067554_graph: !>[X0: $tType, X1: $tType, X2: $tType]:((set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > X2) > labeled_graph @ X0 @ X1 > X2)).
% 3.05/1.75 thf(func_def_23, type, labeled_edges: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X0 @ product_prod @ X1 @ X1)).
% 3.05/1.75 thf(func_def_24, type, labeled_vertices: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ X1)).
% 3.05/1.75 thf(func_def_25, type, map_graph: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > labeled_graph @ X2 @ X0 > labeled_graph @ X2 @ X1)).
% 3.05/1.75 thf(func_def_26, type, restrict: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.05/1.75 thf(func_def_27, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.05/1.75 thf(func_def_28, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.05/1.75 thf(func_def_29, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.05/1.75 thf(func_def_30, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.05/1.75 thf(func_def_31, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 3.05/1.75 thf(func_def_32, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 3.05/1.75 thf(func_def_33, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.05/1.75 thf(func_def_34, type, relcomp: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X0 @ X2)).
% 3.05/1.75 thf(func_def_35, type, agree_on: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.05/1.75 thf(func_def_36, 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)).
% 3.05/1.75 thf(func_def_37, 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)).
% 3.05/1.75 thf(func_def_38, 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)).
% 3.05/1.75 thf(func_def_39, 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)).
% 3.05/1.75 thf(func_def_40, type, maintained: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.05/1.75 thf(func_def_41, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.05/1.75 thf(func_def_42, type, image2: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.05/1.75 thf(func_def_43, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.05/1.75 thf(func_def_44, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.05/1.75 thf(func_def_45, type, standa2002409347_rules: !>[X0: $tType]:(set @ standard_Constant @ X0 > set @ product_prod @ labeled_graph @ standard_Constant @ X0 @ nat @ labeled_graph @ standard_Constant @ X0 @ nat)).
% 3.05/1.75 thf(func_def_46, type, standa244753842y_rule: !>[X0: $tType]:(product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.05/1.75 thf(func_def_47, type, standa245363227y_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.05/1.75 thf(func_def_48, type, standa1805702094y_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.05/1.75 thf(func_def_49, type, standa2114124375e_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.05/1.75 thf(func_def_50, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.05/1.75 thf(func_def_51, type, g: labeled_graph @ standard_Constant @ a @ b).
% 3.05/1.75 thf(func_def_52, type, l: set @ standard_Constant @ a).
% 3.05/1.75 thf(func_def_53, type, p: b > b > $o).
% 3.05/1.75 thf(func_def_54, type, f: b > b).
% 3.05/1.75 thf(func_def_55, type, l2: standard_Constant @ a).
% 3.05/1.75 thf(func_def_56, type, x: b).
% 3.05/1.75 thf(func_def_57, type, y: b).
% 3.05/1.75 thf(func_def_74, type, ph1: !>[X0: $tType]:(X0)).
% 3.05/1.75 thf(f765,plain,(
% 3.05/1.75 $false),
% 3.05/1.75 inference(subsumption_resolution,[status(thm)],[f764,f709])).
% 3.05/1.75 thf(f709,plain,(
% 3.05/1.75 ((member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g)) != $true)),
% 3.05/1.75 inference(cnf_transformation,[status(thm)],[f681])).
% 3.05/1.75 thf(f681,plain,(
% 3.05/1.75 ((member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g)) != $true)),
% 3.05/1.75 inference(flattening,[status(thm)],[f472])).
% 3.05/1.75 thf(f472,plain,(
% 3.05/1.75 ~((member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g)) = $true)),
% 3.05/1.75 inference(fool_elimination,[status(thm)],[f471])).
% 3.05/1.75 thf(f471,plain,(
% 3.05/1.75 ~(member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g))),
% 3.05/1.75 inference(rectify,[status(thm)],[f261])).
% 3.05/1.75 thf(f261,negated_conjecture,(
% 3.05/1.75 ~(member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g))),
% 3.05/1.75 inference(negated_conjecture,[status(cth)],[f260])).
% 3.05/1.75 thf(f260,conjecture,(
% 3.05/1.75 (member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g))),
% 3.05/1.75 file('/export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2THF_32619.p',conj_0)).
% 3.05/1.75 thf(f764,plain,(
% 3.05/1.75 ((member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g)) = $true)),
% 3.05/1.75 inference(trivial_inequality_removal,[status(thm)],[f763])).
% 3.05/1.75 thf(f763,plain,(
% 3.05/1.75 ($true != $true) | ((member @ b @ x @ (labeled_vertices @ standard_Constant @ a @ b @ g)) = $true)),
% 3.05/1.75 inference(superposition,[status(thm)],[f753,f740])).
% 3.05/1.75 thf(f740,plain,(
% 3.05/1.75 ((member @ b @ (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))) @ x @ (hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b)))))) @ (labeled_vertices @ standard_Constant @ a @ b @ g)) = $true)),
% 3.05/1.75 inference(beta_eta_normalization,[status(thm)],[f724])).
% 3.05/1.75 thf(f724,plain,(
% 3.05/1.75 ($true = (member @ b @ ((^[Y0 : b]: (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b))) @ Y0 @ (hilbert_Eps @ b @ ((^[Y1 : b]: ((^[Y2 : b]: (member @ b @ Y2 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y1 @ bot_bot @ set @ b)))))) @ Y0)))) @ x) @ (labeled_vertices @ standard_Constant @ a @ b @ g)))),
% 3.05/1.75 inference(definition_unfolding,[status(thm)],[f699,f721])).
% 3.05/1.75 thf(f721,plain,(
% 3.05/1.75 (f = (^[Y0 : b]: (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b))) @ Y0 @ (hilbert_Eps @ b @ ((^[Y1 : b]: ((^[Y2 : b]: (member @ b @ Y2 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y1 @ bot_bot @ set @ b)))))) @ Y0)))))),
% 3.05/1.75 inference(definition_unfolding,[status(thm)],[f713,f720])).
% 3.05/1.75 thf(f720,plain,(
% 3.05/1.75 (p = (^[Y0 : b]: ((^[Y1 : b]: (member @ b @ Y1 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b)))))))),
% 3.05/1.75 inference(cnf_transformation,[status(thm)],[f478])).
% 3.05/1.75 thf(f478,plain,(
% 3.05/1.75 (p = (^[Y0 : b]: ((^[Y1 : b]: (member @ b @ Y1 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b)))))))),
% 3.05/1.75 inference(fool_elimination,[status(thm)],[f477])).
% 3.05/1.75 thf(f477,plain,(
% 3.05/1.75 (p = (^[X0 : b, X1 : b] : (member @ b @ X1 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ X0 @ bot_bot @ set @ b)))))),
% 3.05/1.75 inference(rectify,[status(thm)],[f236])).
% 3.05/1.75 thf(f236,axiom,(
% 3.05/1.75 (p = (^[X42 : b, X132 : b] : (member @ b @ X132 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ X42 @ bot_bot @ set @ b)))))),
% 3.05/1.75 file('/export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2THF_32619.p',fact_235_P)).
% 3.05/1.75 thf(f713,plain,(
% 3.05/1.75 (f = (^[Y0 : b]: (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b))) @ Y0 @ (hilbert_Eps @ b @ (p @ Y0)))))),
% 3.05/1.75 inference(cnf_transformation,[status(thm)],[f492])).
% 3.05/1.75 thf(f492,plain,(
% 3.05/1.75 (f = (^[Y0 : b]: (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b))) @ Y0 @ (hilbert_Eps @ b @ (p @ Y0)))))),
% 3.05/1.75 inference(fool_elimination,[status(thm)],[f491])).
% 3.05/1.75 thf(f491,plain,(
% 3.05/1.75 ((^[X0 : b] : (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ X0 @ bot_bot @ set @ b))) @ X0 @ (hilbert_Eps @ b @ (p @ X0)))) = f)),
% 3.05/1.75 inference(rectify,[status(thm)],[f231])).
% 3.05/1.75 thf(f231,axiom,(
% 3.05/1.75 ((^[X42 : b] : (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ X42 @ bot_bot @ set @ b))) @ X42 @ (hilbert_Eps @ b @ (p @ X42)))) = f)),
% 3.05/1.75 file('/export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2THF_32619.p',fact_230_f)).
% 3.05/1.75 thf(f699,plain,(
% 3.05/1.75 ((member @ b @ (f @ x) @ (labeled_vertices @ standard_Constant @ a @ b @ g)) = $true)),
% 3.05/1.75 inference(cnf_transformation,[status(thm)],[f278])).
% 3.05/1.75 thf(f278,plain,(
% 3.05/1.75 ((member @ b @ (f @ x) @ (labeled_vertices @ standard_Constant @ a @ b @ g)) = $true)),
% 3.05/1.75 inference(fool_elimination,[status(thm)],[f277])).
% 3.05/1.75 thf(f277,plain,(
% 3.05/1.75 (member @ b @ (f @ x) @ (labeled_vertices @ standard_Constant @ a @ b @ g))),
% 3.05/1.75 inference(rectify,[status(thm)],[f3])).
% 3.05/1.75 thf(f3,axiom,(
% 3.05/1.75 (member @ b @ (f @ x) @ (labeled_vertices @ standard_Constant @ a @ b @ g))),
% 3.05/1.75 file('/export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2THF_32619.p',fact_2_fv_I1_J)).
% 3.05/1.75 thf(f753,plain,(
% 3.05/1.75 ( ! [X0 : b] : (($true != (member @ b @ (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ X0 @ bot_bot @ set @ b))) @ X0 @ (hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ X0 @ bot_bot @ set @ b)))))) @ (labeled_vertices @ standard_Constant @ a @ b @ g))) | ($true = (member @ b @ X0 @ (labeled_vertices @ standard_Constant @ a @ b @ g)))) )),
% 3.05/1.75 inference(beta_eta_normalization,[status(thm)],[f731])).
% 3.05/1.75 thf(f731,plain,(
% 3.05/1.75 ( ! [X0 : b] : (($true = (member @ b @ X0 @ (labeled_vertices @ standard_Constant @ a @ b @ g))) | ($true != (member @ b @ ((^[Y0 : b]: (if @ b @ (bot_bot @ set @ b = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y0 @ bot_bot @ set @ b))) @ Y0 @ (hilbert_Eps @ b @ ((^[Y1 : b]: ((^[Y2 : b]: (member @ b @ Y2 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ Y1 @ bot_bot @ set @ b)))))) @ Y0)))) @ X0) @ (labeled_vertices @ standard_Constant @ a @ b @ g)))) )),
% 3.05/1.75 inference(definition_unfolding,[status(thm)],[f706,f721])).
% 3.05/1.75 thf(f706,plain,(
% 3.05/1.75 ( ! [X0 : b] : (($true = (member @ b @ X0 @ (labeled_vertices @ standard_Constant @ a @ b @ g))) | ($true != (member @ b @ (f @ X0) @ (labeled_vertices @ standard_Constant @ a @ b @ g)))) )),
% 3.05/1.75 inference(cnf_transformation,[status(thm)],[f689])).
% 3.05/1.75 thf(f689,plain,(
% 3.05/1.75 ! [X0 : b] : (($true = (member @ b @ X0 @ (labeled_vertices @ standard_Constant @ a @ b @ g))) | ($true != (member @ b @ (f @ X0) @ (labeled_vertices @ standard_Constant @ a @ b @ g))))),
% 3.05/1.75 inference(ennf_transformation,[status(thm)],[f564])).
% 3.05/1.75 thf(f564,plain,(
% 3.05/1.75 ! [X0 : b] : (($true = (member @ b @ (f @ X0) @ (labeled_vertices @ standard_Constant @ a @ b @ g))) => ($true = (member @ b @ X0 @ (labeled_vertices @ standard_Constant @ a @ b @ g))))),
% 3.05/1.75 inference(fool_elimination,[status(thm)],[f563])).
% 3.05/1.75 thf(f563,plain,(
% 3.05/1.75 ! [X0 : b] : ((member @ b @ (f @ X0) @ (labeled_vertices @ standard_Constant @ a @ b @ g)) => (member @ b @ X0 @ (labeled_vertices @ standard_Constant @ a @ b @ g)))),
% 3.05/1.75 inference(rectify,[status(thm)],[f1])).
% 3.05/1.75 thf(f1,axiom,(
% 3.05/1.75 ! [X7 : b] : ((member @ b @ (f @ X7) @ (labeled_vertices @ standard_Constant @ a @ b @ g)) => (member @ b @ X7 @ (labeled_vertices @ standard_Constant @ a @ b @ g)))),
% 3.05/1.75 file('/export/starexec/sandbox/tmp/tmp.HZtRuWzkiN/DTF2THF_32619.p',fact_0__092_060open_062_092_060And_062xa_O_A_092_060lbrakk_062f_Axa_A_092_060in_062_Avertices_AG_059_Axa_A_092_060notin_062_Avertices_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062)).
% 3.05/1.75 % SZS output end Proof for DTF2THF_32619
% 3.05/1.75 % (387)------------------------------
% 3.05/1.75 % (387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.75 % (387)Termination reason: Refutation
% 3.05/1.75
% 3.05/1.75 % (387)Memory used [KB]: 6908
% 3.05/1.75 % (387)Time elapsed: 0.048 s
% 3.05/1.75 % (387)Instructions burned: 89 (million)
% 3.05/1.75 % (387)------------------------------
% 3.05/1.75 % (387)------------------------------
% 3.05/1.75 % (342)Success in time 0.136 s
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