%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP175^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n003.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 08:56:10 AM UTC 2026
% Result : Theorem 3.26s 1.71s
% Output : Refutation 3.26s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP175^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n003.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:28:41 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.47 thf(spec,logic,$$dhol).
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % File : ITP175^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.47 % Domain : Interactive Theorem Proving
% 0.20/0.47 % Problem : Sledgehammer StandardRules problem prob_201__5389132_1
% 0.20/0.47 % Version : Especial.
% 0.20/0.47 % English :
% 0.20/0.47
% 0.20/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.47 % Source : [Des21]
% 0.20/0.47 % Names : StandardRules/prob_201__5389132_1 [Des21]
% 0.20/0.47
% 0.20/0.47 % Status : Theorem
% 0.20/0.47 % Rating : 0.00 v7.5.0
% 0.20/0.47 % Syntax : Number of formulae : 348 ( 122 unt; 69 typ; 0 def)
% 0.20/0.47 % Number of atoms : 761 ( 235 equ; 0 cnn)
% 0.20/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.20/0.47 % Number of connectives : 5778 ( 65 ~; 7 |; 39 &;5319 @)
% 0.20/0.47 % ( 0 <=>; 348 =>; 0 <=; 0 <~>)
% 0.20/0.47 % Maximal formula depth : 28 ( 9 avg)
% 0.20/0.47 % Number of types : 4 ( 3 usr)
% 0.20/0.47 % Number of type conns : 186 ( 186 >; 0 *; 0 +; 0 <<)
% 0.20/0.47 % Number of symbols : 67 ( 66 usr; 5 con; 0-8 aty)
% 0.20/0.47 % Number of variables : 1303 ( 50 ^;1150 !; 6 ?;1303 :)
% 0.20/0.47 % ( 97 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.47 % SPC : TH1_THM_EQU_NAR
% 0.20/0.47
% 0.20/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:22:17.769
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % Could-be-implicit typings (8)
% 0.20/0.47 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 0.20/0.47 allegorical_term: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 0.20/0.47 labeled_graph: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.20/0.47 product_prod: $tType > $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Set_Oset,type,
% 0.20/0.47 set: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Nat_Onat,type,
% 0.20/0.47 nat: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_itself,type,
% 0.20/0.47 itself: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_b,type,
% 0.20/0.47 b: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_tf_a,type,
% 0.20/0.47 a: $tType ).
% 0.20/0.47
% 0.20/0.47 % Explicit typings (61)
% 0.20/0.47 thf(sy_cl_HOL_Otype,type,
% 0.20/0.47 type:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.20/0.47 bot:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.47 ord:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.47 order:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.47 linorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.47 preorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.20/0.47 order_bot:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
% 0.20/0.47 bNF_Gr1317331620lImage:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
% 0.20/0.47 bNF_Greatest_univ:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 0.20/0.47 equiv_congruent:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Equiv__Relations_Ocongruent2,type,
% 0.20/0.47 equiv_congruent2:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Equiv__Relations_Oequiv,type,
% 0.20/0.47 equiv_equiv:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Equiv__Relations_Oproj,type,
% 0.20/0.47 equiv_proj:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Equiv__Relations_Oquotient,type,
% 0.20/0.47 equiv_quotient:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 0.20/0.47 getRel:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 0.20/0.47 semantics:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 0.20/0.47 graph_homomorphism:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 0.20/0.47 graph_union:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 0.20/0.47 labeled_LG:
% 0.20/0.47 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 0.20/0.47 labele1974067554_graph:
% 0.20/0.47 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 0.20/0.47 labeled_edges:
% 0.20/0.47 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 0.20/0.47 labeled_vertices:
% 0.20/0.47 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Oon__triple,type,
% 0.20/0.47 on_triple:
% 0.20/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_LabeledGraphs_Orestrict,type,
% 0.20/0.47 restrict:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_MissingRelation_Oidempotent,type,
% 0.20/0.47 idempotent:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 0.20/0.47 order_1409979114der_on:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.20/0.47 order_preorder_on:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.20/0.47 bot_bot:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.47 ord_less_eq:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.20/0.47 product_Pair:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.20/0.47 produc2004651681e_prod:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.20/0.47 product_rec_prod:
% 0.20/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_OId__on,type,
% 0.20/0.47 id_on:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Orefl__on,type,
% 0.20/0.47 refl_on:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Osym,type,
% 0.20/0.47 sym:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Ototal__on,type,
% 0.20/0.47 total_on:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Otrans,type,
% 0.20/0.47 trans:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Oagree__on,type,
% 0.20/0.47 agree_on:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 0.20/0.47 consequence_graph:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Oextensible,type,
% 0.20/0.47 extensible:
% 0.20/0.47 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ X @ V ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Ofin__maintained,type,
% 0.20/0.47 fin_maintained:
% 0.20/0.47 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Oleast,type,
% 0.20/0.47 least:
% 0.20/0.47 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 0.20/0.47 least_559130134_graph:
% 0.20/0.47 !>[X: $tType,L: $tType,V: $tType,C: $tType] : ( ( itself @ X ) > ( set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ) ) > ( labeled_graph @ L @ C ) > ( labeled_graph @ L @ C ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_RulesAndChains_Omaintained,type,
% 0.20/0.47 maintained:
% 0.20/0.47 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OCollect,type,
% 0.20/0.47 collect:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Oinsert,type,
% 0.20/0.47 insert:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Ois__empty,type,
% 0.20/0.47 is_empty:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.20/0.47 is_singleton:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.20/0.47 the_elem:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ocongruence__rule,type,
% 0.20/0.47 standa1068660421e_rule:
% 0.20/0.47 !>[L: $tType] : ( L > L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 0.20/0.47 standa244753842y_rule:
% 0.20/0.47 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule,type,
% 0.20/0.47 standa245363227y_rule:
% 0.20/0.47 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule,type,
% 0.20/0.47 standa1805702094y_rule:
% 0.20/0.47 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule,type,
% 0.20/0.47 standa2114124375e_rule:
% 0.20/0.47 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wfrec_Osame__fst,type,
% 0.20/0.47 same_fst:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member,type,
% 0.20/0.47 member:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_G,type,
% 0.20/0.47 g: labeled_graph @ a @ b ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_I,type,
% 0.20/0.47 i: a ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_l,type,
% 0.20/0.47 l: a ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_y____,type,
% 0.20/0.47 y: b ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_z____,type,
% 0.20/0.47 z: b ).
% 0.20/0.47
% 0.20/0.47 % Relevant facts (256)
% 0.20/0.47 thf(fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062,axiom,
% 0.20/0.47 ( ( refl_on @ b @ ( labeled_vertices @ a @ b @ g ) @ ( getRel @ a @ b @ i @ g ) )
% 0.20/0.47 & ! [X2: b,Y: b] :
% 0.20/0.47 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X2 @ Y ) @ ( getRel @ a @ b @ i @ g ) )
% 0.20/0.47 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ Y @ X2 ) @ ( getRel @ a @ b @ i @ g ) ) )
% 0.20/0.47 & ( trans @ b @ ( getRel @ a @ b @ i @ g ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % \<open>refl_on (vertices G) (getRel I G) \<and> (\<forall>x y. (x, y) \<in> getRel I G \<longrightarrow> (y, x) \<in> getRel I G) \<and> trans (getRel I G)\<close>
% 0.20/0.47 thf(fact_1_aI,axiom,
% 0.20/0.47 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ y @ z ) @ ( getRel @ a @ b @ i @ g ) ).
% 0.20/0.47
% 0.20/0.47 % aI
% 0.20/0.47 thf(fact_2_g,axiom,
% 0.20/0.47 ( g
% 0.20/0.47 = ( restrict @ a @ b @ g ) ) ).
% 0.20/0.47
% 0.20/0.47 % g
% 0.20/0.47 thf(fact_3_m,axiom,
% 0.20/0.47 maintained @ a @ nat @ b @ ( standa1068660421e_rule @ a @ i @ l ) @ g ).
% 0.20/0.47
% 0.20/0.47 % m
% 0.20/0.47 thf(fact_4_eq,axiom,
% 0.20/0.47 equiv_equiv @ b @ ( labeled_vertices @ a @ b @ g ) @ ( getRel @ a @ b @ i @ g ) ).
% 0.20/0.47
% 0.20/0.47 % eq
% 0.20/0.47 thf(fact_5_prod_Oinject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.20/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X22 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_6_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.20/0.47 = ( ( A2 = A3 )
% 0.20/0.47 & ( B2 = B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_7_surj__pair,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.20/0.47 ? [X3: A,Y3: B] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_8_prod__cases,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.20/0.47 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.20/0.47 => ( P2 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_9_Pair__inject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.20/0.47 => ~ ( ( A2 = A3 )
% 0.20/0.47 => ( B2 != B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_10_prod__cases3,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.20/0.47 ~ ! [A4: A,B4: B,C2: C] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases3
% 0.20/0.47 thf(fact_11_prod__cases4,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.20/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases4
% 0.20/0.47 thf(fact_12_prod__cases5,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.20/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases5
% 0.20/0.47 thf(fact_13_prod__cases6,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.20/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases6
% 0.20/0.47 thf(fact_14_prod__cases7,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.20/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases7
% 0.20/0.47 thf(fact_15_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.20/0.47 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.20/0.47 => ( P2 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_16_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
% 0.20/0.47 ~ ! [A4: A,B4: B] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_17_prod__induct7,axiom,
% 0.20/0.47 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.20/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct7
% 0.20/0.47 thf(fact_18_prod__induct6,axiom,
% 0.20/0.47 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.20/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct6
% 0.20/0.47 thf(fact_19_prod__induct5,axiom,
% 0.20/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.20/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct5
% 0.20/0.47 thf(fact_20_prod__induct4,axiom,
% 0.20/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.20/0.47 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct4
% 0.20/0.47 thf(fact_21_prod__induct3,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct3
% 0.20/0.47 thf(fact_22_vertices__restrict,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,G3: labeled_graph @ B @ A] :
% 0.20/0.47 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G3 ) )
% 0.20/0.47 = ( labeled_vertices @ B @ A @ G3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % vertices_restrict
% 0.20/0.47 thf(fact_23_getRel__dom_I1_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G3 ) )
% 0.20/0.47 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % getRel_dom(1)
% 0.20/0.47 thf(fact_24_getRel__dom_I2_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G3 ) )
% 0.20/0.47 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % getRel_dom(2)
% 0.20/0.47 thf(fact_25_transitive__rule,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ( maintained @ A @ nat @ B @ ( standa2114124375e_rule @ A @ L2 ) @ G3 )
% 0.20/0.47 => ( trans @ B @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transitive_rule
% 0.20/0.47 thf(fact_26_old_Oprod_Orec,axiom,
% 0.20/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.20/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.20/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.rec
% 0.20/0.47 thf(fact_27_maintained__refl,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,R: labeled_graph @ A @ B,G3: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G3 ) ).
% 0.20/0.47
% 0.20/0.47 % maintained_refl
% 0.20/0.47 thf(fact_28_reflexivity__rule,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ( maintained @ A @ nat @ B @ ( standa245363227y_rule @ A @ L2 ) @ G3 )
% 0.20/0.47 => ( refl_on @ B @ ( labeled_vertices @ A @ B @ G3 ) @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflexivity_rule
% 0.20/0.47 thf(fact_29_restrict__idemp,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X4: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X4 ) )
% 0.20/0.47 = ( restrict @ A @ B @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % restrict_idemp
% 0.20/0.47 thf(fact_30_refl__on__domain,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.20/0.47 ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 0.20/0.47 => ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 & ( member @ A @ B2 @ A5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_31_refl__onD2,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 0.20/0.47 ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member @ A @ Y4 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_32_transD,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A,Z: A] :
% 0.20/0.47 ( ( trans @ A @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transD
% 0.20/0.47 thf(fact_33_transE,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A,Z: A] :
% 0.20/0.47 ( ( trans @ A @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transE
% 0.20/0.47 thf(fact_34_transI,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ! [X3: A,Y3: A,Z2: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) )
% 0.20/0.47 => ( trans @ A @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % transI
% 0.20/0.47 thf(fact_35_Relation_Otrans__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( trans @ A )
% 0.20/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ! [X5: A,Y5: A,Z3: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R3 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Relation.trans_def
% 0.20/0.47 thf(fact_36_refl__onD,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
% 0.20/0.47 ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_37_refl__onD1,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 0.20/0.47 ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member @ A @ X4 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_38_mA,axiom,
% 0.20/0.47 ! [X2: product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat )] :
% 0.20/0.47 ( ( member @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) @ X2 @ ( 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 ) ) ) ) ) ) ) )
% 0.20/0.47 => ( maintained @ a @ nat @ b @ X2 @ g ) ) ).
% 0.20/0.47
% 0.20/0.47 % mA
% 0.20/0.47 thf(fact_39_internal__case__prod__conv,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.20/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.20/0.47 = ( C3 @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % internal_case_prod_conv
% 0.20/0.47 thf(fact_40_preorder__on__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order_preorder_on @ A )
% 0.20/0.47 = ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( refl_on @ A @ A6 @ R3 )
% 0.20/0.47 & ( trans @ A @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_def
% 0.20/0.47 thf(fact_41_on__triple,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 0.20/0.47 = ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L22 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_42_maintained__then__fin__maintained,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G3: labeled_graph @ A @ C] :
% 0.20/0.47 ( ( maintained @ A @ B @ C @ R @ G3 )
% 0.20/0.47 => ( fin_maintained @ A @ B @ C @ R @ G3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintained_then_fin_maintained
% 0.20/0.47 thf(fact_43_refl__trans__impl__idempotent,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( trans @ A @ R2 )
% 0.20/0.47 => ( idempotent @ A @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_trans_impl_idempotent
% 0.20/0.47 thf(fact_44_equivalence,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,I: A] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ! [X3: product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat )] :
% 0.20/0.47 ( ( member @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ X3 @ ( 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 ) ) ) ) ) ) ) )
% 0.20/0.47 => ( maintained @ A @ nat @ B @ X3 @ G3 ) )
% 0.20/0.47 => ( equiv_equiv @ B @ ( labeled_vertices @ A @ B @ G3 ) @ ( getRel @ A @ B @ I @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivalence
% 0.20/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,P2: A > $o] :
% 0.20/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.20/0.47 = ( P2 @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ( collect @ A
% 0.20/0.47 @ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
% 0.20/0.47 = A5 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_47_Collect__cong,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( P2 @ X3 )
% 0.20/0.47 = ( Q @ X3 ) )
% 0.20/0.47 => ( ( collect @ A @ P2 )
% 0.20/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_48_ext,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F3: A > B,G4: A > B] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( F3 @ X3 )
% 0.20/0.47 = ( G4 @ X3 ) )
% 0.20/0.47 => ( F3 = G4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % ext
% 0.20/0.47 thf(fact_49_semantics__in__vertices_I1_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( restrict @ A @ B @ A5 ) )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.20/0.47 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % semantics_in_vertices(1)
% 0.20/0.47 thf(fact_50_semantics__in__vertices_I2_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( restrict @ A @ B @ A5 ) )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.20/0.47 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % semantics_in_vertices(2)
% 0.20/0.47 thf(fact_51_refl__on__singleton,axiom,
% 0.20/0.47 ! [A: $tType,X4: A] : ( refl_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_52_preorder__on__empty,axiom,
% 0.20/0.47 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_empty
% 0.20/0.47 thf(fact_53_refl__on__empty,axiom,
% 0.20/0.47 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_54_trans__singleton,axiom,
% 0.20/0.47 ! [A: $tType,A2: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_singleton
% 0.20/0.47 thf(fact_55_idempotent__impl__trans,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( idempotent @ A @ R2 )
% 0.20/0.47 => ( trans @ A @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % idempotent_impl_trans
% 0.20/0.47 thf(fact_56_trans__empty,axiom,
% 0.20/0.47 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_empty
% 0.20/0.47 thf(fact_57_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 0.20/0.47 => ( L22 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_58_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 0.20/0.47 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_59_on__tripleD_I3_J,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 0.20/0.47 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V2 @ V4 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(3)
% 0.20/0.47 thf(fact_60_singletonI,axiom,
% 0.20/0.47 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % singletonI
% 0.20/0.47 thf(fact_61_nonempty__rule,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,G3: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G3 )
% 0.20/0.47 = ( ( labeled_vertices @ A @ B @ G3 )
% 0.20/0.47 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonempty_rule
% 0.20/0.47 thf(fact_62_insertCI,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.20/0.47 ( ( ~ ( member @ A @ A2 @ B5 )
% 0.20/0.47 => ( A2 = B2 ) )
% 0.20/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertCI
% 0.20/0.47 thf(fact_63_insert__iff,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 0.20/0.47 = ( ( A2 = B2 )
% 0.20/0.47 | ( member @ A @ A2 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_iff
% 0.20/0.47 thf(fact_64_insert__absorb2,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,A5: set @ A] :
% 0.20/0.47 ( ( insert @ A @ X4 @ ( insert @ A @ X4 @ A5 ) )
% 0.20/0.47 = ( insert @ A @ X4 @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_absorb2
% 0.20/0.47 thf(fact_65_empty__iff,axiom,
% 0.20/0.47 ! [A: $tType,C3: A] :
% 0.20/0.47 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % empty_iff
% 0.20/0.47 thf(fact_66_all__not__in__conv,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ( ! [X5: A] :
% 0.20/0.47 ~ ( member @ A @ X5 @ A5 ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_not_in_conv
% 0.20/0.47 thf(fact_67_Collect__empty__eq,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o] :
% 0.20/0.47 ( ( ( collect @ A @ P2 )
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 = ( ! [X5: A] :
% 0.20/0.47 ~ ( P2 @ X5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_empty_eq
% 0.20/0.47 thf(fact_68_empty__Collect__eq,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o] :
% 0.20/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.20/0.47 = ( collect @ A @ P2 ) )
% 0.20/0.47 = ( ! [X5: A] :
% 0.20/0.47 ~ ( P2 @ X5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % empty_Collect_eq
% 0.20/0.47 thf(fact_69_ex__in__conv,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ( ? [X5: A] : ( member @ A @ X5 @ A5 ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ex_in_conv
% 0.20/0.47 thf(fact_70_equals0I,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ! [Y3: A] :
% 0.20/0.47 ~ ( member @ A @ Y3 @ A5 )
% 0.20/0.47 => ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equals0I
% 0.20/0.47 thf(fact_71_equals0D,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,A2: A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ~ ( member @ A @ A2 @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % equals0D
% 0.20/0.47 thf(fact_72_emptyE,axiom,
% 0.20/0.47 ! [A: $tType,A2: A] :
% 0.20/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % emptyE
% 0.20/0.47 thf(fact_73_mk__disjoint__insert,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ? [B6: set @ A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( insert @ A @ A2 @ B6 ) )
% 0.20/0.47 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mk_disjoint_insert
% 0.20/0.47 thf(fact_74_insert__commute,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,Y4: A,A5: set @ A] :
% 0.20/0.47 ( ( insert @ A @ X4 @ ( insert @ A @ Y4 @ A5 ) )
% 0.20/0.47 = ( insert @ A @ Y4 @ ( insert @ A @ X4 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_commute
% 0.20/0.47 thf(fact_75_insert__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,A5: set @ A,B2: A,B5: set @ A] :
% 0.20/0.47 ( ~ ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ( ~ ( member @ A @ B2 @ B5 )
% 0.20/0.47 => ( ( ( insert @ A @ A2 @ A5 )
% 0.20/0.47 = ( insert @ A @ B2 @ B5 ) )
% 0.20/0.47 = ( ( ( A2 = B2 )
% 0.20/0.47 => ( A5 = B5 ) )
% 0.20/0.47 & ( ( A2 != B2 )
% 0.20/0.47 => ? [C4: set @ A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( insert @ A @ B2 @ C4 ) )
% 0.20/0.47 & ~ ( member @ A @ B2 @ C4 )
% 0.20/0.47 & ( B5
% 0.20/0.47 = ( insert @ A @ A2 @ C4 ) )
% 0.20/0.47 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_eq_iff
% 0.20/0.47 thf(fact_76_insert__absorb,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ( ( insert @ A @ A2 @ A5 )
% 0.20/0.47 = A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_absorb
% 0.20/0.47 thf(fact_77_insert__ident,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ~ ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ( ~ ( member @ A @ X4 @ B5 )
% 0.20/0.47 => ( ( ( insert @ A @ X4 @ A5 )
% 0.20/0.47 = ( insert @ A @ X4 @ B5 ) )
% 0.20/0.47 = ( A5 = B5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_ident
% 0.20/0.47 thf(fact_78_Set_Oset__insert,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ~ ! [B6: set @ A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( insert @ A @ X4 @ B6 ) )
% 0.20/0.47 => ( member @ A @ X4 @ B6 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Set.set_insert
% 0.20/0.47 thf(fact_79_insertI2,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.20/0.47 ( ( member @ A @ A2 @ B5 )
% 0.20/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertI2
% 0.20/0.47 thf(fact_80_insertI1,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertI1
% 0.20/0.47 thf(fact_81_insertE,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 0.20/0.47 => ( ( A2 != B2 )
% 0.20/0.47 => ( member @ A @ A2 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertE
% 0.20/0.47 thf(fact_82_singleton__inject,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B2: A] :
% 0.20/0.47 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 => ( A2 = B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_inject
% 0.20/0.47 thf(fact_83_insert__not__empty,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.20/0.47 ( ( insert @ A @ A2 @ A5 )
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_not_empty
% 0.20/0.47 thf(fact_84_doubleton__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
% 0.20/0.47 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 = ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.47 = ( ( ( A2 = C3 )
% 0.20/0.47 & ( B2 = D3 ) )
% 0.20/0.47 | ( ( A2 = D3 )
% 0.20/0.47 & ( B2 = C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % doubleton_eq_iff
% 0.20/0.47 thf(fact_85_singleton__iff,axiom,
% 0.20/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.20/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 = ( B2 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_iff
% 0.20/0.47 thf(fact_86_singletonD,axiom,
% 0.20/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.20/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 => ( B2 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % singletonD
% 0.20/0.47 thf(fact_87_the__elem__eq,axiom,
% 0.20/0.47 ! [A: $tType,X4: A] :
% 0.20/0.47 ( ( the_elem @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 = X4 ) ).
% 0.20/0.47
% 0.20/0.47 % the_elem_eq
% 0.20/0.47 thf(fact_88_bot__apply,axiom,
% 0.20/0.47 ! [C: $tType,D: $tType] :
% 0.20/0.47 ( ( bot @ C )
% 0.20/0.47 => ( ( bot_bot @ ( D > C ) )
% 0.20/0.47 = ( ^ [X5: D] : ( bot_bot @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_apply
% 0.20/0.47 thf(fact_89_graph__single,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,C3: B] :
% 0.20/0.47 ( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C3 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.20/0.47 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C3 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_single
% 0.20/0.47 thf(fact_90_is__singletonI,axiom,
% 0.20/0.47 ! [A: $tType,X4: A] : ( is_singleton @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonI
% 0.20/0.47 thf(fact_91_linear__order__on__singleton,axiom,
% 0.20/0.47 ! [A: $tType,X4: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linear_order_on_singleton
% 0.20/0.47 thf(fact_92_total__on__singleton,axiom,
% 0.20/0.47 ! [A: $tType,X4: A] : ( total_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_singleton
% 0.20/0.47 thf(fact_93_symmetry__rule,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
% 0.20/0.47 ( ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) )
% 0.20/0.47 => ( ( maintained @ A @ nat @ B @ ( standa1805702094y_rule @ A @ L2 ) @ G3 )
% 0.20/0.47 => ( sym @ B @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % symmetry_rule
% 0.20/0.47 thf(fact_94_labeled__graph_Oinject,axiom,
% 0.20/0.47 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V,Y1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),Y2: set @ V] :
% 0.20/0.47 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 0.20/0.47 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X22 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.inject
% 0.20/0.47 thf(fact_95_graph__empty__e,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,V5: set @ B] :
% 0.20/0.47 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V5 )
% 0.20/0.47 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_empty_e
% 0.20/0.47 thf(fact_96_labeled__graph_Oexhaust,axiom,
% 0.20/0.47 ! [L: $tType,V: $tType,Y4: labeled_graph @ L @ V] :
% 0.20/0.47 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.exhaust
% 0.20/0.47 thf(fact_97_labeled__graph_Oinduct,axiom,
% 0.20/0.47 ! [V: $tType,L: $tType,P2: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 0.20/0.47 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P2 @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 0.20/0.47 => ( P2 @ Labeled_graph ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.induct
% 0.20/0.47 thf(fact_98_bot__empty__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.20/0.47 = ( ^ [X5: A] : ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_empty_eq
% 0.20/0.47 thf(fact_99_bot__set__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_set_def
% 0.20/0.47 thf(fact_100_sym__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( sym @ A )
% 0.20/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ! [X5: A,Y5: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sym_def
% 0.20/0.47 thf(fact_101_symI,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ! [A4: A,B4: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 ) )
% 0.20/0.47 => ( sym @ A @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % symI
% 0.20/0.47 thf(fact_102_symE,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
% 0.20/0.47 ( ( sym @ A @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % symE
% 0.20/0.47 thf(fact_103_symD,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
% 0.20/0.47 ( ( sym @ A @ R2 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % symD
% 0.20/0.47 thf(fact_104_total__on__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( total_on @ A )
% 0.20/0.47 = ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A6 )
% 0.20/0.47 => ! [Y5: A] :
% 0.20/0.47 ( ( member @ A @ Y5 @ A6 )
% 0.20/0.47 => ( ( X5 != Y5 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_def
% 0.20/0.47 thf(fact_105_total__onI,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ! [X3: A,Y3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A5 )
% 0.20/0.47 => ( ( member @ A @ Y3 @ A5 )
% 0.20/0.47 => ( ( X3 != Y3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.20/0.47 => ( total_on @ A @ A5 @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_onI
% 0.20/0.47 thf(fact_106_total__on__empty,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_empty
% 0.20/0.47 thf(fact_107_labeled__graph_Osel_I2_J,axiom,
% 0.20/0.47 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.20/0.47 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.20/0.47 = X22 ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.sel(2)
% 0.20/0.47 thf(fact_108_is__singleton__the__elem,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( is_singleton @ A )
% 0.20/0.47 = ( ^ [A6: set @ A] :
% 0.20/0.47 ( A6
% 0.20/0.47 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singleton_the_elem
% 0.20/0.47 thf(fact_109_is__singletonI_H,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ( ! [X3: A,Y3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A5 )
% 0.20/0.47 => ( ( member @ A @ Y3 @ A5 )
% 0.20/0.47 => ( X3 = Y3 ) ) )
% 0.20/0.47 => ( is_singleton @ A @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonI'
% 0.20/0.47 thf(fact_110_lnear__order__on__empty,axiom,
% 0.20/0.47 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lnear_order_on_empty
% 0.20/0.47 thf(fact_111_bot__fun__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( bot @ B )
% 0.20/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.20/0.47 = ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_fun_def
% 0.20/0.47 thf(fact_112_is__singletonE,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ( is_singleton @ A @ A5 )
% 0.20/0.47 => ~ ! [X3: A] :
% 0.20/0.47 ( A5
% 0.20/0.47 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonE
% 0.20/0.47 thf(fact_113_is__singleton__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( is_singleton @ A )
% 0.20/0.47 = ( ^ [A6: set @ A] :
% 0.20/0.47 ? [X5: A] :
% 0.20/0.47 ( A6
% 0.20/0.47 = ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singleton_def
% 0.20/0.47 thf(fact_114_equivE,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ~ ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( sym @ A @ R2 )
% 0.20/0.47 => ~ ( trans @ A @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivE
% 0.20/0.47 thf(fact_115_equivI,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( refl_on @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( sym @ A @ R2 )
% 0.20/0.47 => ( ( trans @ A @ R2 )
% 0.20/0.47 => ( equiv_equiv @ A @ A5 @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivI
% 0.20/0.47 thf(fact_116_equiv__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( equiv_equiv @ A )
% 0.20/0.47 = ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( refl_on @ A @ A6 @ R3 )
% 0.20/0.47 & ( sym @ A @ R3 )
% 0.20/0.47 & ( trans @ A @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equiv_def
% 0.20/0.47 thf(fact_117_agree__on__empty,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,F3: set @ ( product_prod @ B @ C ),G4: 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 ) ) ) @ F3 @ G4 ) ).
% 0.20/0.47
% 0.20/0.47 % agree_on_empty
% 0.20/0.47 thf(fact_118_graph__homomorphism__empty,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G3 @ F3 )
% 0.20/0.47 = ( ( F3
% 0.20/0.47 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 0.20/0.47 & ( G3
% 0.20/0.47 = ( restrict @ A @ C @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_homomorphism_empty
% 0.20/0.47 thf(fact_119_agree__on__comm,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.20/0.47 ( ( agree_on @ A @ B @ C )
% 0.20/0.47 = ( ^ [X6: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X6 @ G5 @ F4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_on_comm
% 0.20/0.47 thf(fact_120_agree__on__refl,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R @ F3 @ F3 ) ).
% 0.20/0.47
% 0.20/0.47 % agree_on_refl
% 0.20/0.47 thf(fact_121_agree__on__trans,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,X7: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C ),G4: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( agree_on @ A @ B @ C @ X7 @ F3 @ G4 )
% 0.20/0.47 => ( ( agree_on @ A @ B @ C @ X7 @ G4 @ H )
% 0.20/0.47 => ( agree_on @ A @ B @ C @ X7 @ F3 @ H ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_on_trans
% 0.20/0.47 thf(fact_122_graph__homomorphism__semantics,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F3 )
% 0.20/0.47 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B5 @ E3 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_homomorphism_semantics
% 0.20/0.47 thf(fact_123_graph__homomorphism__nonempty,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F3 )
% 0.20/0.47 => ( ( ( semantics @ A @ B @ A5 @ E3 )
% 0.20/0.47 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.20/0.47 => ( ( semantics @ A @ C @ B5 @ E3 )
% 0.20/0.47 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_homomorphism_nonempty
% 0.20/0.47 thf(fact_124_Collect__empty__eq__bot,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o] :
% 0.20/0.47 ( ( ( collect @ A @ P2 )
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 = ( P2
% 0.20/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_empty_eq_bot
% 0.20/0.47 thf(fact_125_extensibleI,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,G4: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G3 @ G4 )
% 0.20/0.47 => ( ( agree_on @ A @ B @ C @ R1 @ F3 @ G4 )
% 0.20/0.47 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G3 @ F3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % extensibleI
% 0.20/0.47 thf(fact_126_maintainedI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G3: labeled_graph @ C @ B,B5: labeled_graph @ C @ A] :
% 0.20/0.47 ( ! [F2: set @ ( product_prod @ A @ B )] :
% 0.20/0.47 ( ( graph_homomorphism @ C @ A @ B @ A5 @ G3 @ F2 )
% 0.20/0.47 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G3 @ F2 ) )
% 0.20/0.47 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintainedI
% 0.20/0.47 thf(fact_127_extensible__refl__concr,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V5: set @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V5 ) @ G3 @ F3 )
% 0.20/0.47 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V5 ) @ ( labeled_LG @ A @ B @ E_2 @ V5 ) ) @ G3 @ F3 )
% 0.20/0.47 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V5 ) @ G3 @ F3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % extensible_refl_concr
% 0.20/0.47 thf(fact_128_extensible__refl,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ R @ G3 @ F3 )
% 0.20/0.47 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G3 @ F3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % extensible_refl
% 0.20/0.47 thf(fact_129_maintainedD,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 )
% 0.20/0.47 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G3 @ F3 )
% 0.20/0.47 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 @ F3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintainedD
% 0.20/0.47 thf(fact_130_Set_Ois__empty__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( is_empty @ A )
% 0.20/0.47 = ( ^ [A6: set @ A] :
% 0.20/0.47 ( A6
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Set.is_empty_def
% 0.20/0.47 thf(fact_131_congruent2I,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType,A1: set @ A,R12: set @ ( product_prod @ A @ A ),A22: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A1 @ R12 )
% 0.20/0.47 => ( ( equiv_equiv @ B @ A22 @ R23 )
% 0.20/0.47 => ( ! [Y3: A,Z2: A,W: B] :
% 0.20/0.47 ( ( member @ B @ W @ A22 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R12 )
% 0.20/0.47 => ( ( F3 @ Y3 @ W )
% 0.20/0.47 = ( F3 @ Z2 @ W ) ) ) )
% 0.20/0.47 => ( ! [Y3: B,Z2: B,W: A] :
% 0.20/0.47 ( ( member @ A @ W @ A1 )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z2 ) @ R23 )
% 0.20/0.47 => ( ( F3 @ W @ Y3 )
% 0.20/0.47 = ( F3 @ W @ Z2 ) ) ) )
% 0.20/0.47 => ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruent2I
% 0.20/0.47 thf(fact_132_congruent2__commuteI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > A > B] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ! [Y3: A,Z2: A] :
% 0.20/0.47 ( ( member @ A @ Y3 @ A5 )
% 0.20/0.47 => ( ( member @ A @ Z2 @ A5 )
% 0.20/0.47 => ( ( F3 @ Y3 @ Z2 )
% 0.20/0.47 = ( F3 @ Z2 @ Y3 ) ) ) )
% 0.20/0.47 => ( ! [Y3: A,Z2: A,W: A] :
% 0.20/0.47 ( ( member @ A @ W @ A5 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( ( F3 @ W @ Y3 )
% 0.20/0.47 = ( F3 @ W @ Z2 ) ) ) )
% 0.20/0.47 => ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruent2_commuteI
% 0.20/0.47 thf(fact_133_eq__equiv__class__iff2,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ( ( member @ A @ Y4 @ A5 )
% 0.20/0.47 => ( ( ( equiv_quotient @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
% 0.20/0.47 = ( equiv_quotient @ A @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
% 0.20/0.47 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_equiv_class_iff2
% 0.20/0.47 thf(fact_134_quotient__is__empty2,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( ( bot_bot @ ( set @ ( set @ A ) ) )
% 0.20/0.47 = ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % quotient_is_empty2
% 0.20/0.47 thf(fact_135_quotient__is__empty,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( ( equiv_quotient @ A @ A5 @ R2 )
% 0.20/0.47 = ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % quotient_is_empty
% 0.20/0.47 thf(fact_136_quotient__empty,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
% 0.20/0.47 = ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % quotient_empty
% 0.20/0.47 thf(fact_137_in__quotient__imp__closed,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X4: A,Y4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ( member @ A @ X4 @ X7 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member @ A @ Y4 @ X7 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_quotient_imp_closed
% 0.20/0.47 thf(fact_138_quotient__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y6: set @ A,X4: A,Y4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ( member @ A @ X4 @ X7 )
% 0.20/0.47 => ( ( member @ A @ Y4 @ Y6 )
% 0.20/0.47 => ( ( X7 = Y6 )
% 0.20/0.47 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % quotient_eq_iff
% 0.20/0.47 thf(fact_139_quotient__eqI,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y6: set @ A,X4: A,Y4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ( member @ A @ X4 @ X7 )
% 0.20/0.47 => ( ( member @ A @ Y4 @ Y6 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( X7 = Y6 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % quotient_eqI
% 0.20/0.47 thf(fact_140_in__quotient__imp__non__empty,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( X7
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_quotient_imp_non_empty
% 0.20/0.47 thf(fact_141_congruent2I_H,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
% 0.20/0.47 ( ! [Y12: A,Z1: A,Y22: B,Z22: B] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R12 )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z22 ) @ R23 )
% 0.20/0.47 => ( ( F3 @ Y12 @ Y22 )
% 0.20/0.47 = ( F3 @ Z1 @ Z22 ) ) ) )
% 0.20/0.47 => ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruent2I'
% 0.20/0.47 thf(fact_142_congruent2D,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z12: A,Y2: B,Z23: B] :
% 0.20/0.47 ( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R12 )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z23 ) @ R23 )
% 0.20/0.47 => ( ( F3 @ Y1 @ Y2 )
% 0.20/0.47 = ( F3 @ Z12 @ Z23 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruent2D
% 0.20/0.47 thf(fact_143_congruent2__implies__congruent,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,A5: set @ A,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,A2: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R12 )
% 0.20/0.47 => ( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
% 0.20/0.47 => ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ( equiv_congruent @ B @ C @ R23 @ ( F3 @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruent2_implies_congruent
% 0.20/0.47 thf(fact_144_in__quotient__imp__in__rel,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X4: A,Y4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ X7 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_quotient_imp_in_rel
% 0.20/0.47 thf(fact_145_same__fstI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P2: A > $o,X4: A,Y7: B,Y4: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.20/0.47 ( ( P2 @ X4 )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y4 ) @ ( R @ X4 ) )
% 0.20/0.47 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y7 ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % same_fstI
% 0.20/0.47 thf(fact_146_getRel__subgraph,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Y4: A,Z: A,L2: B,G3: labeled_graph @ B @ A,G6: labeled_graph @ B @ A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( getRel @ B @ A @ L2 @ G3 ) )
% 0.20/0.47 => ( ( graph_homomorphism @ B @ A @ A @ G3 @ G6 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G3 ) ) )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( getRel @ B @ A @ L2 @ G6 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % getRel_subgraph
% 0.20/0.47 thf(fact_147_order__refl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( preorder @ A )
% 0.20/0.47 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_refl
% 0.20/0.47 thf(fact_148_subset__antisym,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.20/0.47 => ( A5 = B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_antisym
% 0.20/0.47 thf(fact_149_subsetI,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A5 )
% 0.20/0.47 => ( member @ A @ X3 @ B5 ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % subsetI
% 0.20/0.47 thf(fact_150_empty__subsetI,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 0.20/0.47
% 0.20/0.47 % empty_subsetI
% 0.20/0.47 thf(fact_151_subset__empty,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_empty
% 0.20/0.47 thf(fact_152_insert__subset,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ A5 ) @ B5 )
% 0.20/0.47 = ( ( member @ A @ X4 @ B5 )
% 0.20/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_subset
% 0.20/0.47 thf(fact_153_Id__onI,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Id_onI
% 0.20/0.47 thf(fact_154_singleton__insert__inj__eq,axiom,
% 0.20/0.47 ! [A: $tType,B2: A,A2: A,A5: set @ A] :
% 0.20/0.47 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 = ( insert @ A @ A2 @ A5 ) )
% 0.20/0.47 = ( ( A2 = B2 )
% 0.20/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_insert_inj_eq
% 0.20/0.47 thf(fact_155_singleton__insert__inj__eq_H,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,A5: set @ A,B2: A] :
% 0.20/0.47 ( ( ( insert @ A @ A2 @ A5 )
% 0.20/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 = ( ( A2 = B2 )
% 0.20/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_insert_inj_eq'
% 0.20/0.47 thf(fact_156_Id__on__empty,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Id_on_empty
% 0.20/0.47 thf(fact_157_subgraph__refl,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ G3 @ G3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G3 ) ) )
% 0.20/0.47 = ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_refl
% 0.20/0.47 thf(fact_158_subgraph__restrict,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ G3 @ ( restrict @ A @ B @ G3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G3 ) ) )
% 0.20/0.47 = ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_restrict
% 0.20/0.47 thf(fact_159_graph__homomorphism__Id,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A2 ) @ ( restrict @ A @ B @ A2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_homomorphism_Id
% 0.20/0.47 thf(fact_160_subgraph__subset_I1_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_subset(1)
% 0.20/0.47 thf(fact_161_Id__onE,axiom,
% 0.20/0.47 ! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
% 0.20/0.47 => ~ ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A5 )
% 0.20/0.47 => ( C3
% 0.20/0.47 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Id_onE
% 0.20/0.47 thf(fact_162_Id__on__eqI,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 0.20/0.47 ( ( A2 = B2 )
% 0.20/0.47 => ( ( member @ A @ A2 @ A5 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Id_on_eqI
% 0.20/0.47 thf(fact_163_Id__on__iff,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,Y4: A,A5: set @ A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( id_on @ A @ A5 ) )
% 0.20/0.47 = ( ( X4 = Y4 )
% 0.20/0.47 & ( member @ A @ X4 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Id_on_iff
% 0.20/0.47 thf(fact_164_bot_Oextremum__uniqueI,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order_bot @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.20/0.47 => ( A2
% 0.20/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot.extremum_uniqueI
% 0.20/0.47 thf(fact_165_bot_Oextremum__unique,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order_bot @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.20/0.47 = ( A2
% 0.20/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot.extremum_unique
% 0.20/0.47 thf(fact_166_bot_Oextremum,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order_bot @ A )
% 0.20/0.47 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot.extremum
% 0.20/0.47 thf(fact_167_subrelI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.20/0.47 ( ! [X3: A,Y3: B] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% 0.20/0.47
% 0.20/0.47 % subrelI
% 0.20/0.47 thf(fact_168_insert__mono,axiom,
% 0.20/0.47 ! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_mono
% 0.20/0.47 thf(fact_169_subset__insert,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ~ ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ B5 ) )
% 0.20/0.47 = ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_insert
% 0.20/0.47 thf(fact_170_subset__insertI,axiom,
% 0.20/0.47 ! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_insertI
% 0.20/0.47 thf(fact_171_subset__insertI2,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,B2: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_insertI2
% 0.20/0.47 thf(fact_172_trans__Id__on,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] : ( trans @ A @ ( id_on @ A @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_Id_on
% 0.20/0.47 thf(fact_173_refl__on__Id__on,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] : ( refl_on @ A @ A5 @ ( id_on @ A @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_Id_on
% 0.20/0.47 thf(fact_174_Collect__mono__iff,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.20/0.47 = ( ! [X5: A] :
% 0.20/0.47 ( ( P2 @ X5 )
% 0.20/0.47 => ( Q @ X5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mono_iff
% 0.20/0.47 thf(fact_175_set__eq__subset,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ^ [Y8: set @ A,Z4: set @ A] : ( Y8 = Z4 ) )
% 0.20/0.47 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.20/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_eq_subset
% 0.20/0.47 thf(fact_176_subset__trans,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C5: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ C5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_trans
% 0.20/0.47 thf(fact_177_Collect__mono,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( P2 @ X3 )
% 0.20/0.47 => ( Q @ X3 ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mono
% 0.20/0.47 thf(fact_178_subset__refl,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.20/0.47
% 0.20/0.47 % subset_refl
% 0.20/0.47 thf(fact_179_subset__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.20/0.47 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.20/0.47 ! [T2: A] :
% 0.20/0.47 ( ( member @ A @ T2 @ A6 )
% 0.20/0.47 => ( member @ A @ T2 @ B7 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_iff
% 0.20/0.47 thf(fact_180_equalityD2,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ( A5 = B5 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % equalityD2
% 0.20/0.47 thf(fact_181_equalityD1,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ( A5 = B5 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % equalityD1
% 0.20/0.47 thf(fact_182_subset__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.20/0.47 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.20/0.47 ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A6 )
% 0.20/0.47 => ( member @ A @ X5 @ B7 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_eq
% 0.20/0.47 thf(fact_183_equalityE,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.20/0.47 ( ( A5 = B5 )
% 0.20/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.20/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equalityE
% 0.20/0.47 thf(fact_184_subsetD,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.20/0.47 => ( ( member @ A @ C3 @ A5 )
% 0.20/0.47 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subsetD
% 0.20/0.47 thf(fact_185_in__mono,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,X4: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.20/0.47 => ( ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ( member @ A @ X4 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_mono
% 0.20/0.47 thf(fact_186_dual__order_Oantisym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [B2: A,A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.antisym
% 0.20/0.47 thf(fact_187_dual__order_Oeq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ( ( ^ [Y8: A,Z4: A] : ( Y8 = Z4 ) )
% 0.20/0.47 = ( ^ [A7: A,B8: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ B8 @ A7 )
% 0.20/0.47 & ( ord_less_eq @ A @ A7 @ B8 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.eq_iff
% 0.20/0.47 thf(fact_188_dual__order_Otrans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.20/0.47 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.trans
% 0.20/0.47 thf(fact_189_linorder__wlog,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [P2: A > A > $o,A2: A,B2: A] :
% 0.20/0.47 ( ! [A4: A,B4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.47 => ( P2 @ A4 @ B4 ) )
% 0.20/0.47 => ( ! [A4: A,B4: A] :
% 0.20/0.47 ( ( P2 @ B4 @ A4 )
% 0.20/0.47 => ( P2 @ A4 @ B4 ) )
% 0.20/0.47 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linorder_wlog
% 0.20/0.47 thf(fact_190_dual__order_Orefl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.refl
% 0.20/0.47 thf(fact_191_order__trans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( preorder @ A )
% 0.20/0.47 => ! [X4: A,Y4: A,Z: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ Y4 @ Z )
% 0.20/0.47 => ( ord_less_eq @ A @ X4 @ Z ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_trans
% 0.20/0.47 thf(fact_192_order__class_Oorder_Oantisym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A,B2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.20/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_class.order.antisym
% 0.20/0.47 thf(fact_193_ord__le__eq__trans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( B2 = C3 )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_le_eq_trans
% 0.20/0.47 thf(fact_194_ord__eq__le__trans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.20/0.47 ( ( A2 = B2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_eq_le_trans
% 0.20/0.47 thf(fact_195_order__class_Oorder_Oeq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ( ( ^ [Y8: A,Z4: A] : ( Y8 = Z4 ) )
% 0.20/0.47 = ( ^ [A7: A,B8: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A7 @ B8 )
% 0.20/0.47 & ( ord_less_eq @ A @ B8 @ A7 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_class.order.eq_iff
% 0.20/0.47 thf(fact_196_antisym__conv,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [Y4: A,X4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ Y4 @ X4 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.20/0.47 = ( X4 = Y4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % antisym_conv
% 0.20/0.47 thf(fact_197_le__cases3,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X4: A,Y4: A,Z: A] :
% 0.20/0.47 ( ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Y4 @ Z ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ X4 )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ X4 @ Z ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ X4 @ Z )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Z @ Y4 ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ Z @ Y4 )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Y4 @ X4 ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ Z )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Z @ X4 ) )
% 0.20/0.47 => ~ ( ( ord_less_eq @ A @ Z @ X4 )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_cases3
% 0.20/0.47 thf(fact_198_order_Otrans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.trans
% 0.20/0.47 thf(fact_199_le__cases,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X4: A,Y4: A] :
% 0.20/0.47 ( ~ ( ord_less_eq @ A @ X4 @ Y4 )
% 0.20/0.47 => ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_cases
% 0.20/0.47 thf(fact_200_eq__refl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( preorder @ A )
% 0.20/0.47 => ! [X4: A,Y4: A] :
% 0.20/0.47 ( ( X4 = Y4 )
% 0.20/0.47 => ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_refl
% 0.20/0.47 thf(fact_201_linear,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X4: A,Y4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.20/0.47 | ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linear
% 0.20/0.47 thf(fact_202_antisym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [X4: A,Y4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ Y4 @ X4 )
% 0.20/0.47 => ( X4 = Y4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % antisym
% 0.20/0.47 thf(fact_203_eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ( ( ^ [Y8: A,Z4: A] : ( Y8 = Z4 ) )
% 0.20/0.47 = ( ^ [X5: A,Y5: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 0.20/0.47 & ( ord_less_eq @ A @ Y5 @ X5 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_iff
% 0.20/0.47 thf(fact_204_ord__le__eq__subst,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( ( ord @ B )
% 0.20/0.47 & ( ord @ A ) )
% 0.20/0.47 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( ( F3 @ B2 )
% 0.20/0.47 = C3 )
% 0.20/0.47 => ( ! [X3: A,Y3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.20/0.47 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_le_eq_subst
% 0.20/0.47 thf(fact_205_ord__eq__le__subst,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( ( ord @ B )
% 0.20/0.47 & ( ord @ A ) )
% 0.20/0.47 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( F3 @ B2 ) )
% 0.20/0.47 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.20/0.47 => ( ! [X3: B,Y3: B] :
% 0.20/0.47 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.20/0.47 => ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_eq_le_subst
% 0.20/0.47 thf(fact_206_order__subst2,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType] :
% 0.20/0.47 ( ( ( order @ C )
% 0.20/0.47 & ( order @ A ) )
% 0.20/0.47 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 0.20/0.47 => ( ! [X3: A,Y3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.20/0.47 => ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_subst2
% 0.20/0.47 thf(fact_207_order__subst1,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( ( order @ B )
% 0.20/0.47 & ( order @ A ) )
% 0.20/0.47 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.20/0.47 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.20/0.47 => ( ! [X3: B,Y3: B] :
% 0.20/0.47 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.20/0.47 => ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_subst1
% 0.20/0.47 thf(fact_208_le__fun__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ord @ B )
% 0.20/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.20/0.47 = ( ^ [F4: A > B,G5: A > B] :
% 0.20/0.47 ! [X5: A] : ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_fun_def
% 0.20/0.47 thf(fact_209_le__funI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ord @ B )
% 0.20/0.47 => ! [F3: A > B,G4: A > B] :
% 0.20/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G4 @ X3 ) )
% 0.20/0.47 => ( ord_less_eq @ ( A > B ) @ F3 @ G4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_funI
% 0.20/0.47 thf(fact_210_le__funE,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ord @ B )
% 0.20/0.47 => ! [F3: A > B,G4: A > B,X4: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G4 )
% 0.20/0.47 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_funE
% 0.20/0.47 thf(fact_211_le__funD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ord @ B )
% 0.20/0.47 => ! [F3: A > B,G4: A > B,X4: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G4 )
% 0.20/0.47 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_funD
% 0.20/0.47 thf(fact_212_sym__Id__on,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] : ( sym @ A @ ( id_on @ A @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sym_Id_on
% 0.20/0.47 thf(fact_213_subset__singletonD,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,X4: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 => ( ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 | ( A5
% 0.20/0.47 = ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_singletonD
% 0.20/0.47 thf(fact_214_subset__singleton__iff,axiom,
% 0.20/0.47 ! [A: $tType,X7: set @ A,A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 = ( ( X7
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 | ( X7
% 0.20/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_singleton_iff
% 0.20/0.47 thf(fact_215_subgraph__trans,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.20/0.47 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 0.20/0.47 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_trans
% 0.20/0.47 thf(fact_216_subgraph__preserves__hom,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,X7: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.20/0.47 => ( ( graph_homomorphism @ A @ C @ B @ X7 @ A5 @ H )
% 0.20/0.47 => ( graph_homomorphism @ A @ C @ B @ X7 @ B5 @ H ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_preserves_hom
% 0.20/0.47 thf(fact_217_congruentD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B,Y4: A,Z: A] :
% 0.20/0.47 ( ( equiv_congruent @ A @ B @ R2 @ F3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( ( F3 @ Y4 )
% 0.20/0.47 = ( F3 @ Z ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruentD
% 0.20/0.47 thf(fact_218_congruentI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
% 0.20/0.47 ( ! [Y3: A,Z2: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( ( F3 @ Y3 )
% 0.20/0.47 = ( F3 @ Z2 ) ) )
% 0.20/0.47 => ( equiv_congruent @ A @ B @ R2 @ F3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % congruentI
% 0.20/0.47 thf(fact_219_in__quotient__imp__subset,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ X7 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_quotient_imp_subset
% 0.20/0.47 thf(fact_220_agree__on__subg__compose,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G4: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F5: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( agree_on @ A @ B @ C @ R @ G4 @ H )
% 0.20/0.47 => ( ( agree_on @ A @ B @ C @ F5 @ F3 @ G4 )
% 0.20/0.47 => ( ( graph_homomorphism @ A @ B @ B @ F5 @ R @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F5 ) ) )
% 0.20/0.47 => ( agree_on @ A @ B @ C @ F5 @ F3 @ H ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % agree_on_subg_compose
% 0.20/0.47 thf(fact_221_maintainedD2,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 )
% 0.20/0.47 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G3 @ F3 )
% 0.20/0.47 => ~ ! [G2: set @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ C @ B5 @ G3 @ G2 )
% 0.20/0.47 => ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F3 @ G2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintainedD2
% 0.20/0.47 thf(fact_222_subgraph__semantics,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 0.20/0.47 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B5 @ E3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_semantics
% 0.20/0.47 thf(fact_223_proj__iff,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
% 0.20/0.47 => ( ( ( equiv_proj @ A @ A @ R2 @ X4 )
% 0.20/0.47 = ( equiv_proj @ A @ A @ R2 @ Y4 ) )
% 0.20/0.47 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % proj_iff
% 0.20/0.47 thf(fact_224_least__consequence__graphI,axiom,
% 0.20/0.47 ! [X: $tType,A: $tType,C: $tType,L: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G3: labeled_graph @ L @ C,S2: labeled_graph @ L @ C,T3: itself @ X] :
% 0.20/0.47 ( ( consequence_graph @ L @ A @ C @ Rs @ G3 )
% 0.20/0.47 => ( ( graph_homomorphism @ L @ C @ C @ S2 @ G3 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 0.20/0.47 => ( ! [C6: labeled_graph @ L @ X] :
% 0.20/0.47 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 0.20/0.47 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G3 ) @ C6 ) )
% 0.20/0.47 => ( least_559130134_graph @ X @ L @ A @ C @ T3 @ Rs @ S2 @ G3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % least_consequence_graphI
% 0.20/0.47 thf(fact_225_leastI,axiom,
% 0.20/0.47 ! [X: $tType,A: $tType,C: $tType,L: $tType,S2: labeled_graph @ L @ C,G3: labeled_graph @ L @ C,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T3: itself @ X] :
% 0.20/0.47 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G3 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 0.20/0.47 => ( ! [C6: labeled_graph @ L @ X] :
% 0.20/0.47 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 0.20/0.47 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G3 ) @ C6 ) )
% 0.20/0.47 => ( least @ X @ L @ A @ C @ T3 @ Rs @ S2 @ G3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % leastI
% 0.20/0.47 thf(fact_226_least__consequence__graph__def,axiom,
% 0.20/0.47 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.20/0.47 ( ( least_559130134_graph @ X @ L @ V @ C )
% 0.20/0.47 = ( ^ [T2: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 0.20/0.47 ( ( consequence_graph @ L @ V @ C @ Rs2 @ G7 )
% 0.20/0.47 & ( least @ X @ L @ V @ C @ T2 @ Rs2 @ S3 @ G7 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % least_consequence_graph_def
% 0.20/0.47 thf(fact_227_consequence__graphD_I3_J,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G3: labeled_graph @ A @ C] :
% 0.20/0.47 ( ( consequence_graph @ A @ B @ C @ Rs @ G3 )
% 0.20/0.47 => ( G3
% 0.20/0.47 = ( restrict @ A @ C @ G3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % consequence_graphD(3)
% 0.20/0.47 thf(fact_228_consequence__graphD_I1_J,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G3: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 0.20/0.47 ( ( consequence_graph @ A @ B @ C @ Rs @ G3 )
% 0.20/0.47 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs )
% 0.20/0.47 => ( maintained @ A @ B @ C @ R @ G3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % consequence_graphD(1)
% 0.20/0.47 thf(fact_229_proj__in__iff,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( member @ ( set @ A ) @ ( equiv_proj @ A @ A @ R2 @ X4 ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 = ( member @ A @ X4 @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % proj_in_iff
% 0.20/0.47 thf(fact_230_least__def,axiom,
% 0.20/0.47 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 0.20/0.47 ( ( least @ X @ L @ V @ C )
% 0.20/0.47 = ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 0.20/0.47 ( ( graph_homomorphism @ L @ C @ C @ S3 @ G7 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S3 ) ) )
% 0.20/0.47 & ! [C4: labeled_graph @ L @ X] :
% 0.20/0.47 ( ( consequence_graph @ L @ V @ X @ Rs2 @ C4 )
% 0.20/0.47 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S3 @ G7 ) @ C4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % least_def
% 0.20/0.47 thf(fact_231_less__by__empty,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_by_empty
% 0.20/0.47 thf(fact_232_insert__subsetI,axiom,
% 0.20/0.47 ! [A: $tType,X4: A,A5: set @ A,X7: set @ A] :
% 0.20/0.47 ( ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ X7 ) @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_subsetI
% 0.20/0.47 thf(fact_233_ssubst__Pair__rhs,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R )
% 0.20/0.47 => ( ( S4 = S )
% 0.20/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ssubst_Pair_rhs
% 0.20/0.47 thf(fact_234_subset__emptyI,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ~ ( member @ A @ X3 @ A5 )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_emptyI
% 0.20/0.47 thf(fact_235_univ__commute,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,X4: A] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( equiv_congruent @ A @ B @ R2 @ F3 )
% 0.20/0.47 => ( ( member @ A @ X4 @ A5 )
% 0.20/0.47 => ( ( bNF_Greatest_univ @ A @ B @ F3 @ ( equiv_proj @ A @ A @ R2 @ X4 ) )
% 0.20/0.47 = ( F3 @ X4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % univ_commute
% 0.20/0.47 thf(fact_236_subgraph__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.20/0.47 = ( ( G_1
% 0.20/0.47 = ( restrict @ A @ B @ G_1 ) )
% 0.20/0.47 & ( G_2
% 0.20/0.47 = ( restrict @ A @ B @ G_2 ) )
% 0.20/0.47 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.20/0.47 = G_2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_def
% 0.20/0.47 thf(fact_237_graph__union__idemp_I3_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B5 @ A5 ) )
% 0.20/0.47 = ( graph_union @ A @ B @ B5 @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_union_idemp(3)
% 0.20/0.47 thf(fact_238_graph__union__idemp_I2_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B5 ) )
% 0.20/0.47 = ( graph_union @ A @ B @ A5 @ B5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_union_idemp(2)
% 0.20/0.47 thf(fact_239_graph__union__idemp_I1_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_union @ A @ B @ A5 @ A5 )
% 0.20/0.47 = A5 ) ).
% 0.20/0.47
% 0.20/0.47 % graph_union_idemp(1)
% 0.20/0.47 thf(fact_240_graph__union__preserves__restrict,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( G_1
% 0.20/0.47 = ( restrict @ A @ B @ G_1 ) )
% 0.20/0.47 => ( ( G_2
% 0.20/0.47 = ( restrict @ A @ B @ G_2 ) )
% 0.20/0.47 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.20/0.47 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_union_preserves_restrict
% 0.20/0.47 thf(fact_241_univ__preserves,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,B5: set @ B] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R2 )
% 0.20/0.47 => ( ( equiv_congruent @ A @ B @ R2 @ F3 )
% 0.20/0.47 => ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A5 )
% 0.20/0.47 => ( member @ B @ ( F3 @ X3 ) @ B5 ) )
% 0.20/0.47 => ! [X2: set @ A] :
% 0.20/0.47 ( ( member @ ( set @ A ) @ X2 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 0.20/0.47 => ( member @ B @ ( bNF_Greatest_univ @ A @ B @ F3 @ X2 ) @ B5 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % univ_preserves
% 0.20/0.47 thf(fact_242_relImage__proj,axiom,
% 0.20/0.47 ! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ( equiv_equiv @ A @ A5 @ R )
% 0.20/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr1317331620lImage @ A @ ( set @ A ) @ R @ ( equiv_proj @ A @ A @ R ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % relImage_proj
% 0.20/0.47 thf(fact_243_subgraph__def2,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( G_1
% 0.20/0.47 = ( restrict @ A @ B @ G_1 ) )
% 0.20/0.47 => ( ( G_2
% 0.20/0.47 = ( restrict @ A @ B @ G_2 ) )
% 0.20/0.47 => ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.20/0.47 = ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.20/0.47 & ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_def2
% 0.20/0.47 thf(fact_244_labeled__graph_Ocollapse,axiom,
% 0.20/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.20/0.47 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 0.20/0.47 = Labeled_graph ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.collapse
% 0.20/0.47 thf(fact_245_graph__unionI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.20/0.47 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.20/0.47 = G_2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_unionI
% 0.20/0.47 thf(fact_246_restrict__subsD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G3 ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G3 ) ) )
% 0.20/0.47 => ( G3
% 0.20/0.47 = ( restrict @ A @ B @ G3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % restrict_subsD
% 0.20/0.47 thf(fact_247_labeled__graph_Oexhaust__sel,axiom,
% 0.20/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.20/0.47 ( Labeled_graph
% 0.20/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.exhaust_sel
% 0.20/0.47 thf(fact_248_labeled__graph_Oexpand,axiom,
% 0.20/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 0.20/0.47 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 0.20/0.47 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 0.20/0.47 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 0.20/0.47 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 0.20/0.47 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.expand
% 0.20/0.47 thf(fact_249_labeled__graph_Osel_I1_J,axiom,
% 0.20/0.47 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.20/0.47 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.20/0.47 = X1 ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.sel(1)
% 0.20/0.47 thf(fact_250_graph__union__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.20/0.47 = G_2 )
% 0.20/0.47 = ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.20/0.47 & ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_union_iff
% 0.20/0.47 thf(fact_251_subgraph__subset_I2_J,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.20/0.47 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G_1 ) ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subgraph_subset(2)
% 0.20/0.47 thf(fact_252_labeled__graph_Osplit__sel__asm,axiom,
% 0.20/0.47 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.20/0.47 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F3 @ Labeled_graph ) )
% 0.20/0.47 = ( ~ ( ( Labeled_graph
% 0.20/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.20/0.47 & ~ ( P2 @ ( F3 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.split_sel_asm
% 0.20/0.47 thf(fact_253_labeled__graph_Osplit__sel,axiom,
% 0.20/0.47 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.20/0.47 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F3 @ Labeled_graph ) )
% 0.20/0.47 = ( ( Labeled_graph
% 0.20/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.20/0.47 => ( P2 @ ( F3 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.split_sel
% 0.20/0.47 thf(fact_254_labeled__graph_Ocase,axiom,
% 0.20/0.47 ! [L: $tType,A: $tType,V: $tType,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.20/0.47 ( ( labele1974067554_graph @ L @ V @ A @ F3 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.20/0.47 = ( F3 @ X1 @ X22 ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.case
% 0.20/0.47 thf(fact_255_labeled__graph_Ocase__eq__if,axiom,
% 0.20/0.47 ! [A: $tType,V: $tType,L: $tType] :
% 0.20/0.47 ( ( labele1974067554_graph @ L @ V @ A )
% 0.20/0.47 = ( ^ [F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.case_eq_if
% 0.20/0.47
% 0.20/0.47 % Type constructors (22)
% 0.20/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( order_bot @ A9 )
% 0.20/0.47 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( preorder @ A9 )
% 0.20/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( order @ A9 )
% 0.20/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( ord @ A9 )
% 0.20/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( bot @ A9 )
% 0.20/0.47 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.20/0.47 order_bot @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.20/0.47 preorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.20/0.47 linorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.20/0.47 order @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.20/0.47 ord @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.20/0.47 bot @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.20/0.47 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.20/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 0.20/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 0.20/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
% 0.20/0.47 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 0.20/0.47 order_bot @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.20/0.47 preorder @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
% 0.20/0.47 linorder @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 0.20/0.47 order @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.20/0.47 ord @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 0.20/0.47 bot @ $o ).
% 0.20/0.47
% 0.20/0.47 % Conjectures (1)
% 0.20/0.47 thf(conj_0,conjecture,
% 0.20/0.47 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ z @ y ) @ ( getRel @ a @ b @ i @ g ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 3.18/1.62 ---- Embedded in THF ---
% 3.18/1.62 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.18/1.62 %%% Generated on Tue Feb 24 08:28:42 EST 2026
% 3.18/1.62 %%% using '$$dhol' embedding, version 1.3.0.
% 3.18/1.62 %%% Logic specification used:
% 3.18/1.62 %%% thf(spec, logic, $$dhol).
% 3.18/1.62
% 3.18/1.62 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ANXxQNlu00/DTF2DTF30214.p
% 3.18/1.62 thf(ty_t_LabeledGraphSemantics_Oallegorical__term, type, allegorical_term: ($tType > $tType)).
% 3.18/1.62 thf(ty_t_LabeledGraphs_Olabeled__graph, type, labeled_graph: ($tType > ($tType > $tType))).
% 3.18/1.62 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.18/1.62 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.18/1.62 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.18/1.62 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.18/1.62 thf(ty_tf_b, type, b: $tType).
% 3.18/1.62 thf(ty_tf_a, type, a: $tType).
% 3.18/1.62 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 3.18/1.62 thf(sy_c_BNF__Greatest__Fixpoint_OrelImage, type, bNF_Gr1317331620lImage: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((B > A) > (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv, type, bNF_Greatest_univ: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 3.18/1.62 thf(sy_c_Equiv__Relations_Ocongruent, type, equiv_congruent: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 3.18/1.62 thf(sy_c_Equiv__Relations_Ocongruent2, type, equiv_congruent2: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > ((A > (B > C)) > $o)))))).
% 3.18/1.62 thf(sy_c_Equiv__Relations_Oequiv, type, equiv_equiv: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.18/1.62 thf(sy_c_Equiv__Relations_Oproj, type, equiv_proj: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > (set @ A)))))).
% 3.18/1.62 thf(sy_c_Equiv__Relations_Oquotient, type, equiv_quotient: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ (set @ A))))))).
% 3.18/1.62 thf(sy_c_LabeledGraphSemantics_OgetRel, type, getRel: (!> [B:$tType,A:$tType]: ((B > (((labeled_graph @ B) @ A) > (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(sy_c_LabeledGraphSemantics_Osemantics, type, semantics: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((allegorical_term @ A) > (set @ ((product_prod @ B) @ B))))))).
% 3.18/1.62 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)) > $o)))))).
% 3.18/1.62 thf(sy_c_LabeledGraphs_Ograph__union, type, graph_union: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B)))))).
% 3.18/1.62 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG, type, labeled_LG: (!> [L:$tType,V:$tType]: (((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > ((labeled_graph @ L) @ V)))))).
% 3.18/1.62 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph, type, labele1974067554_graph: (!> [L:$tType,V:$tType,A:$tType]: ((((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)) > (((labeled_graph @ L) @ V) > A))))).
% 3.18/1.62 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))))))).
% 3.18/1.62 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices, type, labeled_vertices: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ V))))).
% 3.18/1.62 thf(sy_c_LabeledGraphs_Oon__triple, type, on_triple: (!> [B:$tType,C:$tType,A:$tType]: (((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C)))))))).
% 3.18/1.62 thf(sy_c_LabeledGraphs_Orestrict, type, restrict: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B))))).
% 3.18/1.62 thf(sy_c_MissingRelation_Oidempotent, type, idempotent: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.18/1.62 thf(sy_c_Order__Relation_Olinear__order__on, type, order_1409979114der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.18/1.62 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.18/1.62 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.18/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.18/1.62 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.18/1.62 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.18/1.62 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.18/1.62 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 3.18/1.62 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.18/1.62 thf(sy_c_Relation_Osym, type, sym: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.18/1.62 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.18/1.62 thf(sy_c_Relation_Otrans, type, trans: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Oagree__on, type, agree_on: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > ((set @ ((product_prod @ B) @ C)) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Oconsequence__graph, type, consequence_graph: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) > (((labeled_graph @ A) @ C) > $o))))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Oextensible, type, extensible: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > ((set @ ((product_prod @ X) @ V)) > $o)))))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Ofin__maintained, type, fin_maintained: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > $o))))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Oleast, type, least: (!> [X:$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) > $o))))))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Oleast__consequence__graph, type, least_559130134_graph: (!> [X:$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) > $o))))))).
% 3.18/1.62 thf(sy_c_RulesAndChains_Omaintained, type, maintained: (!> [L:$tType,X:$tType,V:$tType]: ((((product_prod @ ((labeled_graph @ L) @ X)) @ ((labeled_graph @ L) @ X)) > (((labeled_graph @ L) @ V) > $o))))).
% 3.18/1.62 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.18/1.62 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 3.18/1.62 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.18/1.62 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.18/1.62 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 3.18/1.62 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ocongruence__rule, type, standa1068660421e_rule: (!> [L:$tType]: ((L > (L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat))))))).
% 3.18/1.62 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule, type, standa244753842y_rule: (!> [L:$tType]: (((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat))))).
% 3.18/1.62 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule, type, standa245363227y_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 3.18/1.62 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule, type, standa1805702094y_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 3.18/1.62 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule, type, standa2114124375e_rule: (!> [L:$tType]: ((L > ((product_prod @ ((labeled_graph @ L) @ nat)) @ ((labeled_graph @ L) @ nat)))))).
% 3.18/1.62 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.18/1.62 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.18/1.62 thf(sy_v_G, type, g: ((labeled_graph @ a) @ b)).
% 3.18/1.62 thf(sy_v_I, type, i: a).
% 3.18/1.62 thf(sy_v_l, type, l: a).
% 3.18/1.62 thf(sy_v_y____, type, y: b).
% 3.18/1.62 thf(sy_v_z____, type, z: b).
% 3.18/1.62 thf(fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062, axiom, ((((refl_on @ b) @ (((labeled_vertices @ a) @ b) @ g)) @ ((((getRel @ a) @ b) @ i) @ g)) & ((! [X2:b,Y:b]: (((((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X2) @ Y)) @ ((((getRel @ a) @ b) @ i) @ g)) => (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ Y) @ X2)) @ ((((getRel @ a) @ b) @ i) @ g))))) & ((trans @ b) @ ((((getRel @ a) @ b) @ i) @ g))))).
% 3.18/1.62 thf(fact_1_aI, axiom, (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ y) @ z)) @ ((((getRel @ a) @ b) @ i) @ g))).
% 3.18/1.62 thf(fact_2_g, axiom, (g = (((restrict @ a) @ b) @ g))).
% 3.18/1.62 thf(fact_3_m, axiom, (((((maintained @ a) @ nat) @ b) @ (((standa1068660421e_rule @ a) @ i) @ l)) @ g)).
% 3.18/1.62 thf(fact_4_eq, axiom, (((equiv_equiv @ b) @ (((labeled_vertices @ a) @ b) @ g)) @ ((((getRel @ a) @ b) @ i) @ g))).
% 3.18/1.62 thf(fact_5_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 3.18/1.62 thf(fact_6_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 3.18/1.62 thf(fact_7_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 3.18/1.62 thf(fact_8_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P2 @ P))))).
% 3.18/1.62 thf(fact_9_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 3.18/1.62 thf(fact_10_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C2:C]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 3.18/1.62 thf(fact_11_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 3.18/1.62 thf(fact_12_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 3.18/1.62 thf(fact_13_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 3.18/1.62 thf(fact_14_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 3.18/1.62 thf(fact_15_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P2 @ Prod))))).
% 3.18/1.62 thf(fact_16_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y4:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y4 != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 3.18/1.62 thf(fact_17_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P2 @ X4))))).
% 3.18/1.62 thf(fact_18_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P2 @ X4))))).
% 3.18/1.62 thf(fact_19_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X4))))).
% 3.18/1.62 thf(fact_20_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C2:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P2 @ X4))))).
% 3.18/1.62 thf(fact_21_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C2:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P2 @ X4))))).
% 3.18/1.62 thf(fact_22_vertices__restrict, axiom, (! [A:$tType,B:$tType,G3:((labeled_graph @ B) @ A)]: (((((labeled_vertices @ B) @ A) @ (((restrict @ B) @ A) @ G3)) = (((labeled_vertices @ B) @ A) @ G3))))).
% 3.18/1.62 thf(fact_23_getRel__dom_I1_J, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B),A2:B,B2:B,L2:A]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((getRel @ A) @ B) @ L2) @ G3)) => (((member @ B) @ A2) @ (((labeled_vertices @ A) @ B) @ G3))))))).
% 3.18/1.62 thf(fact_24_getRel__dom_I2_J, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B),A2:B,B2:B,L2:A]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((getRel @ A) @ B) @ L2) @ G3)) => (((member @ B) @ B2) @ (((labeled_vertices @ A) @ B) @ G3))))))).
% 3.18/1.62 thf(fact_25_transitive__rule, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B),L2:A]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((((((maintained @ A) @ nat) @ B) @ ((standa2114124375e_rule @ A) @ L2)) @ G3) => ((trans @ B) @ ((((getRel @ A) @ B) @ L2) @ G3))))))).
% 3.18/1.62 thf(fact_26_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 3.18/1.62 thf(fact_27_maintained__refl, axiom, (! [B:$tType,C:$tType,A:$tType,R:((labeled_graph @ A) @ B),G3:((labeled_graph @ A) @ C)]: ((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) @ R)) @ G3)))).
% 3.18/1.62 thf(fact_28_reflexivity__rule, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B),L2:A]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((((((maintained @ A) @ nat) @ B) @ ((standa245363227y_rule @ A) @ L2)) @ G3) => (((refl_on @ B) @ (((labeled_vertices @ A) @ B) @ G3)) @ ((((getRel @ A) @ B) @ L2) @ G3))))))).
% 3.18/1.62 thf(fact_29_restrict__idemp, axiom, (! [B:$tType,A:$tType,X4:((labeled_graph @ A) @ B)]: (((((restrict @ A) @ B) @ (((restrict @ A) @ B) @ X4)) = (((restrict @ A) @ B) @ X4))))).
% 3.18/1.62 thf(fact_30_refl__on__domain, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((refl_on @ A) @ A5) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ A) @ A2) @ A5) & (((member @ A) @ B2) @ A5))))))).
% 3.18/1.62 thf(fact_31_refl__onD2, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X4:A,Y4:A]: (((((refl_on @ A) @ A5) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2) => (((member @ A) @ Y4) @ A5)))))).
% 3.18/1.62 thf(fact_32_transD, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X4:A,Y4:A,Z:A]: ((((trans @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z)) @ R2))))))).
% 3.18/1.62 thf(fact_33_transE, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X4:A,Y4:A,Z:A]: ((((trans @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z)) @ R2))))))).
% 3.18/1.62 thf(fact_34_transI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y3:A,Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ R2))))) => ((trans @ A) @ R2))))).
% 3.18/1.62 thf(fact_35_Relation_Otrans__def, axiom, (! [A:$tType]: (((trans @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X5:A,Y5:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y5)) @ R3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ R3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Z3)) @ R3))))))))))).
% 3.18/1.62 thf(fact_36_refl__onD, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ A5) @ R2) => ((((member @ A) @ A2) @ A5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R2)))))).
% 3.18/1.62 thf(fact_37_refl__onD1, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X4:A,Y4:A]: (((((refl_on @ A) @ A5) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2) => (((member @ A) @ X4) @ A5)))))).
% 3.18/1.62 thf(fact_38_mA, axiom, (! [X2:((product_prod @ ((labeled_graph @ a) @ nat)) @ ((labeled_graph @ a) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ a) @ nat)) @ ((labeled_graph @ a) @ nat))) @ X2) @ (((insert @ ((product_prod @ ((labeled_graph @ a) @ nat)) @ ((labeled_graph @ a) @ nat))) @ ((standa245363227y_rule @ a) @ i)) @ (((insert @ ((product_prod @ ((labeled_graph @ a) @ nat)) @ ((labeled_graph @ a) @ nat))) @ ((standa2114124375e_rule @ a) @ i)) @ (((insert @ ((product_prod @ ((labeled_graph @ a) @ nat)) @ ((labeled_graph @ a) @ nat))) @ ((standa1805702094y_rule @ a) @ i)) @ (bot_bot @ (set @ ((product_prod @ ((labeled_graph @ a) @ nat)) @ ((labeled_graph @ a) @ nat)))))))) => (((((maintained @ a) @ nat) @ b) @ X2) @ g))))).
% 3.18/1.62 thf(fact_39_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 3.18/1.62 thf(fact_40_preorder__on__def, axiom, (! [A:$tType]: (((order_preorder_on @ A) = (^ [A6:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A6) @ R3) & ((trans @ A) @ R3)))))))).
% 3.18/1.62 thf(fact_41_on__triple, axiom, (! [B:$tType,C:$tType,A:$tType,L1:A,V1:B,V2:B,L22:A,V3:C,V4:C,R:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ L1) @ ((((product_Pair @ B) @ B) @ V1) @ V2))) @ ((((product_Pair @ A) @ ((product_prod @ C) @ C)) @ L22) @ ((((product_Pair @ C) @ C) @ V3) @ V4)))) @ ((((on_triple @ B) @ C) @ A) @ R)) = ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ V1) @ V3)) @ R) & ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ V2) @ V4)) @ R) & (L1 = L22))))))).
% 3.18/1.62 thf(fact_42_maintained__then__fin__maintained, axiom, (! [B:$tType,C:$tType,A:$tType,R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)),G3:((labeled_graph @ A) @ C)]: (((((((maintained @ A) @ B) @ C) @ R) @ G3) => (((((fin_maintained @ A) @ B) @ C) @ R) @ G3))))).
% 3.18/1.62 thf(fact_43_refl__trans__impl__idempotent, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A5) @ R2) => (((trans @ A) @ R2) => ((idempotent @ A) @ R2)))))).
% 3.18/1.62 thf(fact_44_equivalence, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B),I:A]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((! [X3:((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))]: (((((member @ ((product_prod @ ((labeled_graph @ A) @ nat)) @ ((labeled_graph @ A) @ nat))) @ X3) @ (((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) @ X3) @ G3)))) => (((equiv_equiv @ B) @ (((labeled_vertices @ A) @ B) @ G3)) @ ((((getRel @ A) @ B) @ I) @ G3))))))).
% 3.18/1.62 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 3.18/1.62 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A5)))) = A5)))).
% 3.18/1.62 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) = (Q @ X3)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 3.18/1.62 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G4:(A > B)]: (((! [X3:A]: (((F3 @ X3) = (G4 @ X3)))) => (F3 = G4))))).
% 3.18/1.62 thf(fact_49_semantics__in__vertices_I1_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: (((A5 = (((restrict @ A) @ B) @ A5)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ B) @ A2) @ (((labeled_vertices @ A) @ B) @ A5))))))).
% 3.18/1.62 thf(fact_50_semantics__in__vertices_I2_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: (((A5 = (((restrict @ A) @ B) @ A5)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ B) @ B2) @ (((labeled_vertices @ A) @ B) @ A5))))))).
% 3.18/1.62 thf(fact_51_refl__on__singleton, axiom, (! [A:$tType,X4:A]: ((((refl_on @ A) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ X4)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.18/1.62 thf(fact_52_preorder__on__empty, axiom, (! [A:$tType]: ((((order_preorder_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(fact_53_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(fact_54_trans__singleton, axiom, (! [A:$tType,A2:A]: (((trans @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.18/1.62 thf(fact_55_idempotent__impl__trans, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((idempotent @ A) @ R2) => ((trans @ A) @ R2))))).
% 3.18/1.62 thf(fact_56_trans__empty, axiom, (! [A:$tType]: (((trans @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(fact_57_on__tripleD_I1_J, axiom, (! [B:$tType,C:$tType,A:$tType,L1:A,V1:B,V2:B,L22:A,V3:C,V4:C,R:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ L1) @ ((((product_Pair @ B) @ B) @ V1) @ V2))) @ ((((product_Pair @ A) @ ((product_prod @ C) @ C)) @ L22) @ ((((product_Pair @ C) @ C) @ V3) @ V4)))) @ ((((on_triple @ B) @ C) @ A) @ R)) => (L22 = L1))))).
% 3.18/1.62 thf(fact_58_on__tripleD_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,L1:A,V1:B,V2:B,L22:A,V3:C,V4:C,R:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ L1) @ ((((product_Pair @ B) @ B) @ V1) @ V2))) @ ((((product_Pair @ A) @ ((product_prod @ C) @ C)) @ L22) @ ((((product_Pair @ C) @ C) @ V3) @ V4)))) @ ((((on_triple @ B) @ C) @ A) @ R)) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ V1) @ V3)) @ R))))).
% 3.18/1.62 thf(fact_59_on__tripleD_I3_J, axiom, (! [A:$tType,C:$tType,B:$tType,L1:A,V1:B,V2:B,L22:A,V3:C,V4:C,R:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((product_prod @ A) @ ((product_prod @ C) @ C))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ L1) @ ((((product_Pair @ B) @ B) @ V1) @ V2))) @ ((((product_Pair @ A) @ ((product_prod @ C) @ C)) @ L22) @ ((((product_Pair @ C) @ C) @ V3) @ V4)))) @ ((((on_triple @ B) @ C) @ A) @ R)) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ V2) @ V4)) @ R))))).
% 3.18/1.62 thf(fact_60_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_61_nonempty__rule, axiom, (! [A:$tType,B:$tType,G3:((labeled_graph @ A) @ B)]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((((((maintained @ A) @ nat) @ B) @ (standa244753842y_rule @ A)) @ G3) = ((((labeled_vertices @ A) @ B) @ G3) != (bot_bot @ (set @ B)))))))).
% 3.18/1.62 thf(fact_62_insertCI, axiom, (! [A:$tType,A2:A,B5:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B5))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B5)))))).
% 3.18/1.62 thf(fact_63_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A5)) = ((A2 = B2) | (((member @ A) @ A2) @ A5)))))).
% 3.18/1.62 thf(fact_64_insert__absorb2, axiom, (! [A:$tType,X4:A,A5:(set @ A)]: (((((insert @ A) @ X4) @ (((insert @ A) @ X4) @ A5)) = (((insert @ A) @ X4) @ A5))))).
% 3.18/1.62 thf(fact_65_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 3.18/1.62 thf(fact_66_all__not__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A5))))) = (A5 = (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_67_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X5:A]: ((~ ((P2 @ X5))))))))).
% 3.18/1.62 thf(fact_68_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X5:A]: ((~ ((P2 @ X5))))))))).
% 3.18/1.62 thf(fact_69_ex__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((? [X5:A]: ((((member @ A) @ X5) @ A5))) = (A5 != (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_70_equals0I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A5))))) => (A5 = (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_71_equals0D, axiom, (! [A:$tType,A5:(set @ A),A2:A]: (((A5 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A5))))))).
% 3.18/1.62 thf(fact_72_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 3.18/1.62 thf(fact_73_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ A5) => (? [B6:(set @ A)]: (((A5 = (((insert @ A) @ A2) @ B6)) & (~ ((((member @ A) @ A2) @ B6)))))))))).
% 3.18/1.62 thf(fact_74_insert__commute, axiom, (! [A:$tType,X4:A,Y4:A,A5:(set @ A)]: (((((insert @ A) @ X4) @ (((insert @ A) @ Y4) @ A5)) = (((insert @ A) @ Y4) @ (((insert @ A) @ X4) @ A5)))))).
% 3.18/1.62 thf(fact_75_insert__eq__iff, axiom, (! [A:$tType,A2:A,A5:(set @ A),B2:A,B5:(set @ A)]: (((~ ((((member @ A) @ A2) @ A5))) => ((~ ((((member @ A) @ B2) @ B5))) => (((((insert @ A) @ A2) @ A5) = (((insert @ A) @ B2) @ B5)) = (((A2 = B2) => (A5 = B5)) & ((A2 != B2) => (? [C4:(set @ A)]: (((A5 = (((insert @ A) @ B2) @ C4)) & ((~ ((((member @ A) @ B2) @ C4))) & ((B5 = (((insert @ A) @ A2) @ C4)) & (~ ((((member @ A) @ A2) @ C4)))))))))))))))).
% 3.18/1.62 thf(fact_76_insert__absorb, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ A5) => ((((insert @ A) @ A2) @ A5) = A5))))).
% 3.18/1.62 thf(fact_77_insert__ident, axiom, (! [A:$tType,X4:A,A5:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X4) @ A5))) => ((~ ((((member @ A) @ X4) @ B5))) => (((((insert @ A) @ X4) @ A5) = (((insert @ A) @ X4) @ B5)) = (A5 = B5))))))).
% 3.18/1.62 thf(fact_78_Set_Oset__insert, axiom, (! [A:$tType,X4:A,A5:(set @ A)]: (((((member @ A) @ X4) @ A5) => (~ ((! [B6:(set @ A)]: (((A5 = (((insert @ A) @ X4) @ B6)) => (((member @ A) @ X4) @ B6)))))))))).
% 3.18/1.62 thf(fact_79_insertI2, axiom, (! [A:$tType,A2:A,B5:(set @ A),B2:A]: (((((member @ A) @ A2) @ B5) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B5)))))).
% 3.18/1.62 thf(fact_80_insertI1, axiom, (! [A:$tType,A2:A,B5:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B5))))).
% 3.18/1.62 thf(fact_81_insertE, axiom, (! [A:$tType,A2:A,B2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A5)) => ((A2 != B2) => (((member @ A) @ A2) @ A5)))))).
% 3.18/1.62 thf(fact_82_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 3.18/1.62 thf(fact_83_insert__not__empty, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((insert @ A) @ A2) @ A5) != (bot_bot @ (set @ A)))))).
% 3.18/1.62 thf(fact_84_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C3:A,D3:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D3) @ (bot_bot @ (set @ A))))) = (((A2 = C3) & (B2 = D3)) | ((A2 = D3) & (B2 = C3))))))).
% 3.18/1.62 thf(fact_85_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 3.18/1.62 thf(fact_86_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 3.18/1.62 thf(fact_87_the__elem__eq, axiom, (! [A:$tType,X4:A]: ((((the_elem @ A) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))) = X4)))).
% 3.18/1.62 thf(fact_88_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X5:D]: ((bot_bot @ C)))))))).
% 3.18/1.62 thf(fact_89_graph__single, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,C3:B]: ((((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A2) @ ((((product_Pair @ B) @ B) @ B2) @ C3))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B2) @ (((insert @ B) @ C3) @ (bot_bot @ (set @ B))))) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A2) @ ((((product_Pair @ B) @ B) @ B2) @ C3))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B2) @ (((insert @ B) @ C3) @ (bot_bot @ (set @ B)))))))))).
% 3.18/1.62 thf(fact_90_is__singletonI, axiom, (! [A:$tType,X4:A]: (((is_singleton @ A) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_91_linear__order__on__singleton, axiom, (! [A:$tType,X4:A]: ((((order_1409979114der_on @ A) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ X4)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.18/1.62 thf(fact_92_total__on__singleton, axiom, (! [A:$tType,X4:A]: ((((total_on @ A) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ X4)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.18/1.62 thf(fact_93_symmetry__rule, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B),L2:A]: (((G3 = (((restrict @ A) @ B) @ G3)) => ((((((maintained @ A) @ nat) @ B) @ ((standa1805702094y_rule @ A) @ L2)) @ G3) => ((sym @ B) @ ((((getRel @ A) @ B) @ L2) @ G3))))))).
% 3.18/1.62 thf(fact_94_labeled__graph_Oinject, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V),Y1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),Y2:(set @ V)]: (((((((labeled_LG @ L) @ V) @ X1) @ X22) = ((((labeled_LG @ L) @ V) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 3.18/1.62 thf(fact_95_graph__empty__e, axiom, (! [A:$tType,B:$tType,V5:(set @ B)]: ((((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V5) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V5)))))).
% 3.18/1.62 thf(fact_96_labeled__graph_Oexhaust, axiom, (! [L:$tType,V:$tType,Y4:((labeled_graph @ L) @ V)]: ((~ ((! [X12:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X23:(set @ V)]: ((Y4 != ((((labeled_LG @ L) @ V) @ X12) @ X23))))))))).
% 3.18/1.62 thf(fact_97_labeled__graph_Oinduct, axiom, (! [V:$tType,L:$tType,P2:(((labeled_graph @ L) @ V) > $o),Labeled_graph:((labeled_graph @ L) @ V)]: (((! [X1a:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X2a:(set @ V)]: ((P2 @ ((((labeled_LG @ L) @ V) @ X1a) @ X2a)))) => (P2 @ Labeled_graph))))).
% 3.18/1.62 thf(fact_98_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X5:A]: ((((member @ A) @ X5) @ (bot_bot @ (set @ A))))))))).
% 3.18/1.62 thf(fact_99_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 3.18/1.62 thf(fact_100_sym__def, axiom, (! [A:$tType]: (((sym @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X5:A,Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y5)) @ R3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X5)) @ R3)))))))))).
% 3.18/1.62 thf(fact_101_symI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [A4:A,B4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ A4)) @ R2)))) => ((sym @ A) @ R2))))).
% 3.18/1.62 thf(fact_102_symE, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),B2:A,A2:A]: ((((sym @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2)))))).
% 3.18/1.62 thf(fact_103_symD, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),B2:A,A2:A]: ((((sym @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2)))))).
% 3.18/1.62 thf(fact_104_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A6:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X5:A]: (((((member @ A) @ X5) @ A6) => (! [Y5:A]: (((((member @ A) @ Y5) @ A6) => ((X5 != Y5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y5)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X5)) @ R3))))))))))))))).
% 3.18/1.62 thf(fact_105_total__onI, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y3:A]: (((((member @ A) @ X3) @ A5) => ((((member @ A) @ Y3) @ A5) => ((X3 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R2) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R2))))))) => (((total_on @ A) @ A5) @ R2))))).
% 3.18/1.62 thf(fact_106_total__on__empty, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R2)))).
% 3.18/1.62 thf(fact_107_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)))).
% 3.18/1.62 thf(fact_108_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((A6 = (((insert @ A) @ ((the_elem @ A) @ A6)) @ (bot_bot @ (set @ A)))))))))).
% 3.18/1.62 thf(fact_109_is__singletonI_H, axiom, (! [A:$tType,A5:(set @ A)]: (((A5 != (bot_bot @ (set @ A))) => ((! [X3:A,Y3:A]: (((((member @ A) @ X3) @ A5) => ((((member @ A) @ Y3) @ A5) => (X3 = Y3))))) => ((is_singleton @ A) @ A5)))))).
% 3.18/1.62 thf(fact_110_lnear__order__on__empty, axiom, (! [A:$tType]: ((((order_1409979114der_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(fact_111_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X5:A]: ((bot_bot @ B)))))))).
% 3.18/1.62 thf(fact_112_is__singletonE, axiom, (! [A:$tType,A5:(set @ A)]: ((((is_singleton @ A) @ A5) => (~ ((! [X3:A]: ((A5 != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 3.18/1.62 thf(fact_113_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((? [X5:A]: ((A6 = (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))))).
% 3.18/1.62 thf(fact_114_equivE, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((equiv_equiv @ A) @ A5) @ R2) => (~ (((((refl_on @ A) @ A5) @ R2) => (((sym @ A) @ R2) => (~ (((trans @ A) @ R2))))))))))).
% 3.18/1.62 thf(fact_115_equivI, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A5) @ R2) => (((sym @ A) @ R2) => (((trans @ A) @ R2) => (((equiv_equiv @ A) @ A5) @ R2))))))).
% 3.18/1.62 thf(fact_116_equiv__def, axiom, (! [A:$tType]: (((equiv_equiv @ A) = (^ [A6:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A6) @ R3) & (((sym @ A) @ R3) & ((trans @ A) @ R3))))))))).
% 3.18/1.62 thf(fact_117_agree__on__empty, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(set @ ((product_prod @ B) @ C)),G4:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ (bot_bot @ (set @ B)))) @ F3) @ G4)))).
% 3.18/1.62 thf(fact_118_graph__homomorphism__empty, axiom, (! [B:$tType,C:$tType,A:$tType,G3:((labeled_graph @ A) @ C),F3:(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)))) @ G3) @ F3) = ((F3 = (bot_bot @ (set @ ((product_prod @ B) @ C)))) & (G3 = (((restrict @ A) @ C) @ G3))))))).
% 3.18/1.62 thf(fact_119_agree__on__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((agree_on @ A) @ B) @ C) = (^ [X6:((labeled_graph @ A) @ B),F4:(set @ ((product_prod @ B) @ C)),G5:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ X6) @ G5) @ F4))))))).
% 3.18/1.62 thf(fact_120_agree__on__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R:((labeled_graph @ A) @ B),F3:(set @ ((product_prod @ B) @ C))]: (((((((agree_on @ A) @ B) @ C) @ R) @ F3) @ F3)))).
% 3.18/1.62 thf(fact_121_agree__on__trans, axiom, (! [A:$tType,C:$tType,B:$tType,X7:((labeled_graph @ A) @ B),F3:(set @ ((product_prod @ B) @ C)),G4:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ X7) @ F3) @ G4) => (((((((agree_on @ A) @ B) @ C) @ X7) @ G4) @ H) => ((((((agree_on @ A) @ B) @ C) @ X7) @ F3) @ H)))))).
% 3.18/1.62 thf(fact_122_graph__homomorphism__semantics, axiom, (! [B:$tType,C:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ C),F3:(set @ ((product_prod @ B) @ C)),A2:B,B2:B,E3:(allegorical_term @ A),A3:C,B3:C]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ B5) @ F3) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ A2) @ A3)) @ F3) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B2) @ B3)) @ F3) => (((member @ ((product_prod @ C) @ C)) @ ((((product_Pair @ C) @ C) @ A3) @ B3)) @ ((((semantics @ A) @ C) @ B5) @ E3))))))))).
% 3.18/1.62 thf(fact_123_graph__homomorphism__nonempty, axiom, (! [B:$tType,A:$tType,C:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ C),F3:(set @ ((product_prod @ B) @ C)),E3:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ B5) @ F3) => ((((((semantics @ A) @ B) @ A5) @ E3) != (bot_bot @ (set @ ((product_prod @ B) @ B)))) => (((((semantics @ A) @ C) @ B5) @ E3) != (bot_bot @ (set @ ((product_prod @ C) @ C))))))))).
% 3.18/1.62 thf(fact_124_Collect__empty__eq__bot, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (P2 = (bot_bot @ (A > $o))))))).
% 3.18/1.62 thf(fact_125_extensibleI, axiom, (! [A:$tType,C:$tType,B:$tType,R22:((labeled_graph @ A) @ B),G3:((labeled_graph @ A) @ C),G4:(set @ ((product_prod @ B) @ C)),R1:((labeled_graph @ A) @ B),F3:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R22) @ G3) @ G4) => (((((((agree_on @ A) @ B) @ C) @ R1) @ F3) @ G4) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R1) @ R22)) @ G3) @ F3)))))).
% 3.18/1.62 thf(fact_126_maintainedI, axiom, (! [A:$tType,B:$tType,C:$tType,A5:((labeled_graph @ C) @ A),G3:((labeled_graph @ C) @ B),B5:((labeled_graph @ C) @ A)]: (((! [F2:(set @ ((product_prod @ A) @ B))]: ((((((((graph_homomorphism @ C) @ A) @ B) @ A5) @ G3) @ F2) => ((((((extensible @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A5) @ B5)) @ G3) @ F2)))) => (((((maintained @ C) @ A) @ B) @ ((((product_Pair @ ((labeled_graph @ C) @ A)) @ ((labeled_graph @ C) @ A)) @ A5) @ B5)) @ G3))))).
% 3.18/1.62 thf(fact_127_extensible__refl__concr, axiom, (! [A:$tType,C:$tType,B:$tType,E_1:(set @ ((product_prod @ A) @ ((product_prod @ B) @ B))),V5:(set @ B),G3:((labeled_graph @ A) @ C),F3:(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) @ V5)) @ G3) @ F3) => (((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ ((((labeled_LG @ A) @ B) @ E_1) @ V5)) @ ((((labeled_LG @ A) @ B) @ E_2) @ V5))) @ G3) @ F3) = ((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ E_2) @ V5)) @ G3) @ F3)))))).
% 3.18/1.62 thf(fact_128_extensible__refl, axiom, (! [A:$tType,C:$tType,B:$tType,R:((labeled_graph @ A) @ B),G3:((labeled_graph @ A) @ C),F3:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ R) @ G3) @ F3) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ R) @ R)) @ G3) @ F3))))).
% 3.18/1.62 thf(fact_129_maintainedD, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),G3:((labeled_graph @ A) @ C),F3:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B5)) @ G3) => (((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ G3) @ F3) => ((((((extensible @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B5)) @ G3) @ F3)))))).
% 3.18/1.62 thf(fact_130_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A6:(set @ A)]: ((A6 = (bot_bot @ (set @ A))))))))).
% 3.18/1.62 thf(fact_131_congruent2I, axiom, (! [C:$tType,B:$tType,A:$tType,A1:(set @ A),R12:(set @ ((product_prod @ A) @ A)),A22:(set @ B),R23:(set @ ((product_prod @ B) @ B)),F3:(A > (B > C))]: (((((equiv_equiv @ A) @ A1) @ R12) => ((((equiv_equiv @ B) @ A22) @ R23) => ((! [Y3:A,Z2:A,W:B]: (((((member @ B) @ W) @ A22) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z2)) @ R12) => (((F3 @ Y3) @ W) = ((F3 @ Z2) @ W)))))) => ((! [Y3:B,Z2:B,W:A]: (((((member @ A) @ W) @ A1) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y3) @ Z2)) @ R23) => (((F3 @ W) @ Y3) = ((F3 @ W) @ Z2)))))) => ((((((equiv_congruent2 @ A) @ B) @ C) @ R12) @ R23) @ F3)))))))).
% 3.18/1.62 thf(fact_132_congruent2__commuteI, axiom, (! [B:$tType,A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),F3:(A > (A > B))]: (((((equiv_equiv @ A) @ A5) @ R2) => ((! [Y3:A,Z2:A]: (((((member @ A) @ Y3) @ A5) => ((((member @ A) @ Z2) @ A5) => (((F3 @ Y3) @ Z2) = ((F3 @ Z2) @ Y3)))))) => ((! [Y3:A,Z2:A,W:A]: (((((member @ A) @ W) @ A5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z2)) @ R2) => (((F3 @ W) @ Y3) = ((F3 @ W) @ Z2)))))) => ((((((equiv_congruent2 @ A) @ A) @ B) @ R2) @ R2) @ F3))))))).
% 3.18/1.62 thf(fact_133_eq__equiv__class__iff2, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X4:A,Y4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ A) @ X4) @ A5) => ((((member @ A) @ Y4) @ A5) => (((((equiv_quotient @ A) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))) @ R2) = (((equiv_quotient @ A) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A)))) @ R2)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2)))))))).
% 3.18/1.62 thf(fact_134_quotient__is__empty2, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: ((((bot_bot @ (set @ (set @ A))) = (((equiv_quotient @ A) @ A5) @ R2)) = (A5 = (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_135_quotient__is__empty, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: ((((((equiv_quotient @ A) @ A5) @ R2) = (bot_bot @ (set @ (set @ A)))) = (A5 = (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_136_quotient__empty, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((equiv_quotient @ A) @ (bot_bot @ (set @ A))) @ R2) = (bot_bot @ (set @ (set @ A))))))).
% 3.18/1.62 thf(fact_137_in__quotient__imp__closed, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X7:(set @ A),X4:A,Y4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ X7) @ (((equiv_quotient @ A) @ A5) @ R2)) => ((((member @ A) @ X4) @ X7) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2) => (((member @ A) @ Y4) @ X7)))))))).
% 3.18/1.62 thf(fact_138_quotient__eq__iff, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X7:(set @ A),Y6:(set @ A),X4:A,Y4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ X7) @ (((equiv_quotient @ A) @ A5) @ R2)) => ((((member @ (set @ A)) @ Y6) @ (((equiv_quotient @ A) @ A5) @ R2)) => ((((member @ A) @ X4) @ X7) => ((((member @ A) @ Y4) @ Y6) => ((X7 = Y6) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2)))))))))).
% 3.18/1.62 thf(fact_139_quotient__eqI, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X7:(set @ A),Y6:(set @ A),X4:A,Y4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ X7) @ (((equiv_quotient @ A) @ A5) @ R2)) => ((((member @ (set @ A)) @ Y6) @ (((equiv_quotient @ A) @ A5) @ R2)) => ((((member @ A) @ X4) @ X7) => ((((member @ A) @ Y4) @ Y6) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2) => (X7 = Y6)))))))))).
% 3.18/1.62 thf(fact_140_in__quotient__imp__non__empty, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X7:(set @ A)]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ X7) @ (((equiv_quotient @ A) @ A5) @ R2)) => (X7 != (bot_bot @ (set @ A)))))))).
% 3.18/1.62 thf(fact_141_congruent2I_H, axiom, (! [C:$tType,B:$tType,A:$tType,R12:(set @ ((product_prod @ A) @ A)),R23:(set @ ((product_prod @ B) @ B)),F3:(A > (B > C))]: (((! [Y12:A,Z1:A,Y22:B,Z22:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y12) @ Z1)) @ R12) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y22) @ Z22)) @ R23) => (((F3 @ Y12) @ Y22) = ((F3 @ Z1) @ Z22)))))) => ((((((equiv_congruent2 @ A) @ B) @ C) @ R12) @ R23) @ F3))))).
% 3.18/1.62 thf(fact_142_congruent2D, axiom, (! [A:$tType,C:$tType,B:$tType,R12:(set @ ((product_prod @ A) @ A)),R23:(set @ ((product_prod @ B) @ B)),F3:(A > (B > C)),Y1:A,Z12:A,Y2:B,Z23:B]: ((((((((equiv_congruent2 @ A) @ B) @ C) @ R12) @ R23) @ F3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y1) @ Z12)) @ R12) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y2) @ Z23)) @ R23) => (((F3 @ Y1) @ Y2) = ((F3 @ Z12) @ Z23)))))))).
% 3.18/1.62 thf(fact_143_congruent2__implies__congruent, axiom, (! [B:$tType,C:$tType,A:$tType,A5:(set @ A),R12:(set @ ((product_prod @ A) @ A)),R23:(set @ ((product_prod @ B) @ B)),F3:(A > (B > C)),A2:A]: (((((equiv_equiv @ A) @ A5) @ R12) => (((((((equiv_congruent2 @ A) @ B) @ C) @ R12) @ R23) @ F3) => ((((member @ A) @ A2) @ A5) => ((((equiv_congruent @ B) @ C) @ R23) @ (F3 @ A2)))))))).
% 3.18/1.62 thf(fact_144_in__quotient__imp__in__rel, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X7:(set @ A),X4:A,Y4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ X7) @ (((equiv_quotient @ A) @ A5) @ R2)) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X4) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) @ X7) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2))))))).
% 3.18/1.62 thf(fact_145_same__fstI, axiom, (! [B:$tType,A:$tType,P2:(A > $o),X4:A,Y7:B,Y4:B,R:(A > (set @ ((product_prod @ B) @ B)))]: (((P2 @ X4) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y4)) @ (R @ X4)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y7)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4))) @ ((((same_fst @ A) @ B) @ P2) @ R))))))).
% 3.18/1.62 thf(fact_146_getRel__subgraph, axiom, (! [A:$tType,B:$tType,Y4:A,Z:A,L2:B,G3:((labeled_graph @ B) @ A),G6:((labeled_graph @ B) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ ((((getRel @ B) @ A) @ L2) @ G3)) => (((((((graph_homomorphism @ B) @ A) @ A) @ G3) @ G6) @ ((id_on @ A) @ (((labeled_vertices @ B) @ A) @ G3))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ ((((getRel @ B) @ A) @ L2) @ G6))))))).
% 3.18/1.62 thf(fact_147_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 3.18/1.62 thf(fact_148_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A5) => (A5 = B5)))))).
% 3.18/1.62 thf(fact_149_subsetI, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A5) => (((member @ A) @ X3) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 3.18/1.62 thf(fact_150_empty__subsetI, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A5)))).
% 3.18/1.62 thf(fact_151_subset__empty, axiom, (! [A:$tType,A5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_152_insert__subset, axiom, (! [A:$tType,X4:A,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X4) @ A5)) @ B5) = ((((member @ A) @ X4) @ B5) & (((ord_less_eq @ (set @ A)) @ A5) @ B5)))))).
% 3.18/1.62 thf(fact_153_Id__onI, axiom, (! [A:$tType,A2:A,A5:(set @ A)]: (((((member @ A) @ A2) @ A5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ ((id_on @ A) @ A5)))))).
% 3.18/1.62 thf(fact_154_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A5:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A5)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 3.18/1.62 thf(fact_155_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A5:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A5) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 3.18/1.62 thf(fact_156_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.18/1.62 thf(fact_157_subgraph__refl, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G3) @ G3) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G3))) = (G3 = (((restrict @ A) @ B) @ G3)))))).
% 3.18/1.62 thf(fact_158_subgraph__restrict, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G3) @ (((restrict @ A) @ B) @ G3)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G3))) = (G3 = (((restrict @ A) @ B) @ G3)))))).
% 3.18/1.62 thf(fact_159_graph__homomorphism__Id, axiom, (! [B:$tType,A:$tType,A2:((labeled_graph @ A) @ B)]: (((((((graph_homomorphism @ A) @ B) @ B) @ (((restrict @ A) @ B) @ A2)) @ (((restrict @ A) @ B) @ A2)) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A2)))))).
% 3.18/1.62 thf(fact_160_subgraph__subset_I1_J, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) => (((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)))))).
% 3.18/1.62 thf(fact_161_Id__onE, axiom, (! [A:$tType,C3:((product_prod @ A) @ A),A5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C3) @ ((id_on @ A) @ A5)) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ A5) => (C3 != ((((product_Pair @ A) @ A) @ X3) @ X3))))))))))).
% 3.18/1.62 thf(fact_162_Id__on__eqI, axiom, (! [A:$tType,A2:A,B2:A,A5:(set @ A)]: (((A2 = B2) => ((((member @ A) @ A2) @ A5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((id_on @ A) @ A5))))))).
% 3.18/1.62 thf(fact_163_Id__on__iff, axiom, (! [A:$tType,X4:A,Y4:A,A5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ ((id_on @ A) @ A5)) = ((X4 = Y4) & (((member @ A) @ X4) @ A5)))))).
% 3.18/1.62 thf(fact_164_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 3.18/1.62 thf(fact_165_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 3.18/1.62 thf(fact_166_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 3.18/1.62 thf(fact_167_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S))))).
% 3.18/1.62 thf(fact_168_insert__mono, axiom, (! [A:$tType,C5:(set @ A),D4:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C5) @ D4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C5)) @ (((insert @ A) @ A2) @ D4)))))).
% 3.18/1.62 thf(fact_169_subset__insert, axiom, (! [A:$tType,X4:A,A5:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X4) @ A5))) => ((((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ X4) @ B5)) = (((ord_less_eq @ (set @ A)) @ A5) @ B5)))))).
% 3.18/1.62 thf(fact_170_subset__insertI, axiom, (! [A:$tType,B5:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B5) @ (((insert @ A) @ A2) @ B5))))).
% 3.18/1.62 thf(fact_171_subset__insertI2, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B2) @ B5)))))).
% 3.18/1.62 thf(fact_172_trans__Id__on, axiom, (! [A:$tType,A5:(set @ A)]: (((trans @ A) @ ((id_on @ A) @ A5))))).
% 3.18/1.62 thf(fact_173_refl__on__Id__on, axiom, (! [A:$tType,A5:(set @ A)]: ((((refl_on @ A) @ A5) @ ((id_on @ A) @ A5))))).
% 3.18/1.62 thf(fact_174_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X5:A]: (((P2 @ X5) => (Q @ X5)))))))).
% 3.18/1.62 thf(fact_175_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y8:(set @ A),Z4:(set @ A)]: ((Y8 = Z4))) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 3.18/1.62 thf(fact_176_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C5) => (((ord_less_eq @ (set @ A)) @ A5) @ C5)))))).
% 3.18/1.62 thf(fact_177_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)))))).
% 3.18/1.62 thf(fact_178_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 3.18/1.62 thf(fact_179_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A6) => (((member @ A) @ T2) @ B7)))))))))).
% 3.18/1.62 thf(fact_180_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A5))))).
% 3.18/1.62 thf(fact_181_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 3.18/1.62 thf(fact_182_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [X5:A]: (((((member @ A) @ X5) @ A6) => (((member @ A) @ X5) @ B7)))))))))).
% 3.18/1.62 thf(fact_183_equalityE, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 3.18/1.62 thf(fact_184_subsetD, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B5)))))).
% 3.18/1.62 thf(fact_185_in__mono, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B5)))))).
% 3.18/1.62 thf(fact_186_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 3.18/1.62 thf(fact_187_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y8:A,Z4:A]: ((Y8 = Z4))) = (^ [A7:A,B8:A]: (((((ord_less_eq @ A) @ B8) @ A7) & (((ord_less_eq @ A) @ A7) @ B8))))))))).
% 3.18/1.62 thf(fact_188_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C3) @ B2) => (((ord_less_eq @ A) @ C3) @ A2))))))))).
% 3.18/1.62 thf(fact_189_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P2:(A > (A > $o)),A2:A,B2:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P2 @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P2 @ B4) @ A4) => ((P2 @ A4) @ B4)))) => ((P2 @ A2) @ B2))))))))).
% 3.18/1.62 thf(fact_190_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 3.18/1.62 thf(fact_191_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y4:A,Z:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z) => (((ord_less_eq @ A) @ X4) @ Z))))))))).
% 3.18/1.62 thf(fact_192_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 3.18/1.62 thf(fact_193_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 3.18/1.62 thf(fact_194_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 3.18/1.62 thf(fact_195_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y8:A,Z4:A]: ((Y8 = Z4))) = (^ [A7:A,B8:A]: (((((ord_less_eq @ A) @ A7) @ B8) & (((ord_less_eq @ A) @ B8) @ A7))))))))).
% 3.18/1.62 thf(fact_196_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y4:A,X4:A]: (((((ord_less_eq @ A) @ Y4) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y4) = (X4 = Y4))))))))).
% 3.18/1.62 thf(fact_197_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y4:A,Z:A]: ((((((ord_less_eq @ A) @ X4) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ Z)))) => (((((ord_less_eq @ A) @ Y4) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z)))) => (((((ord_less_eq @ A) @ X4) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y4)))) => (((((ord_less_eq @ A) @ Z) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ X4)))) => (((((ord_less_eq @ A) @ Y4) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y4))))))))))))))))).
% 3.18/1.62 thf(fact_198_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 3.18/1.62 thf(fact_199_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X4)))))))).
% 3.18/1.62 thf(fact_200_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y4:A]: (((X4 = Y4) => (((ord_less_eq @ A) @ X4) @ Y4)))))))).
% 3.18/1.62 thf(fact_201_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X4)))))))).
% 3.18/1.62 thf(fact_202_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X4) => (X4 = Y4))))))))).
% 3.18/1.62 thf(fact_203_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y8:A,Z4:A]: ((Y8 = Z4))) = (^ [X5:A,Y5:A]: (((((ord_less_eq @ A) @ X5) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X5))))))))).
% 3.18/1.62 thf(fact_204_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F3:(A > B),C3:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F3 @ B2) = C3) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (F3 @ Y3))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C3)))))))))).
% 3.18/1.62 thf(fact_205_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((A2 = (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F3 @ X3)) @ (F3 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 3.18/1.62 thf(fact_206_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F3:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F3 @ B2)) @ C3) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ C) @ (F3 @ X3)) @ (F3 @ Y3))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C3)))))))))).
% 3.18/1.62 thf(fact_207_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F3 @ X3)) @ (F3 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 3.18/1.62 thf(fact_208_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G5:(A > B)]: ((! [X5:A]: ((((ord_less_eq @ B) @ (F4 @ X5)) @ (G5 @ X5))))))))))).
% 3.18/1.62 thf(fact_209_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G4:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F3 @ X3)) @ (G4 @ X3)))) => (((ord_less_eq @ (A > B)) @ F3) @ G4)))))))).
% 3.18/1.62 thf(fact_210_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G4:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G4) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (G4 @ X4))))))))).
% 3.18/1.62 thf(fact_211_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G4:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G4) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (G4 @ X4))))))))).
% 3.18/1.62 thf(fact_212_sym__Id__on, axiom, (! [A:$tType,A5:(set @ A)]: (((sym @ A) @ ((id_on @ A) @ A5))))).
% 3.18/1.62 thf(fact_213_subset__singletonD, axiom, (! [A:$tType,A5:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))) => ((A5 = (bot_bot @ (set @ A))) | (A5 = (((insert @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 3.18/1.62 thf(fact_214_subset__singleton__iff, axiom, (! [A:$tType,X7:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X7) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X7 = (bot_bot @ (set @ A))) | (X7 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 3.18/1.62 thf(fact_215_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)))))))).
% 3.18/1.62 thf(fact_216_subgraph__preserves__hom, axiom, (! [A:$tType,B:$tType,C:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),X7:((labeled_graph @ A) @ C),H:(set @ ((product_prod @ C) @ B))]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B5) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => (((((((graph_homomorphism @ A) @ C) @ B) @ X7) @ A5) @ H) => ((((((graph_homomorphism @ A) @ C) @ B) @ X7) @ B5) @ H)))))).
% 3.18/1.62 thf(fact_217_congruentD, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),F3:(A > B),Y4:A,Z:A]: ((((((equiv_congruent @ A) @ B) @ R2) @ F3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ R2) => ((F3 @ Y4) = (F3 @ Z))))))).
% 3.18/1.62 thf(fact_218_congruentI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),F3:(A > B)]: (((! [Y3:A,Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z2)) @ R2) => ((F3 @ Y3) = (F3 @ Z2))))) => ((((equiv_congruent @ A) @ B) @ R2) @ F3))))).
% 3.18/1.62 thf(fact_219_in__quotient__imp__subset, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X7:(set @ A)]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ X7) @ (((equiv_quotient @ A) @ A5) @ R2)) => (((ord_less_eq @ (set @ A)) @ X7) @ A5)))))).
% 3.18/1.62 thf(fact_220_agree__on__subg__compose, axiom, (! [A:$tType,C:$tType,B:$tType,R:((labeled_graph @ A) @ B),G4:(set @ ((product_prod @ B) @ C)),H:(set @ ((product_prod @ B) @ C)),F5:((labeled_graph @ A) @ B),F3:(set @ ((product_prod @ B) @ C))]: ((((((((agree_on @ A) @ B) @ C) @ R) @ G4) @ H) => (((((((agree_on @ A) @ B) @ C) @ F5) @ F3) @ G4) => (((((((graph_homomorphism @ A) @ B) @ B) @ F5) @ R) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ F5))) => ((((((agree_on @ A) @ B) @ C) @ F5) @ F3) @ H))))))).
% 3.18/1.62 thf(fact_221_maintainedD2, axiom, (! [A:$tType,C:$tType,B:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),G3:((labeled_graph @ A) @ C),F3:(set @ ((product_prod @ B) @ C))]: (((((((maintained @ A) @ B) @ C) @ ((((product_Pair @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B)) @ A5) @ B5)) @ G3) => (((((((graph_homomorphism @ A) @ B) @ C) @ A5) @ G3) @ F3) => (~ ((! [G2:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ B5) @ G3) @ G2) => (~ ((((ord_less_eq @ (set @ ((product_prod @ B) @ C))) @ F3) @ G2))))))))))))).
% 3.18/1.62 thf(fact_222_subgraph__semantics, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B),A2:B,B2:B,E3:(allegorical_term @ A)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ A5) @ B5) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ A5))) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ A5) @ E3)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((semantics @ A) @ B) @ B5) @ E3))))))).
% 3.18/1.62 thf(fact_223_proj__iff, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X4:A,Y4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X4) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) @ A5) => ((((((equiv_proj @ A) @ A) @ R2) @ X4) = ((((equiv_proj @ A) @ A) @ R2) @ Y4)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R2))))))).
% 3.18/1.62 thf(fact_224_least__consequence__graphI, axiom, (! [X:$tType,A:$tType,C:$tType,L:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ A)) @ ((labeled_graph @ L) @ A))),G3:((labeled_graph @ L) @ C),S2:((labeled_graph @ L) @ C),T3:(itself @ X)]: (((((((consequence_graph @ L) @ A) @ C) @ Rs) @ G3) => (((((((graph_homomorphism @ L) @ C) @ C) @ S2) @ G3) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S2))) => ((! [C6:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs) @ C6) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S2) @ G3)) @ C6)))) => ((((((((least_559130134_graph @ X) @ L) @ A) @ C) @ T3) @ Rs) @ S2) @ G3))))))).
% 3.18/1.62 thf(fact_225_leastI, axiom, (! [X:$tType,A:$tType,C:$tType,L:$tType,S2:((labeled_graph @ L) @ C),G3:((labeled_graph @ L) @ C),Rs:(set @ ((product_prod @ ((labeled_graph @ L) @ A)) @ ((labeled_graph @ L) @ A))),T3:(itself @ X)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S2) @ G3) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S2))) => ((! [C6:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ A) @ X) @ Rs) @ C6) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S2) @ G3)) @ C6)))) => ((((((((least @ X) @ L) @ A) @ C) @ T3) @ Rs) @ S2) @ G3)))))).
% 3.18/1.62 thf(fact_226_least__consequence__graph__def, axiom, (! [L:$tType,V:$tType,C:$tType,X:$tType]: ((((((least_559130134_graph @ X) @ L) @ V) @ C) = (^ [T2:(itself @ X),Rs2:(set @ ((product_prod @ ((labeled_graph @ L) @ V)) @ ((labeled_graph @ L) @ V))),S3:((labeled_graph @ L) @ C),G7:((labeled_graph @ L) @ C)]: (((((((consequence_graph @ L) @ V) @ C) @ Rs2) @ G7) & ((((((((least @ X) @ L) @ V) @ C) @ T2) @ Rs2) @ S3) @ G7)))))))).
% 3.18/1.62 thf(fact_227_consequence__graphD_I3_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G3:((labeled_graph @ A) @ C)]: (((((((consequence_graph @ A) @ B) @ C) @ Rs) @ G3) => (G3 = (((restrict @ A) @ C) @ G3)))))).
% 3.18/1.62 thf(fact_228_consequence__graphD_I1_J, axiom, (! [B:$tType,C:$tType,A:$tType,Rs:(set @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))),G3:((labeled_graph @ A) @ C),R:((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))]: (((((((consequence_graph @ A) @ B) @ C) @ Rs) @ G3) => ((((member @ ((product_prod @ ((labeled_graph @ A) @ B)) @ ((labeled_graph @ A) @ B))) @ R) @ Rs) => (((((maintained @ A) @ B) @ C) @ R) @ G3)))))).
% 3.18/1.62 thf(fact_229_proj__in__iff, axiom, (! [A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => ((((member @ (set @ A)) @ ((((equiv_proj @ A) @ A) @ R2) @ X4)) @ (((equiv_quotient @ A) @ A5) @ R2)) = (((member @ A) @ X4) @ A5)))))).
% 3.18/1.62 thf(fact_230_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))),S3:((labeled_graph @ L) @ C),G7:((labeled_graph @ L) @ C)]: ((((((((graph_homomorphism @ L) @ C) @ C) @ S3) @ G7) @ ((id_on @ C) @ (((labeled_vertices @ L) @ C) @ S3))) & (! [C4:((labeled_graph @ L) @ X)]: (((((((consequence_graph @ L) @ V) @ X) @ Rs2) @ C4) => (((((maintained @ L) @ C) @ X) @ ((((product_Pair @ ((labeled_graph @ L) @ C)) @ ((labeled_graph @ L) @ C)) @ S3) @ G7)) @ C4))))))))))).
% 3.18/1.62 thf(fact_231_less__by__empty, axiom, (! [A:$tType,A5:(set @ ((product_prod @ A) @ A)),B5:(set @ ((product_prod @ A) @ A))]: (((A5 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A5) @ B5))))).
% 3.18/1.62 thf(fact_232_insert__subsetI, axiom, (! [A:$tType,X4:A,A5:(set @ A),X7:(set @ A)]: (((((member @ A) @ X4) @ A5) => ((((ord_less_eq @ (set @ A)) @ X7) @ A5) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X4) @ X7)) @ A5)))))).
% 3.18/1.62 thf(fact_233_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S:B,R:(set @ ((product_prod @ A) @ B)),S4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S)) @ R) => ((S4 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S4)) @ R)))))).
% 3.18/1.62 thf(fact_234_subset__emptyI, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A5))))) => (((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))))))).
% 3.18/1.62 thf(fact_235_univ__commute, axiom, (! [B:$tType,A:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),F3:(A > B),X4:A]: (((((equiv_equiv @ A) @ A5) @ R2) => (((((equiv_congruent @ A) @ B) @ R2) @ F3) => ((((member @ A) @ X4) @ A5) => (((((bNF_Greatest_univ @ A) @ B) @ F3) @ ((((equiv_proj @ A) @ A) @ R2) @ X4)) = (F3 @ X4)))))))).
% 3.18/1.62 thf(fact_236_subgraph__def, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: ((((((((graph_homomorphism @ A) @ B) @ B) @ G_1) @ G_2) @ ((id_on @ B) @ (((labeled_vertices @ A) @ B) @ G_1))) = ((G_1 = (((restrict @ A) @ B) @ G_1)) & ((G_2 = (((restrict @ A) @ B) @ G_2)) & (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2))))))).
% 3.18/1.62 thf(fact_237_graph__union__idemp_I3_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ ((((graph_union @ A) @ B) @ B5) @ A5)) = ((((graph_union @ A) @ B) @ B5) @ A5))))).
% 3.18/1.62 thf(fact_238_graph__union__idemp_I2_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B),B5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ ((((graph_union @ A) @ B) @ A5) @ B5)) = ((((graph_union @ A) @ B) @ A5) @ B5))))).
% 3.18/1.62 thf(fact_239_graph__union__idemp_I1_J, axiom, (! [B:$tType,A:$tType,A5:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A5) @ A5) = A5)))).
% 3.18/1.62 thf(fact_240_graph__union__preserves__restrict, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((G_1 = (((restrict @ A) @ B) @ G_1)) => ((G_2 = (((restrict @ A) @ B) @ G_2)) => (((((graph_union @ A) @ B) @ G_1) @ G_2) = (((restrict @ A) @ B) @ ((((graph_union @ A) @ B) @ G_1) @ G_2)))))))).
% 3.18/1.62 thf(fact_241_univ__preserves, axiom, (! [A:$tType,B:$tType,A5:(set @ A),R2:(set @ ((product_prod @ A) @ A)),F3:(A > B),B5:(set @ B)]: (((((equiv_equiv @ A) @ A5) @ R2) => (((((equiv_congruent @ A) @ B) @ R2) @ F3) => ((! [X3:A]: (((((member @ A) @ X3) @ A5) => (((member @ B) @ (F3 @ X3)) @ B5)))) => (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ (((equiv_quotient @ A) @ A5) @ R2)) => (((member @ B) @ ((((bNF_Greatest_univ @ A) @ B) @ F3) @ X2)) @ B5)))))))))).
% 3.18/1.62 thf(fact_242_relImage__proj, axiom, (! [A:$tType,A5:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((((equiv_equiv @ A) @ A5) @ R) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((((bNF_Gr1317331620lImage @ A) @ (set @ A)) @ R) @ (((equiv_proj @ A) @ A) @ R))) @ ((id_on @ (set @ A)) @ (((equiv_quotient @ A) @ A5) @ R))))))).
% 3.18/1.62 thf(fact_243_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))))))))).
% 3.18/1.62 thf(fact_244_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)))).
% 3.18/1.62 thf(fact_245_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)))))).
% 3.18/1.62 thf(fact_246_restrict__subsD, axiom, (! [B:$tType,A:$tType,G3:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G3)) @ (((labeled_edges @ A) @ B) @ (((restrict @ A) @ B) @ G3))) => (G3 = (((restrict @ A) @ B) @ G3)))))).
% 3.18/1.62 thf(fact_247_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)))))).
% 3.18/1.62 thf(fact_248_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))))).
% 3.18/1.62 thf(fact_249_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)))).
% 3.18/1.62 thf(fact_250_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))))))).
% 3.18/1.62 thf(fact_251_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)))))).
% 3.18/1.62 thf(fact_252_labeled__graph_Osplit__sel__asm, axiom, (! [A:$tType,V:$tType,L:$tType,P2:(A > $o),F3:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P2 @ (((((labele1974067554_graph @ L) @ V) @ A) @ F3) @ Labeled_graph)) = (~ (((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) & (~ ((P2 @ ((F3 @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))))))).
% 3.18/1.62 thf(fact_253_labeled__graph_Osplit__sel, axiom, (! [A:$tType,V:$tType,L:$tType,P2:(A > $o),F3:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P2 @ (((((labele1974067554_graph @ L) @ V) @ A) @ F3) @ Labeled_graph)) = ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) => (P2 @ ((F3 @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))).
% 3.18/1.62 thf(fact_254_labeled__graph_Ocase, axiom, (! [L:$tType,A:$tType,V:$tType,F3:((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) @ F3) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = ((F3 @ X1) @ X22))))).
% 3.18/1.62 thf(fact_255_labeled__graph_Ocase__eq__if, axiom, (! [A:$tType,V:$tType,L:$tType]: (((((labele1974067554_graph @ L) @ V) @ A) = (^ [F4:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph3:((labeled_graph @ L) @ V)]: (((F4 @ (((labeled_edges @ L) @ V) @ Labeled_graph3)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph3)))))))).
% 3.18/1.62 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 3.18/1.62 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 3.18/1.62 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 3.18/1.62 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 3.18/1.62 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 3.18/1.62 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1, axiom, (order_bot @ nat)).
% 3.18/1.62 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, (preorder @ nat)).
% 3.18/1.62 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 3.18/1.62 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, (order @ nat)).
% 3.18/1.62 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, (ord @ nat)).
% 3.18/1.62 thf(tcon_Nat_Onat___Orderings_Obot_5, axiom, (bot @ nat)).
% 3.18/1.62 thf(tcon_Set_Oset___Orderings_Oorder__bot_6, axiom, (! [A8:$tType]: ((order_bot @ (set @ A8))))).
% 3.18/1.62 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 3.18/1.62 thf(tcon_Set_Oset___Orderings_Oorder_8, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 3.18/1.62 thf(tcon_Set_Oset___Orderings_Oord_9, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 3.18/1.62 thf(tcon_Set_Oset___Orderings_Obot_10, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 3.18/1.62 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11, axiom, (order_bot @ $o)).
% 3.18/1.62 thf(tcon_HOL_Obool___Orderings_Opreorder_12, axiom, (preorder @ $o)).
% 3.18/1.62 thf(tcon_HOL_Obool___Orderings_Olinorder_13, axiom, (linorder @ $o)).
% 3.18/1.62 thf(tcon_HOL_Obool___Orderings_Oorder_14, axiom, (order @ $o)).
% 3.18/1.62 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, (ord @ $o)).
% 3.18/1.62 thf(tcon_HOL_Obool___Orderings_Obot_16, axiom, (bot @ $o)).
% 3.18/1.62 thf(conj_0, conjecture, (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ z) @ y)) @ ((((getRel @ a) @ b) @ i) @ g))).
% 3.18/1.62 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ANXxQNlu00/DTF2DTF30214.p
% 3.18/1.62 ------------------------
% 3.18/1.66 ---- Cleaned THF ---
% 3.18/1.66 thf(ty_t_LabeledGraphSemantics_Oallegorical__term,type,
% 3.18/1.66 allegorical_term: $tType > $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 3.18/1.66 labeled_graph: $tType > $tType > $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_t_Product__Type_Oprod,type,
% 3.18/1.66 product_prod: $tType > $tType > $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_t_Set_Oset,type,
% 3.18/1.66 set: $tType > $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_t_Nat_Onat,type,
% 3.18/1.66 nat: $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_t_itself,type,
% 3.18/1.66 itself: $tType > $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_tf_b,type,
% 3.18/1.66 b: $tType ).
% 3.18/1.66
% 3.18/1.66 thf(ty_tf_a,type,
% 3.18/1.66 a: $tType ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_HOL_Otype,type,
% 3.18/1.66 type:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_Orderings_Obot,type,
% 3.18/1.66 bot:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_Orderings_Oord,type,
% 3.18/1.66 ord:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_Orderings_Oorder,type,
% 3.18/1.66 order:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_Orderings_Olinorder,type,
% 3.18/1.66 linorder:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_Orderings_Opreorder,type,
% 3.18/1.66 preorder:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_cl_Orderings_Oorder__bot,type,
% 3.18/1.66 order_bot:
% 3.18/1.66 !>[A: $tType] : $o ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
% 3.18/1.66 bNF_Gr1317331620lImage:
% 3.18/1.66 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
% 3.18/1.66 bNF_Greatest_univ:
% 3.18/1.66 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 3.18/1.66 equiv_congruent:
% 3.18/1.66 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Equiv__Relations_Ocongruent2,type,
% 3.18/1.66 equiv_congruent2:
% 3.18/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Equiv__Relations_Oequiv,type,
% 3.18/1.66 equiv_equiv:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Equiv__Relations_Oproj,type,
% 3.18/1.66 equiv_proj:
% 3.18/1.66 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Equiv__Relations_Oquotient,type,
% 3.18/1.66 equiv_quotient:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 3.18/1.66 getRel:
% 3.18/1.66 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphSemantics_Osemantics,type,
% 3.18/1.66 semantics:
% 3.18/1.66 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( allegorical_term @ A ) > ( set @ ( product_prod @ B @ B ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 3.18/1.66 graph_homomorphism:
% 3.18/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 3.18/1.66 graph_union:
% 3.18/1.66 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 3.18/1.66 labeled_LG:
% 3.18/1.66 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 3.18/1.66 labele1974067554_graph:
% 3.18/1.66 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 3.18/1.66 labeled_edges:
% 3.18/1.66 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 3.18/1.66 labeled_vertices:
% 3.18/1.66 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Oon__triple,type,
% 3.18/1.66 on_triple:
% 3.18/1.66 !>[B: $tType,C: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_LabeledGraphs_Orestrict,type,
% 3.18/1.66 restrict:
% 3.18/1.66 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_MissingRelation_Oidempotent,type,
% 3.18/1.66 idempotent:
% 3.18/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 3.18/1.66 order_1409979114der_on:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Order__Relation_Opreorder__on,type,
% 3.18/1.66 order_preorder_on:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.18/1.66 bot_bot:
% 3.18/1.66 !>[A: $tType] : A ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.18/1.66 ord_less_eq:
% 3.18/1.66 !>[A: $tType] : ( A > A > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Product__Type_OPair,type,
% 3.18/1.66 product_Pair:
% 3.18/1.66 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.18/1.66 produc2004651681e_prod:
% 3.18/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.18/1.66 product_rec_prod:
% 3.18/1.66 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Relation_OId__on,type,
% 3.18/1.66 id_on:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Relation_Orefl__on,type,
% 3.18/1.66 refl_on:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Relation_Osym,type,
% 3.18/1.66 sym:
% 3.18/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Relation_Ototal__on,type,
% 3.18/1.66 total_on:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Relation_Otrans,type,
% 3.18/1.66 trans:
% 3.18/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Oagree__on,type,
% 3.18/1.66 agree_on:
% 3.18/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Oconsequence__graph,type,
% 3.18/1.66 consequence_graph:
% 3.18/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) ) > ( labeled_graph @ A @ C ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Oextensible,type,
% 3.18/1.66 extensible:
% 3.18/1.66 !>[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 ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Ofin__maintained,type,
% 3.18/1.66 fin_maintained:
% 3.18/1.66 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Oleast,type,
% 3.18/1.66 least:
% 3.18/1.66 !>[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 ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Oleast__consequence__graph,type,
% 3.18/1.66 least_559130134_graph:
% 3.18/1.66 !>[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 ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_RulesAndChains_Omaintained,type,
% 3.18/1.66 maintained:
% 3.18/1.66 !>[L: $tType,X: $tType,V: $tType] : ( ( product_prod @ ( labeled_graph @ L @ X ) @ ( labeled_graph @ L @ X ) ) > ( labeled_graph @ L @ V ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Set_OCollect,type,
% 3.18/1.66 collect:
% 3.18/1.66 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Set_Oinsert,type,
% 3.18/1.66 insert:
% 3.18/1.66 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Set_Ois__empty,type,
% 3.18/1.66 is_empty:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Set_Ois__singleton,type,
% 3.18/1.66 is_singleton:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Set_Othe__elem,type,
% 3.18/1.66 the_elem:
% 3.18/1.66 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ocongruence__rule,type,
% 3.18/1.66 standa1068660421e_rule:
% 3.18/1.66 !>[L: $tType] : ( L > L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Ononempty__rule,type,
% 3.18/1.66 standa244753842y_rule:
% 3.18/1.66 !>[L: $tType] : ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Oreflexivity__rule,type,
% 3.18/1.66 standa245363227y_rule:
% 3.18/1.66 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Osymmetry__rule,type,
% 3.18/1.66 standa1805702094y_rule:
% 3.18/1.66 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_StandardRules__Mirabelle__xlrcbcygrg_Otransitive__rule,type,
% 3.18/1.66 standa2114124375e_rule:
% 3.18/1.66 !>[L: $tType] : ( L > ( product_prod @ ( labeled_graph @ L @ nat ) @ ( labeled_graph @ L @ nat ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_Wfrec_Osame__fst,type,
% 3.18/1.66 same_fst:
% 3.18/1.66 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_c_member,type,
% 3.18/1.66 member:
% 3.18/1.66 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.18/1.66
% 3.18/1.66 thf(sy_v_G,type,
% 3.18/1.66 g: labeled_graph @ a @ b ).
% 3.18/1.66
% 3.18/1.66 thf(sy_v_I,type,
% 3.18/1.66 i: a ).
% 3.18/1.66
% 3.18/1.66 thf(sy_v_l,type,
% 3.18/1.66 l: a ).
% 3.18/1.66
% 3.18/1.66 thf(sy_v_y____,type,
% 3.18/1.66 y: b ).
% 3.18/1.66
% 3.18/1.66 thf(sy_v_z____,type,
% 3.18/1.66 z: b ).
% 3.18/1.66
% 3.18/1.66 thf(fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062,axiom,
% 3.18/1.66 ( ( refl_on @ b @ ( labeled_vertices @ a @ b @ g ) @ ( getRel @ a @ b @ i @ g ) )
% 3.18/1.66 & ! [X2: b,Y: b] :
% 3.18/1.66 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X2 @ Y ) @ ( getRel @ a @ b @ i @ g ) )
% 3.18/1.66 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ Y @ X2 ) @ ( getRel @ a @ b @ i @ g ) ) )
% 3.18/1.66 & ( trans @ b @ ( getRel @ a @ b @ i @ g ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_1_aI,axiom,
% 3.18/1.66 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ y @ z ) @ ( getRel @ a @ b @ i @ g ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_2_g,axiom,
% 3.18/1.66 ( g
% 3.18/1.66 = ( restrict @ a @ b @ g ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_3_m,axiom,
% 3.18/1.66 maintained @ a @ nat @ b @ ( standa1068660421e_rule @ a @ i @ l ) @ g ).
% 3.18/1.66
% 3.18/1.66 thf(fact_4_eq,axiom,
% 3.18/1.66 equiv_equiv @ b @ ( labeled_vertices @ a @ b @ g ) @ ( getRel @ a @ b @ i @ g ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_5_prod_Oinject,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 3.18/1.66 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 3.18/1.66 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.18/1.66 = ( ( X1 = Y1 )
% 3.18/1.66 & ( X22 = Y2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_6_old_Oprod_Oinject,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 3.18/1.66 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.18/1.66 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.18/1.66 = ( ( A2 = A3 )
% 3.18/1.66 & ( B2 = B3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_7_surj__pair,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 3.18/1.66 ? [X3: A,Y3: B] :
% 3.18/1.66 ( P
% 3.18/1.66 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_8_prod__cases,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 3.18/1.66 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.18/1.66 => ( P2 @ P ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_9_Pair__inject,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 3.18/1.66 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.18/1.66 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.18/1.66 => ~ ( ( A2 = A3 )
% 3.18/1.66 => ( B2 != B3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_10_prod__cases3,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.18/1.66 ~ ! [A4: A,B4: B,C2: C] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_11_prod__cases4,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.18/1.66 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_12_prod__cases5,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.18/1.66 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_13_prod__cases6,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.18/1.66 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_14_prod__cases7,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 3.18/1.66 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_15_old_Oprod_Oinducts,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.18/1.66 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.18/1.66 => ( P2 @ Prod ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_16_old_Oprod_Oexhaust,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
% 3.18/1.66 ~ ! [A4: A,B4: B] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_17_prod__induct7,axiom,
% 3.18/1.66 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 3.18/1.66 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 3.18/1.66 => ( P2 @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_18_prod__induct6,axiom,
% 3.18/1.66 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.18/1.66 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 3.18/1.66 => ( P2 @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_19_prod__induct5,axiom,
% 3.18/1.66 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.18/1.66 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.18/1.66 => ( P2 @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_20_prod__induct4,axiom,
% 3.18/1.66 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.18/1.66 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 3.18/1.66 => ( P2 @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_21_prod__induct3,axiom,
% 3.18/1.66 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 3.18/1.66 => ( P2 @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_22_vertices__restrict,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,G3: labeled_graph @ B @ A] :
% 3.18/1.66 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G3 ) )
% 3.18/1.66 = ( labeled_vertices @ B @ A @ G3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_23_getRel__dom_I1_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G3 ) )
% 3.18/1.66 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_24_getRel__dom_I2_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,A2: B,B2: B,L2: A] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L2 @ G3 ) )
% 3.18/1.66 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_25_transitive__rule,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ( maintained @ A @ nat @ B @ ( standa2114124375e_rule @ A @ L2 ) @ G3 )
% 3.18/1.66 => ( trans @ B @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_26_old_Oprod_Orec,axiom,
% 3.18/1.66 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.18/1.66 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.18/1.66 = ( F1 @ A2 @ B2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_27_maintained__refl,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,R: labeled_graph @ A @ B,G3: labeled_graph @ A @ C] : ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G3 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_28_reflexivity__rule,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ( maintained @ A @ nat @ B @ ( standa245363227y_rule @ A @ L2 ) @ G3 )
% 3.18/1.66 => ( refl_on @ B @ ( labeled_vertices @ A @ B @ G3 ) @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_29_restrict__idemp,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,X4: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X4 ) )
% 3.18/1.66 = ( restrict @ A @ B @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_30_refl__on__domain,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 3.18/1.66 ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 3.18/1.66 => ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 & ( member @ A @ B2 @ A5 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_31_refl__onD2,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 3.18/1.66 ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 3.18/1.66 => ( member @ A @ Y4 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_32_transD,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A,Z: A] :
% 3.18/1.66 ( ( trans @ A @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_33_transE,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A,Z: A] :
% 3.18/1.66 ( ( trans @ A @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_34_transI,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ! [X3: A,Y3: A,Z2: A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) )
% 3.18/1.66 => ( trans @ A @ R2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_35_Relation_Otrans__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( trans @ A )
% 3.18/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ! [X5: A,Y5: A,Z3: A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R3 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_36_refl__onD,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
% 3.18/1.66 ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_37_refl__onD1,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 3.18/1.66 ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 3.18/1.66 => ( member @ A @ X4 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_38_mA,axiom,
% 3.18/1.66 ! [X2: product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat )] :
% 3.18/1.66 ( ( member @ ( product_prod @ ( labeled_graph @ a @ nat ) @ ( labeled_graph @ a @ nat ) ) @ X2 @ ( 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 ) ) ) ) ) ) ) )
% 3.18/1.66 => ( maintained @ a @ nat @ b @ X2 @ g ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_39_internal__case__prod__conv,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 3.18/1.66 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.18/1.66 = ( C3 @ A2 @ B2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_40_preorder__on__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order_preorder_on @ A )
% 3.18/1.66 = ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( refl_on @ A @ A6 @ R3 )
% 3.18/1.66 & ( trans @ A @ R3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_41_on__triple,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 3.18/1.66 = ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V1 @ V3 ) @ R )
% 3.18/1.66 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V2 @ V4 ) @ R )
% 3.18/1.66 & ( L1 = L22 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_42_maintained__then__fin__maintained,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ),G3: labeled_graph @ A @ C] :
% 3.18/1.66 ( ( maintained @ A @ B @ C @ R @ G3 )
% 3.18/1.66 => ( fin_maintained @ A @ B @ C @ R @ G3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_43_refl__trans__impl__idempotent,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( trans @ A @ R2 )
% 3.18/1.66 => ( idempotent @ A @ R2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_44_equivalence,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,I: A] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ! [X3: product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat )] :
% 3.18/1.66 ( ( member @ ( product_prod @ ( labeled_graph @ A @ nat ) @ ( labeled_graph @ A @ nat ) ) @ X3 @ ( 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 ) ) ) ) ) ) ) )
% 3.18/1.66 => ( maintained @ A @ nat @ B @ X3 @ G3 ) )
% 3.18/1.66 => ( equiv_equiv @ B @ ( labeled_vertices @ A @ B @ G3 ) @ ( getRel @ A @ B @ I @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_45_mem__Collect__eq,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,P2: A > $o] :
% 3.18/1.66 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 3.18/1.66 = ( P2 @ A2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_46_Collect__mem__eq,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ( collect @ A
% 3.18/1.66 @ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
% 3.18/1.66 = A5 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_47_Collect__cong,axiom,
% 3.18/1.66 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.18/1.66 ( ! [X3: A] :
% 3.18/1.66 ( ( P2 @ X3 )
% 3.18/1.66 = ( Q @ X3 ) )
% 3.18/1.66 => ( ( collect @ A @ P2 )
% 3.18/1.66 = ( collect @ A @ Q ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_48_ext,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,F3: A > B,G4: A > B] :
% 3.18/1.66 ( ! [X3: A] :
% 3.18/1.66 ( ( F3 @ X3 )
% 3.18/1.66 = ( G4 @ X3 ) )
% 3.18/1.66 => ( F3 = G4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_49_semantics__in__vertices_I1_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( restrict @ A @ B @ A5 ) )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 3.18/1.66 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_50_semantics__in__vertices_I2_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( restrict @ A @ B @ A5 ) )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 3.18/1.66 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ A5 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_51_refl__on__singleton,axiom,
% 3.18/1.66 ! [A: $tType,X4: A] : ( refl_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_52_preorder__on__empty,axiom,
% 3.18/1.66 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_53_refl__on__empty,axiom,
% 3.18/1.66 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_54_trans__singleton,axiom,
% 3.18/1.66 ! [A: $tType,A2: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_55_idempotent__impl__trans,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( idempotent @ A @ R2 )
% 3.18/1.66 => ( trans @ A @ R2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_56_trans__empty,axiom,
% 3.18/1.66 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_57_on__tripleD_I1_J,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 3.18/1.66 => ( L22 = L1 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_58_on__tripleD_I2_J,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 3.18/1.66 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V1 @ V3 ) @ R ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_59_on__tripleD_I3_J,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,L1: A,V1: B,V2: B,L22: A,V3: C,V4: C,R: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( member @ ( product_prod @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_prod @ A @ ( product_prod @ C @ C ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ L1 @ ( product_Pair @ B @ B @ V1 @ V2 ) ) @ ( product_Pair @ A @ ( product_prod @ C @ C ) @ L22 @ ( product_Pair @ C @ C @ V3 @ V4 ) ) ) @ ( on_triple @ B @ C @ A @ R ) )
% 3.18/1.66 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ V2 @ V4 ) @ R ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_60_singletonI,axiom,
% 3.18/1.66 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_61_nonempty__rule,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,G3: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ( maintained @ A @ nat @ B @ ( standa244753842y_rule @ A ) @ G3 )
% 3.18/1.66 = ( ( labeled_vertices @ A @ B @ G3 )
% 3.18/1.66 != ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_62_insertCI,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 3.18/1.66 ( ( ~ ( member @ A @ A2 @ B5 )
% 3.18/1.66 => ( A2 = B2 ) )
% 3.18/1.66 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_63_insert__iff,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 3.18/1.66 = ( ( A2 = B2 )
% 3.18/1.66 | ( member @ A @ A2 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_64_insert__absorb2,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,A5: set @ A] :
% 3.18/1.66 ( ( insert @ A @ X4 @ ( insert @ A @ X4 @ A5 ) )
% 3.18/1.66 = ( insert @ A @ X4 @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_65_empty__iff,axiom,
% 3.18/1.66 ! [A: $tType,C3: A] :
% 3.18/1.66 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_66_all__not__in__conv,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ( ! [X5: A] :
% 3.18/1.66 ~ ( member @ A @ X5 @ A5 ) )
% 3.18/1.66 = ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_67_Collect__empty__eq,axiom,
% 3.18/1.66 ! [A: $tType,P2: A > $o] :
% 3.18/1.66 ( ( ( collect @ A @ P2 )
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 = ( ! [X5: A] :
% 3.18/1.66 ~ ( P2 @ X5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_68_empty__Collect__eq,axiom,
% 3.18/1.66 ! [A: $tType,P2: A > $o] :
% 3.18/1.66 ( ( ( bot_bot @ ( set @ A ) )
% 3.18/1.66 = ( collect @ A @ P2 ) )
% 3.18/1.66 = ( ! [X5: A] :
% 3.18/1.66 ~ ( P2 @ X5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_69_ex__in__conv,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ( ? [X5: A] : ( member @ A @ X5 @ A5 ) )
% 3.18/1.66 = ( A5
% 3.18/1.66 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_70_equals0I,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ! [Y3: A] :
% 3.18/1.66 ~ ( member @ A @ Y3 @ A5 )
% 3.18/1.66 => ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_71_equals0D,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,A2: A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 => ~ ( member @ A @ A2 @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_72_emptyE,axiom,
% 3.18/1.66 ! [A: $tType,A2: A] :
% 3.18/1.66 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_73_mk__disjoint__insert,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ? [B6: set @ A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( insert @ A @ A2 @ B6 ) )
% 3.18/1.66 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_74_insert__commute,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,Y4: A,A5: set @ A] :
% 3.18/1.66 ( ( insert @ A @ X4 @ ( insert @ A @ Y4 @ A5 ) )
% 3.18/1.66 = ( insert @ A @ Y4 @ ( insert @ A @ X4 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_75_insert__eq__iff,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,A5: set @ A,B2: A,B5: set @ A] :
% 3.18/1.66 ( ~ ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ( ~ ( member @ A @ B2 @ B5 )
% 3.18/1.66 => ( ( ( insert @ A @ A2 @ A5 )
% 3.18/1.66 = ( insert @ A @ B2 @ B5 ) )
% 3.18/1.66 = ( ( ( A2 = B2 )
% 3.18/1.66 => ( A5 = B5 ) )
% 3.18/1.66 & ( ( A2 != B2 )
% 3.18/1.66 => ? [C4: set @ A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( insert @ A @ B2 @ C4 ) )
% 3.18/1.66 & ~ ( member @ A @ B2 @ C4 )
% 3.18/1.66 & ( B5
% 3.18/1.66 = ( insert @ A @ A2 @ C4 ) )
% 3.18/1.66 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_76_insert__absorb,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ( ( insert @ A @ A2 @ A5 )
% 3.18/1.66 = A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_77_insert__ident,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ~ ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ( ~ ( member @ A @ X4 @ B5 )
% 3.18/1.66 => ( ( ( insert @ A @ X4 @ A5 )
% 3.18/1.66 = ( insert @ A @ X4 @ B5 ) )
% 3.18/1.66 = ( A5 = B5 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_78_Set_Oset__insert,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ~ ! [B6: set @ A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( insert @ A @ X4 @ B6 ) )
% 3.18/1.66 => ( member @ A @ X4 @ B6 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_79_insertI2,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 3.18/1.66 ( ( member @ A @ A2 @ B5 )
% 3.18/1.66 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_80_insertI1,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_81_insertE,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
% 3.18/1.66 => ( ( A2 != B2 )
% 3.18/1.66 => ( member @ A @ A2 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_82_singleton__inject,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B2: A] :
% 3.18/1.66 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 => ( A2 = B2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_83_insert__not__empty,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,A5: set @ A] :
% 3.18/1.66 ( ( insert @ A @ A2 @ A5 )
% 3.18/1.66 != ( bot_bot @ ( set @ A ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_84_doubleton__eq__iff,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
% 3.18/1.66 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 = ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.18/1.66 = ( ( ( A2 = C3 )
% 3.18/1.66 & ( B2 = D3 ) )
% 3.18/1.66 | ( ( A2 = D3 )
% 3.18/1.66 & ( B2 = C3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_85_singleton__iff,axiom,
% 3.18/1.66 ! [A: $tType,B2: A,A2: A] :
% 3.18/1.66 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 = ( B2 = A2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_86_singletonD,axiom,
% 3.18/1.66 ! [A: $tType,B2: A,A2: A] :
% 3.18/1.66 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 => ( B2 = A2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_87_the__elem__eq,axiom,
% 3.18/1.66 ! [A: $tType,X4: A] :
% 3.18/1.66 ( ( the_elem @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 = X4 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_88_bot__apply,axiom,
% 3.18/1.66 ! [C: $tType,D: $tType] :
% 3.18/1.66 ( ( bot @ C )
% 3.18/1.66 => ( ( bot_bot @ ( D > C ) )
% 3.18/1.66 = ( ^ [X5: D] : ( bot_bot @ C ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_89_graph__single,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,A2: A,B2: B,C3: B] :
% 3.18/1.66 ( ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C3 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 3.18/1.66 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( insert @ ( product_prod @ A @ ( product_prod @ B @ B ) ) @ ( product_Pair @ A @ ( product_prod @ B @ B ) @ A2 @ ( product_Pair @ B @ B @ B2 @ C3 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_90_is__singletonI,axiom,
% 3.18/1.66 ! [A: $tType,X4: A] : ( is_singleton @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_91_linear__order__on__singleton,axiom,
% 3.18/1.66 ! [A: $tType,X4: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_92_total__on__singleton,axiom,
% 3.18/1.66 ! [A: $tType,X4: A] : ( total_on @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_93_symmetry__rule,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B,L2: A] :
% 3.18/1.66 ( ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) )
% 3.18/1.66 => ( ( maintained @ A @ nat @ B @ ( standa1805702094y_rule @ A @ L2 ) @ G3 )
% 3.18/1.66 => ( sym @ B @ ( getRel @ A @ B @ L2 @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_94_labeled__graph_Oinject,axiom,
% 3.18/1.66 ! [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] :
% 3.18/1.66 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 3.18/1.66 = ( labeled_LG @ L @ V @ Y1 @ Y2 ) )
% 3.18/1.66 = ( ( X1 = Y1 )
% 3.18/1.66 & ( X22 = Y2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_95_graph__empty__e,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,V5: set @ B] :
% 3.18/1.66 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V5 )
% 3.18/1.66 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_96_labeled__graph_Oexhaust,axiom,
% 3.18/1.66 ! [L: $tType,V: $tType,Y4: labeled_graph @ L @ V] :
% 3.18/1.66 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 3.18/1.66 ( Y4
% 3.18/1.66 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_97_labeled__graph_Oinduct,axiom,
% 3.18/1.66 ! [V: $tType,L: $tType,P2: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 3.18/1.66 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P2 @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 3.18/1.66 => ( P2 @ Labeled_graph ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_98_bot__empty__eq,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( bot_bot @ ( A > $o ) )
% 3.18/1.66 = ( ^ [X5: A] : ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_99_bot__set__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( bot_bot @ ( set @ A ) )
% 3.18/1.66 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_100_sym__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( sym @ A )
% 3.18/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ! [X5: A,Y5: A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_101_symI,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ! [A4: A,B4: A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 ) )
% 3.18/1.66 => ( sym @ A @ R2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_102_symE,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
% 3.18/1.66 ( ( sym @ A @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_103_symD,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
% 3.18/1.66 ( ( sym @ A @ R2 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_104_total__on__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( total_on @ A )
% 3.18/1.66 = ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ! [X5: A] :
% 3.18/1.66 ( ( member @ A @ X5 @ A6 )
% 3.18/1.66 => ! [Y5: A] :
% 3.18/1.66 ( ( member @ A @ Y5 @ A6 )
% 3.18/1.66 => ( ( X5 != Y5 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R3 )
% 3.18/1.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R3 ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_105_total__onI,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ! [X3: A,Y3: A] :
% 3.18/1.66 ( ( member @ A @ X3 @ A5 )
% 3.18/1.66 => ( ( member @ A @ Y3 @ A5 )
% 3.18/1.66 => ( ( X3 != Y3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
% 3.18/1.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 3.18/1.66 => ( total_on @ A @ A5 @ R2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_106_total__on__empty,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_107_labeled__graph_Osel_I2_J,axiom,
% 3.18/1.66 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 3.18/1.66 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 3.18/1.66 = X22 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_108_is__singleton__the__elem,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( is_singleton @ A )
% 3.18/1.66 = ( ^ [A6: set @ A] :
% 3.18/1.66 ( A6
% 3.18/1.66 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_109_is__singletonI_H,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 != ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 => ( ! [X3: A,Y3: A] :
% 3.18/1.66 ( ( member @ A @ X3 @ A5 )
% 3.18/1.66 => ( ( member @ A @ Y3 @ A5 )
% 3.18/1.66 => ( X3 = Y3 ) ) )
% 3.18/1.66 => ( is_singleton @ A @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_110_lnear__order__on__empty,axiom,
% 3.18/1.66 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_111_bot__fun__def,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType] :
% 3.18/1.66 ( ( bot @ B )
% 3.18/1.66 => ( ( bot_bot @ ( A > B ) )
% 3.18/1.66 = ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_112_is__singletonE,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ( is_singleton @ A @ A5 )
% 3.18/1.66 => ~ ! [X3: A] :
% 3.18/1.66 ( A5
% 3.18/1.66 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_113_is__singleton__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( is_singleton @ A )
% 3.18/1.66 = ( ^ [A6: set @ A] :
% 3.18/1.66 ? [X5: A] :
% 3.18/1.66 ( A6
% 3.18/1.66 = ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_114_equivE,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ~ ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( sym @ A @ R2 )
% 3.18/1.66 => ~ ( trans @ A @ R2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_115_equivI,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( refl_on @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( sym @ A @ R2 )
% 3.18/1.66 => ( ( trans @ A @ R2 )
% 3.18/1.66 => ( equiv_equiv @ A @ A5 @ R2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_116_equiv__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( equiv_equiv @ A )
% 3.18/1.66 = ( ^ [A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( refl_on @ A @ A6 @ R3 )
% 3.18/1.66 & ( sym @ A @ R3 )
% 3.18/1.66 & ( trans @ A @ R3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_117_agree__on__empty,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,F3: set @ ( product_prod @ B @ C ),G4: 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 ) ) ) @ F3 @ G4 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_118_graph__homomorphism__empty,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G3 @ F3 )
% 3.18/1.66 = ( ( F3
% 3.18/1.66 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 3.18/1.66 & ( G3
% 3.18/1.66 = ( restrict @ A @ C @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_119_agree__on__comm,axiom,
% 3.18/1.66 ! [C: $tType,B: $tType,A: $tType] :
% 3.18/1.66 ( ( agree_on @ A @ B @ C )
% 3.18/1.66 = ( ^ [X6: labeled_graph @ A @ B,F4: set @ ( product_prod @ B @ C ),G5: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ X6 @ G5 @ F4 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_120_agree__on__refl,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] : ( agree_on @ A @ B @ C @ R @ F3 @ F3 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_121_agree__on__trans,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,X7: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C ),G4: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( agree_on @ A @ B @ C @ X7 @ F3 @ G4 )
% 3.18/1.66 => ( ( agree_on @ A @ B @ C @ X7 @ G4 @ H )
% 3.18/1.66 => ( agree_on @ A @ B @ C @ X7 @ F3 @ H ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_122_graph__homomorphism__semantics,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),A2: B,B2: B,E3: allegorical_term @ A,A3: C,B3: C] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ A3 ) @ F3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ B3 ) @ F3 )
% 3.18/1.66 => ( member @ ( product_prod @ C @ C ) @ ( product_Pair @ C @ C @ A3 @ B3 ) @ ( semantics @ A @ C @ B5 @ E3 ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_123_graph__homomorphism__nonempty,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),E3: allegorical_term @ A] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ A5 @ B5 @ F3 )
% 3.18/1.66 => ( ( ( semantics @ A @ B @ A5 @ E3 )
% 3.18/1.66 != ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 3.18/1.66 => ( ( semantics @ A @ C @ B5 @ E3 )
% 3.18/1.66 != ( bot_bot @ ( set @ ( product_prod @ C @ C ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_124_Collect__empty__eq__bot,axiom,
% 3.18/1.66 ! [A: $tType,P2: A > $o] :
% 3.18/1.66 ( ( ( collect @ A @ P2 )
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 = ( P2
% 3.18/1.66 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_125_extensibleI,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,R22: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,G4: set @ ( product_prod @ B @ C ),R1: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ R22 @ G3 @ G4 )
% 3.18/1.66 => ( ( agree_on @ A @ B @ C @ R1 @ F3 @ G4 )
% 3.18/1.66 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R1 @ R22 ) @ G3 @ F3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_126_maintainedI,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ C @ A,G3: labeled_graph @ C @ B,B5: labeled_graph @ C @ A] :
% 3.18/1.66 ( ! [F2: set @ ( product_prod @ A @ B )] :
% 3.18/1.66 ( ( graph_homomorphism @ C @ A @ B @ A5 @ G3 @ F2 )
% 3.18/1.66 => ( extensible @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G3 @ F2 ) )
% 3.18/1.66 => ( maintained @ C @ A @ B @ ( product_Pair @ ( labeled_graph @ C @ A ) @ ( labeled_graph @ C @ A ) @ A5 @ B5 ) @ G3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_127_extensible__refl__concr,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,E_1: set @ ( product_prod @ A @ ( product_prod @ B @ B ) ),V5: set @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C ),E_2: set @ ( product_prod @ A @ ( product_prod @ B @ B ) )] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_1 @ V5 ) @ G3 @ F3 )
% 3.18/1.66 => ( ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ ( labeled_LG @ A @ B @ E_1 @ V5 ) @ ( labeled_LG @ A @ B @ E_2 @ V5 ) ) @ G3 @ F3 )
% 3.18/1.66 = ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ E_2 @ V5 ) @ G3 @ F3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_128_extensible__refl,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ R @ G3 @ F3 )
% 3.18/1.66 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ R @ R ) @ G3 @ F3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_129_maintainedD,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 )
% 3.18/1.66 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G3 @ F3 )
% 3.18/1.66 => ( extensible @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 @ F3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_130_Set_Ois__empty__def,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( is_empty @ A )
% 3.18/1.66 = ( ^ [A6: set @ A] :
% 3.18/1.66 ( A6
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_131_congruent2I,axiom,
% 3.18/1.66 ! [C: $tType,B: $tType,A: $tType,A1: set @ A,R12: set @ ( product_prod @ A @ A ),A22: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A1 @ R12 )
% 3.18/1.66 => ( ( equiv_equiv @ B @ A22 @ R23 )
% 3.18/1.66 => ( ! [Y3: A,Z2: A,W: B] :
% 3.18/1.66 ( ( member @ B @ W @ A22 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R12 )
% 3.18/1.66 => ( ( F3 @ Y3 @ W )
% 3.18/1.66 = ( F3 @ Z2 @ W ) ) ) )
% 3.18/1.66 => ( ! [Y3: B,Z2: B,W: A] :
% 3.18/1.66 ( ( member @ A @ W @ A1 )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z2 ) @ R23 )
% 3.18/1.66 => ( ( F3 @ W @ Y3 )
% 3.18/1.66 = ( F3 @ W @ Z2 ) ) ) )
% 3.18/1.66 => ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_132_congruent2__commuteI,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > A > B] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ! [Y3: A,Z2: A] :
% 3.18/1.66 ( ( member @ A @ Y3 @ A5 )
% 3.18/1.66 => ( ( member @ A @ Z2 @ A5 )
% 3.18/1.66 => ( ( F3 @ Y3 @ Z2 )
% 3.18/1.66 = ( F3 @ Z2 @ Y3 ) ) ) )
% 3.18/1.66 => ( ! [Y3: A,Z2: A,W: A] :
% 3.18/1.66 ( ( member @ A @ W @ A5 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
% 3.18/1.66 => ( ( F3 @ W @ Y3 )
% 3.18/1.66 = ( F3 @ W @ Z2 ) ) ) )
% 3.18/1.66 => ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_133_eq__equiv__class__iff2,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ( ( member @ A @ Y4 @ A5 )
% 3.18/1.66 => ( ( ( equiv_quotient @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
% 3.18/1.66 = ( equiv_quotient @ A @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
% 3.18/1.66 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_134_quotient__is__empty2,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( ( bot_bot @ ( set @ ( set @ A ) ) )
% 3.18/1.66 = ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 = ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_135_quotient__is__empty,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( ( equiv_quotient @ A @ A5 @ R2 )
% 3.18/1.66 = ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 3.18/1.66 = ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_136_quotient__empty,axiom,
% 3.18/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
% 3.18/1.66 = ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_137_in__quotient__imp__closed,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X4: A,Y4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ( member @ A @ X4 @ X7 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 3.18/1.66 => ( member @ A @ Y4 @ X7 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_138_quotient__eq__iff,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y6: set @ A,X4: A,Y4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ( member @ A @ X4 @ X7 )
% 3.18/1.66 => ( ( member @ A @ Y4 @ Y6 )
% 3.18/1.66 => ( ( X7 = Y6 )
% 3.18/1.66 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_139_quotient__eqI,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y6: set @ A,X4: A,Y4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ( member @ A @ X4 @ X7 )
% 3.18/1.66 => ( ( member @ A @ Y4 @ Y6 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 )
% 3.18/1.66 => ( X7 = Y6 ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_140_in__quotient__imp__non__empty,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( X7
% 3.18/1.66 != ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_141_congruent2I_H,axiom,
% 3.18/1.66 ! [C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
% 3.18/1.66 ( ! [Y12: A,Z1: A,Y22: B,Z22: B] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R12 )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z22 ) @ R23 )
% 3.18/1.66 => ( ( F3 @ Y12 @ Y22 )
% 3.18/1.66 = ( F3 @ Z1 @ Z22 ) ) ) )
% 3.18/1.66 => ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_142_congruent2D,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z12: A,Y2: B,Z23: B] :
% 3.18/1.66 ( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R12 )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z23 ) @ R23 )
% 3.18/1.66 => ( ( F3 @ Y1 @ Y2 )
% 3.18/1.66 = ( F3 @ Z12 @ Z23 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_143_congruent2__implies__congruent,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,A5: set @ A,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,A2: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R12 )
% 3.18/1.66 => ( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
% 3.18/1.66 => ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ( equiv_congruent @ B @ C @ R23 @ ( F3 @ A2 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_144_in__quotient__imp__in__rel,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X4: A,Y4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ X7 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_145_same__fstI,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,P2: A > $o,X4: A,Y7: B,Y4: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.18/1.66 ( ( P2 @ X4 )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y4 ) @ ( R @ X4 ) )
% 3.18/1.66 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y7 ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_146_getRel__subgraph,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,Y4: A,Z: A,L2: B,G3: labeled_graph @ B @ A,G6: labeled_graph @ B @ A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( getRel @ B @ A @ L2 @ G3 ) )
% 3.18/1.66 => ( ( graph_homomorphism @ B @ A @ A @ G3 @ G6 @ ( id_on @ A @ ( labeled_vertices @ B @ A @ G3 ) ) )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( getRel @ B @ A @ L2 @ G6 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_147_order__refl,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( preorder @ A )
% 3.18/1.66 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_148_subset__antisym,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 3.18/1.66 => ( A5 = B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_149_subsetI,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ! [X3: A] :
% 3.18/1.66 ( ( member @ A @ X3 @ A5 )
% 3.18/1.66 => ( member @ A @ X3 @ B5 ) )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_150_empty__subsetI,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_151_subset__empty,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 = ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_152_insert__subset,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ A5 ) @ B5 )
% 3.18/1.66 = ( ( member @ A @ X4 @ B5 )
% 3.18/1.66 & ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_153_Id__onI,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_154_singleton__insert__inj__eq,axiom,
% 3.18/1.66 ! [A: $tType,B2: A,A2: A,A5: set @ A] :
% 3.18/1.66 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 = ( insert @ A @ A2 @ A5 ) )
% 3.18/1.66 = ( ( A2 = B2 )
% 3.18/1.66 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_155_singleton__insert__inj__eq_H,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,A5: set @ A,B2: A] :
% 3.18/1.66 ( ( ( insert @ A @ A2 @ A5 )
% 3.18/1.66 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 = ( ( A2 = B2 )
% 3.18/1.66 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_156_Id__on__empty,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_157_subgraph__refl,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ G3 @ G3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G3 ) ) )
% 3.18/1.66 = ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_158_subgraph__restrict,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ G3 @ ( restrict @ A @ B @ G3 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G3 ) ) )
% 3.18/1.66 = ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_159_graph__homomorphism__Id,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A2: labeled_graph @ A @ B] : ( graph_homomorphism @ A @ B @ B @ ( restrict @ A @ B @ A2 ) @ ( restrict @ A @ B @ A2 ) @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_160_subgraph__subset_I1_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 3.18/1.66 => ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_161_Id__onE,axiom,
% 3.18/1.66 ! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
% 3.18/1.66 => ~ ! [X3: A] :
% 3.18/1.66 ( ( member @ A @ X3 @ A5 )
% 3.18/1.66 => ( C3
% 3.18/1.66 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_162_Id__on__eqI,axiom,
% 3.18/1.66 ! [A: $tType,A2: A,B2: A,A5: set @ A] :
% 3.18/1.66 ( ( A2 = B2 )
% 3.18/1.66 => ( ( member @ A @ A2 @ A5 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_163_Id__on__iff,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,Y4: A,A5: set @ A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( id_on @ A @ A5 ) )
% 3.18/1.66 = ( ( X4 = Y4 )
% 3.18/1.66 & ( member @ A @ X4 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_164_bot_Oextremum__uniqueI,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order_bot @ A )
% 3.18/1.66 => ! [A2: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 3.18/1.66 => ( A2
% 3.18/1.66 = ( bot_bot @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_165_bot_Oextremum__unique,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order_bot @ A )
% 3.18/1.66 => ! [A2: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 3.18/1.66 = ( A2
% 3.18/1.66 = ( bot_bot @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_166_bot_Oextremum,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order_bot @ A )
% 3.18/1.66 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_167_subrelI,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 3.18/1.66 ( ! [X3: A,Y3: B] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
% 3.18/1.66 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 3.18/1.66 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_168_insert__mono,axiom,
% 3.18/1.66 ! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_169_subset__insert,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ~ ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ B5 ) )
% 3.18/1.66 = ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_170_subset__insertI,axiom,
% 3.18/1.66 ! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_171_subset__insertI2,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A,B2: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_172_trans__Id__on,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] : ( trans @ A @ ( id_on @ A @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_173_refl__on__Id__on,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] : ( refl_on @ A @ A5 @ ( id_on @ A @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_174_Collect__mono__iff,axiom,
% 3.18/1.66 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 3.18/1.66 = ( ! [X5: A] :
% 3.18/1.66 ( ( P2 @ X5 )
% 3.18/1.66 => ( Q @ X5 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_175_set__eq__subset,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( ^ [Y8: set @ A,Z4: set @ A] : ( Y8 = Z4 ) )
% 3.18/1.66 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 3.18/1.66 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_176_subset__trans,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A,C5: set @ A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ A5 @ C5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_177_Collect__mono,axiom,
% 3.18/1.66 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.18/1.66 ( ! [X3: A] :
% 3.18/1.66 ( ( P2 @ X3 )
% 3.18/1.66 => ( Q @ X3 ) )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_178_subset__refl,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_179_subset__iff,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) )
% 3.18/1.66 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.18/1.66 ! [T2: A] :
% 3.18/1.66 ( ( member @ A @ T2 @ A6 )
% 3.18/1.66 => ( member @ A @ T2 @ B7 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_180_equalityD2,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ( A5 = B5 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_181_equalityD1,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ( A5 = B5 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_182_subset__eq,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) )
% 3.18/1.66 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.18/1.66 ! [X5: A] :
% 3.18/1.66 ( ( member @ A @ X5 @ A6 )
% 3.18/1.66 => ( member @ A @ X5 @ B7 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_183_equalityE,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 3.18/1.66 ( ( A5 = B5 )
% 3.18/1.66 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.18/1.66 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_184_subsetD,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.18/1.66 => ( ( member @ A @ C3 @ A5 )
% 3.18/1.66 => ( member @ A @ C3 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_185_in__mono,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,B5: set @ A,X4: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.18/1.66 => ( ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ( member @ A @ X4 @ B5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_186_dual__order_Oantisym,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [B2: A,A2: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.18/1.66 => ( A2 = B2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_187_dual__order_Oeq__iff,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ( ( ^ [Y8: A,Z4: A] : ( Y8 = Z4 ) )
% 3.18/1.66 = ( ^ [A7: A,B8: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ B8 @ A7 )
% 3.18/1.66 & ( ord_less_eq @ A @ A7 @ B8 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_188_dual__order_Otrans,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [B2: A,A2: A,C3: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 3.18/1.66 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_189_linorder__wlog,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( linorder @ A )
% 3.18/1.66 => ! [P2: A > A > $o,A2: A,B2: A] :
% 3.18/1.66 ( ! [A4: A,B4: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.18/1.66 => ( P2 @ A4 @ B4 ) )
% 3.18/1.66 => ( ! [A4: A,B4: A] :
% 3.18/1.66 ( ( P2 @ B4 @ A4 )
% 3.18/1.66 => ( P2 @ A4 @ B4 ) )
% 3.18/1.66 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_190_dual__order_Orefl,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_191_order__trans,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( preorder @ A )
% 3.18/1.66 => ! [X4: A,Y4: A,Z: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ Y4 @ Z )
% 3.18/1.66 => ( ord_less_eq @ A @ X4 @ Z ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_192_order__class_Oorder_Oantisym,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [A2: A,B2: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.18/1.66 => ( A2 = B2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_193_ord__le__eq__trans,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( ord @ A )
% 3.18/1.66 => ! [A2: A,B2: A,C3: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.18/1.66 => ( ( B2 = C3 )
% 3.18/1.66 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_194_ord__eq__le__trans,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( ord @ A )
% 3.18/1.66 => ! [A2: A,B2: A,C3: A] :
% 3.18/1.66 ( ( A2 = B2 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 3.18/1.66 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_195_order__class_Oorder_Oeq__iff,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ( ( ^ [Y8: A,Z4: A] : ( Y8 = Z4 ) )
% 3.18/1.66 = ( ^ [A7: A,B8: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A7 @ B8 )
% 3.18/1.66 & ( ord_less_eq @ A @ B8 @ A7 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_196_antisym__conv,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [Y4: A,X4: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ Y4 @ X4 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.18/1.66 = ( X4 = Y4 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_197_le__cases3,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( linorder @ A )
% 3.18/1.66 => ! [X4: A,Y4: A,Z: A] :
% 3.18/1.66 ( ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.18/1.66 => ~ ( ord_less_eq @ A @ Y4 @ Z ) )
% 3.18/1.66 => ( ( ( ord_less_eq @ A @ Y4 @ X4 )
% 3.18/1.66 => ~ ( ord_less_eq @ A @ X4 @ Z ) )
% 3.18/1.66 => ( ( ( ord_less_eq @ A @ X4 @ Z )
% 3.18/1.66 => ~ ( ord_less_eq @ A @ Z @ Y4 ) )
% 3.18/1.66 => ( ( ( ord_less_eq @ A @ Z @ Y4 )
% 3.18/1.66 => ~ ( ord_less_eq @ A @ Y4 @ X4 ) )
% 3.18/1.66 => ( ( ( ord_less_eq @ A @ Y4 @ Z )
% 3.18/1.66 => ~ ( ord_less_eq @ A @ Z @ X4 ) )
% 3.18/1.66 => ~ ( ( ord_less_eq @ A @ Z @ X4 )
% 3.18/1.66 => ~ ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_198_order_Otrans,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [A2: A,B2: A,C3: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 3.18/1.66 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_199_le__cases,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( linorder @ A )
% 3.18/1.66 => ! [X4: A,Y4: A] :
% 3.18/1.66 ( ~ ( ord_less_eq @ A @ X4 @ Y4 )
% 3.18/1.66 => ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_200_eq__refl,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( preorder @ A )
% 3.18/1.66 => ! [X4: A,Y4: A] :
% 3.18/1.66 ( ( X4 = Y4 )
% 3.18/1.66 => ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_201_linear,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( linorder @ A )
% 3.18/1.66 => ! [X4: A,Y4: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.18/1.66 | ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_202_antisym,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ! [X4: A,Y4: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.18/1.66 => ( ( ord_less_eq @ A @ Y4 @ X4 )
% 3.18/1.66 => ( X4 = Y4 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_203_eq__iff,axiom,
% 3.18/1.66 ! [A: $tType] :
% 3.18/1.66 ( ( order @ A )
% 3.18/1.66 => ( ( ^ [Y8: A,Z4: A] : ( Y8 = Z4 ) )
% 3.18/1.66 = ( ^ [X5: A,Y5: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 3.18/1.66 & ( ord_less_eq @ A @ Y5 @ X5 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_204_ord__le__eq__subst,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType] :
% 3.18/1.66 ( ( ( ord @ B )
% 3.18/1.66 & ( ord @ A ) )
% 3.18/1.66 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.18/1.66 => ( ( ( F3 @ B2 )
% 3.18/1.66 = C3 )
% 3.18/1.66 => ( ! [X3: A,Y3: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.18/1.66 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 3.18/1.66 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_205_ord__eq__le__subst,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType] :
% 3.18/1.66 ( ( ( ord @ B )
% 3.18/1.66 & ( ord @ A ) )
% 3.18/1.66 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 3.18/1.66 ( ( A2
% 3.18/1.66 = ( F3 @ B2 ) )
% 3.18/1.66 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 3.18/1.66 => ( ! [X3: B,Y3: B] :
% 3.18/1.66 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 3.18/1.66 => ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 3.18/1.66 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_206_order__subst2,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType] :
% 3.18/1.66 ( ( ( order @ C )
% 3.18/1.66 & ( order @ A ) )
% 3.18/1.66 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.18/1.66 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 3.18/1.66 => ( ! [X3: A,Y3: A] :
% 3.18/1.66 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.18/1.66 => ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 3.18/1.66 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_207_order__subst1,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType] :
% 3.18/1.66 ( ( ( order @ B )
% 3.18/1.66 & ( order @ A ) )
% 3.18/1.66 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 3.18/1.66 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 3.18/1.66 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 3.18/1.66 => ( ! [X3: B,Y3: B] :
% 3.18/1.66 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 3.18/1.66 => ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 3.18/1.66 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_208_le__fun__def,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType] :
% 3.18/1.66 ( ( ord @ B )
% 3.18/1.66 => ( ( ord_less_eq @ ( A > B ) )
% 3.18/1.66 = ( ^ [F4: A > B,G5: A > B] :
% 3.18/1.66 ! [X5: A] : ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_209_le__funI,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType] :
% 3.18/1.66 ( ( ord @ B )
% 3.18/1.66 => ! [F3: A > B,G4: A > B] :
% 3.18/1.66 ( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G4 @ X3 ) )
% 3.18/1.66 => ( ord_less_eq @ ( A > B ) @ F3 @ G4 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_210_le__funE,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType] :
% 3.18/1.66 ( ( ord @ B )
% 3.18/1.66 => ! [F3: A > B,G4: A > B,X4: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( A > B ) @ F3 @ G4 )
% 3.18/1.66 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_211_le__funD,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType] :
% 3.18/1.66 ( ( ord @ B )
% 3.18/1.66 => ! [F3: A > B,G4: A > B,X4: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( A > B ) @ F3 @ G4 )
% 3.18/1.66 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_212_sym__Id__on,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] : ( sym @ A @ ( id_on @ A @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_213_subset__singletonD,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,X4: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 => ( ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 | ( A5
% 3.18/1.66 = ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_214_subset__singleton__iff,axiom,
% 3.18/1.66 ! [A: $tType,X7: set @ A,A2: A] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.18/1.66 = ( ( X7
% 3.18/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.18/1.66 | ( X7
% 3.18/1.66 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_215_subgraph__trans,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B,G_3: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 3.18/1.66 => ( ( graph_homomorphism @ A @ B @ B @ G_2 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_2 ) ) )
% 3.18/1.66 => ( graph_homomorphism @ A @ B @ B @ G_1 @ G_3 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_216_subgraph__preserves__hom,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,C: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,X7: labeled_graph @ A @ C,H: set @ ( product_prod @ C @ B )] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 3.18/1.66 => ( ( graph_homomorphism @ A @ C @ B @ X7 @ A5 @ H )
% 3.18/1.66 => ( graph_homomorphism @ A @ C @ B @ X7 @ B5 @ H ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_217_congruentD,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B,Y4: A,Z: A] :
% 3.18/1.66 ( ( equiv_congruent @ A @ B @ R2 @ F3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 3.18/1.66 => ( ( F3 @ Y4 )
% 3.18/1.66 = ( F3 @ Z ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_218_congruentI,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
% 3.18/1.66 ( ! [Y3: A,Z2: A] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 )
% 3.18/1.66 => ( ( F3 @ Y3 )
% 3.18/1.66 = ( F3 @ Z2 ) ) )
% 3.18/1.66 => ( equiv_congruent @ A @ B @ R2 @ F3 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_219_in__quotient__imp__subset,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ X7 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_220_agree__on__subg__compose,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,R: labeled_graph @ A @ B,G4: set @ ( product_prod @ B @ C ),H: set @ ( product_prod @ B @ C ),F5: labeled_graph @ A @ B,F3: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( agree_on @ A @ B @ C @ R @ G4 @ H )
% 3.18/1.66 => ( ( agree_on @ A @ B @ C @ F5 @ F3 @ G4 )
% 3.18/1.66 => ( ( graph_homomorphism @ A @ B @ B @ F5 @ R @ ( id_on @ B @ ( labeled_vertices @ A @ B @ F5 ) ) )
% 3.18/1.66 => ( agree_on @ A @ B @ C @ F5 @ F3 @ H ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_221_maintainedD2,axiom,
% 3.18/1.66 ! [A: $tType,C: $tType,B: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,G3: labeled_graph @ A @ C,F3: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( maintained @ A @ B @ C @ ( product_Pair @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) @ A5 @ B5 ) @ G3 )
% 3.18/1.66 => ( ( graph_homomorphism @ A @ B @ C @ A5 @ G3 @ F3 )
% 3.18/1.66 => ~ ! [G2: set @ ( product_prod @ B @ C )] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ C @ B5 @ G3 @ G2 )
% 3.18/1.66 => ~ ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ F3 @ G2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_222_subgraph__semantics,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B,A2: B,B2: B,E3: allegorical_term @ A] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ A5 @ B5 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ A5 ) ) )
% 3.18/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ A5 @ E3 ) )
% 3.18/1.66 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( semantics @ A @ B @ B5 @ E3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_223_proj__iff,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A,Y4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
% 3.18/1.66 => ( ( ( equiv_proj @ A @ A @ R2 @ X4 )
% 3.18/1.66 = ( equiv_proj @ A @ A @ R2 @ Y4 ) )
% 3.18/1.66 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_224_least__consequence__graphI,axiom,
% 3.18/1.66 ! [X: $tType,A: $tType,C: $tType,L: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),G3: labeled_graph @ L @ C,S2: labeled_graph @ L @ C,T3: itself @ X] :
% 3.18/1.66 ( ( consequence_graph @ L @ A @ C @ Rs @ G3 )
% 3.18/1.66 => ( ( graph_homomorphism @ L @ C @ C @ S2 @ G3 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 3.18/1.66 => ( ! [C6: labeled_graph @ L @ X] :
% 3.18/1.66 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 3.18/1.66 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G3 ) @ C6 ) )
% 3.18/1.66 => ( least_559130134_graph @ X @ L @ A @ C @ T3 @ Rs @ S2 @ G3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_225_leastI,axiom,
% 3.18/1.66 ! [X: $tType,A: $tType,C: $tType,L: $tType,S2: labeled_graph @ L @ C,G3: labeled_graph @ L @ C,Rs: set @ ( product_prod @ ( labeled_graph @ L @ A ) @ ( labeled_graph @ L @ A ) ),T3: itself @ X] :
% 3.18/1.66 ( ( graph_homomorphism @ L @ C @ C @ S2 @ G3 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S2 ) ) )
% 3.18/1.66 => ( ! [C6: labeled_graph @ L @ X] :
% 3.18/1.66 ( ( consequence_graph @ L @ A @ X @ Rs @ C6 )
% 3.18/1.66 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S2 @ G3 ) @ C6 ) )
% 3.18/1.66 => ( least @ X @ L @ A @ C @ T3 @ Rs @ S2 @ G3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_226_least__consequence__graph__def,axiom,
% 3.18/1.66 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 3.18/1.66 ( ( least_559130134_graph @ X @ L @ V @ C )
% 3.18/1.66 = ( ^ [T2: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 3.18/1.66 ( ( consequence_graph @ L @ V @ C @ Rs2 @ G7 )
% 3.18/1.66 & ( least @ X @ L @ V @ C @ T2 @ Rs2 @ S3 @ G7 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_227_consequence__graphD_I3_J,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G3: labeled_graph @ A @ C] :
% 3.18/1.66 ( ( consequence_graph @ A @ B @ C @ Rs @ G3 )
% 3.18/1.66 => ( G3
% 3.18/1.66 = ( restrict @ A @ C @ G3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_228_consequence__graphD_I1_J,axiom,
% 3.18/1.66 ! [B: $tType,C: $tType,A: $tType,Rs: set @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ),G3: labeled_graph @ A @ C,R: product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B )] :
% 3.18/1.66 ( ( consequence_graph @ A @ B @ C @ Rs @ G3 )
% 3.18/1.66 => ( ( member @ ( product_prod @ ( labeled_graph @ A @ B ) @ ( labeled_graph @ A @ B ) ) @ R @ Rs )
% 3.18/1.66 => ( maintained @ A @ B @ C @ R @ G3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_229_proj__in__iff,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( member @ ( set @ A ) @ ( equiv_proj @ A @ A @ R2 @ X4 ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 = ( member @ A @ X4 @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_230_least__def,axiom,
% 3.18/1.66 ! [L: $tType,V: $tType,C: $tType,X: $tType] :
% 3.18/1.66 ( ( least @ X @ L @ V @ C )
% 3.18/1.66 = ( ^ [Uu: itself @ X,Rs2: set @ ( product_prod @ ( labeled_graph @ L @ V ) @ ( labeled_graph @ L @ V ) ),S3: labeled_graph @ L @ C,G7: labeled_graph @ L @ C] :
% 3.18/1.66 ( ( graph_homomorphism @ L @ C @ C @ S3 @ G7 @ ( id_on @ C @ ( labeled_vertices @ L @ C @ S3 ) ) )
% 3.18/1.66 & ! [C4: labeled_graph @ L @ X] :
% 3.18/1.66 ( ( consequence_graph @ L @ V @ X @ Rs2 @ C4 )
% 3.18/1.66 => ( maintained @ L @ C @ X @ ( product_Pair @ ( labeled_graph @ L @ C ) @ ( labeled_graph @ L @ C ) @ S3 @ G7 ) @ C4 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_231_less__by__empty,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( A5
% 3.18/1.66 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 3.18/1.66 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_232_insert__subsetI,axiom,
% 3.18/1.66 ! [A: $tType,X4: A,A5: set @ A,X7: set @ A] :
% 3.18/1.66 ( ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ X7 ) @ A5 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_233_ssubst__Pair__rhs,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] :
% 3.18/1.66 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R )
% 3.18/1.66 => ( ( S4 = S )
% 3.18/1.66 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_234_subset__emptyI,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A] :
% 3.18/1.66 ( ! [X3: A] :
% 3.18/1.66 ~ ( member @ A @ X3 @ A5 )
% 3.18/1.66 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_235_univ__commute,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,X4: A] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( equiv_congruent @ A @ B @ R2 @ F3 )
% 3.18/1.66 => ( ( member @ A @ X4 @ A5 )
% 3.18/1.66 => ( ( bNF_Greatest_univ @ A @ B @ F3 @ ( equiv_proj @ A @ A @ R2 @ X4 ) )
% 3.18/1.66 = ( F3 @ X4 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_236_subgraph__def,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 3.18/1.66 = ( ( G_1
% 3.18/1.66 = ( restrict @ A @ B @ G_1 ) )
% 3.18/1.66 & ( G_2
% 3.18/1.66 = ( restrict @ A @ B @ G_2 ) )
% 3.18/1.66 & ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 3.18/1.66 = G_2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_237_graph__union__idemp_I3_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ B5 @ A5 ) )
% 3.18/1.66 = ( graph_union @ A @ B @ B5 @ A5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_238_graph__union__idemp_I2_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B,B5: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_union @ A @ B @ A5 @ ( graph_union @ A @ B @ A5 @ B5 ) )
% 3.18/1.66 = ( graph_union @ A @ B @ A5 @ B5 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_239_graph__union__idemp_I1_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,A5: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_union @ A @ B @ A5 @ A5 )
% 3.18/1.66 = A5 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_240_graph__union__preserves__restrict,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( G_1
% 3.18/1.66 = ( restrict @ A @ B @ G_1 ) )
% 3.18/1.66 => ( ( G_2
% 3.18/1.66 = ( restrict @ A @ B @ G_2 ) )
% 3.18/1.66 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 3.18/1.66 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_241_univ__preserves,axiom,
% 3.18/1.66 ! [A: $tType,B: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,B5: set @ B] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R2 )
% 3.18/1.66 => ( ( equiv_congruent @ A @ B @ R2 @ F3 )
% 3.18/1.66 => ( ! [X3: A] :
% 3.18/1.66 ( ( member @ A @ X3 @ A5 )
% 3.18/1.66 => ( member @ B @ ( F3 @ X3 ) @ B5 ) )
% 3.18/1.66 => ! [X2: set @ A] :
% 3.18/1.66 ( ( member @ ( set @ A ) @ X2 @ ( equiv_quotient @ A @ A5 @ R2 ) )
% 3.18/1.66 => ( member @ B @ ( bNF_Greatest_univ @ A @ B @ F3 @ X2 ) @ B5 ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_242_relImage__proj,axiom,
% 3.18/1.66 ! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
% 3.18/1.66 ( ( equiv_equiv @ A @ A5 @ R )
% 3.18/1.66 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr1317331620lImage @ A @ ( set @ A ) @ R @ ( equiv_proj @ A @ A @ R ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_243_subgraph__def2,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( G_1
% 3.18/1.66 = ( restrict @ A @ B @ G_1 ) )
% 3.18/1.66 => ( ( G_2
% 3.18/1.66 = ( restrict @ A @ B @ G_2 ) )
% 3.18/1.66 => ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 3.18/1.66 = ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 3.18/1.66 & ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_244_labeled__graph_Ocollapse,axiom,
% 3.18/1.66 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 3.18/1.66 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 3.18/1.66 = Labeled_graph ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_245_graph__unionI,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 3.18/1.66 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 3.18/1.66 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 3.18/1.66 = G_2 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_246_restrict__subsD,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G3: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G3 ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G3 ) ) )
% 3.18/1.66 => ( G3
% 3.18/1.66 = ( restrict @ A @ B @ G3 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_247_labeled__graph_Oexhaust__sel,axiom,
% 3.18/1.66 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 3.18/1.66 ( Labeled_graph
% 3.18/1.66 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_248_labeled__graph_Oexpand,axiom,
% 3.18/1.66 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 3.18/1.66 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 3.18/1.66 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 3.18/1.66 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 3.18/1.66 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 3.18/1.66 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_249_labeled__graph_Osel_I1_J,axiom,
% 3.18/1.66 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 3.18/1.66 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 3.18/1.66 = X1 ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_250_graph__union__iff,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 3.18/1.66 = G_2 )
% 3.18/1.66 = ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 3.18/1.66 & ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_251_subgraph__subset_I2_J,axiom,
% 3.18/1.66 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 3.18/1.66 ( ( graph_homomorphism @ A @ B @ B @ G_1 @ G_2 @ ( id_on @ B @ ( labeled_vertices @ A @ B @ G_1 ) ) )
% 3.18/1.66 => ( 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 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_252_labeled__graph_Osplit__sel__asm,axiom,
% 3.18/1.66 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 3.18/1.66 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F3 @ Labeled_graph ) )
% 3.18/1.66 = ( ~ ( ( Labeled_graph
% 3.18/1.66 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 3.18/1.66 & ~ ( P2 @ ( F3 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_253_labeled__graph_Osplit__sel,axiom,
% 3.18/1.66 ! [A: $tType,V: $tType,L: $tType,P2: A > $o,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 3.18/1.66 ( ( P2 @ ( labele1974067554_graph @ L @ V @ A @ F3 @ Labeled_graph ) )
% 3.18/1.66 = ( ( Labeled_graph
% 3.18/1.66 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 3.18/1.66 => ( P2 @ ( F3 @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_254_labeled__graph_Ocase,axiom,
% 3.18/1.66 ! [L: $tType,A: $tType,V: $tType,F3: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 3.18/1.66 ( ( labele1974067554_graph @ L @ V @ A @ F3 @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 3.18/1.66 = ( F3 @ X1 @ X22 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(fact_255_labeled__graph_Ocase__eq__if,axiom,
% 3.18/1.66 ! [A: $tType,V: $tType,L: $tType] :
% 3.18/1.66 ( ( labele1974067554_graph @ L @ V @ A )
% 3.18/1.66 = ( ^ [F4: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph3: labeled_graph @ L @ V] : ( F4 @ ( labeled_edges @ L @ V @ Labeled_graph3 ) @ ( labeled_vertices @ L @ V @ Labeled_graph3 ) ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 3.18/1.66 ! [A8: $tType,A9: $tType] :
% 3.18/1.66 ( ( order_bot @ A9 )
% 3.18/1.66 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.18/1.66 ! [A8: $tType,A9: $tType] :
% 3.18/1.66 ( ( preorder @ A9 )
% 3.18/1.66 => ( preorder @ ( A8 > A9 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.18/1.66 ! [A8: $tType,A9: $tType] :
% 3.18/1.66 ( ( order @ A9 )
% 3.18/1.66 => ( order @ ( A8 > A9 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_fun___Orderings_Oord,axiom,
% 3.18/1.66 ! [A8: $tType,A9: $tType] :
% 3.18/1.66 ( ( ord @ A9 )
% 3.18/1.66 => ( ord @ ( A8 > A9 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_fun___Orderings_Obot,axiom,
% 3.18/1.66 ! [A8: $tType,A9: $tType] :
% 3.18/1.66 ( ( bot @ A9 )
% 3.18/1.66 => ( bot @ ( A8 > A9 ) ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 3.18/1.66 order_bot @ nat ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 3.18/1.66 preorder @ nat ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.18/1.66 linorder @ nat ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 3.18/1.66 order @ nat ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 3.18/1.66 ord @ nat ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 3.18/1.66 bot @ nat ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 3.18/1.66 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 3.18/1.66 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 3.18/1.66 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 3.18/1.66 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
% 3.18/1.66 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 3.18/1.66 order_bot @ $o ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 3.18/1.66 preorder @ $o ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
% 3.18/1.66 linorder @ $o ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 3.18/1.66 order @ $o ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 3.18/1.66 ord @ $o ).
% 3.18/1.66
% 3.18/1.66 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 3.18/1.66 bot @ $o ).
% 3.18/1.66
% 3.18/1.66 thf(conj_0,conjecture,
% 3.18/1.66 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ z @ y ) @ ( getRel @ a @ b @ i @ g ) ).
% 3.18/1.66 ------------------------
% 3.18/1.68 % (30360)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_30214 for (2999ds/4Mi)
% 3.18/1.68 % (30365)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30214 for (2999ds/18Mi)
% 3.18/1.68 % (30361)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_30214 for (2999ds/27Mi)
% 3.18/1.68 % (30362)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30214 for (2999ds/2Mi)
% 3.18/1.68 % (30363)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_30214 for (2999ds/2Mi)
% 3.18/1.68 % (30362)Instruction limit reached!
% 3.18/1.68 % (30362)------------------------------
% 3.18/1.68 % (30362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (30362)Termination reason: Unknown
% 3.18/1.68 % (30362)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (30362)Memory used [KB]: 1407
% 3.18/1.68 % (30362)Time elapsed: 0.002 s
% 3.18/1.68 % (30363)Instruction limit reached!
% 3.18/1.68 % (30363)------------------------------
% 3.18/1.68 % (30363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (30363)Termination reason: Unknown
% 3.18/1.68 % (30363)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (30363)Memory used [KB]: 1407
% 3.18/1.68 % (30363)Time elapsed: 0.002 s
% 3.18/1.68 % (30363)Instructions burned: 2 (million)
% 3.18/1.68 % (30363)------------------------------
% 3.18/1.68 % (30363)------------------------------
% 3.18/1.68 % (30362)Instructions burned: 2 (million)
% 3.18/1.68 % (30364)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_30214 for (2999ds/275Mi)
% 3.18/1.68 % (30362)------------------------------
% 3.18/1.68 % (30362)------------------------------
% 3.18/1.68 % (30360)Instruction limit reached!
% 3.18/1.68 % (30360)------------------------------
% 3.18/1.68 % (30360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (30360)Termination reason: Unknown
% 3.18/1.68 % (30360)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (30360)Memory used [KB]: 1407
% 3.18/1.68 % (30360)Time elapsed: 0.003 s
% 3.18/1.68 % (30360)Instructions burned: 5 (million)
% 3.18/1.68 % (30360)------------------------------
% 3.18/1.68 % (30360)------------------------------
% 3.18/1.69 % (30365)Instruction limit reached!
% 3.18/1.69 % (30365)------------------------------
% 3.18/1.69 % (30365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.69 % (30365)Termination reason: Unknown
% 3.18/1.69 % (30365)Termination phase: shuffling
% 3.18/1.69
% 3.18/1.69 % (30365)Memory used [KB]: 1791
% 3.18/1.69 % (30365)Time elapsed: 0.008 s
% 3.18/1.69 % (30365)Instructions burned: 20 (million)
% 3.18/1.69 % (30365)------------------------------
% 3.18/1.69 % (30365)------------------------------
% 3.18/1.69 % (30359)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30214 for (2999ds/183Mi)
% 3.18/1.69 % (30361)Instruction limit reached!
% 3.18/1.69 % (30361)------------------------------
% 3.18/1.69 % (30361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.69 % (30361)Termination reason: Unknown
% 3.18/1.69 % (30361)Termination phase: shuffling
% 3.18/1.69
% 3.18/1.69 % (30361)Memory used [KB]: 2046
% 3.18/1.69 % (30361)Time elapsed: 0.011 s
% 3.18/1.69 % (30361)Instructions burned: 29 (million)
% 3.18/1.69 % (30361)------------------------------
% 3.18/1.69 % (30361)------------------------------
% 3.18/1.69 % (30366)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30214 for (2999ds/3Mi)
% 3.18/1.69 % (30367)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_30214 for (2999ds/37Mi)
% 3.18/1.69 % (30368)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_30214 for (2999ds/15Mi)
% 3.18/1.69 % (30366)Instruction limit reached!
% 3.18/1.69 % (30366)------------------------------
% 3.18/1.69 % (30366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.69 % (30366)Termination reason: Unknown
% 3.18/1.69 % (30366)Termination phase: shuffling
% 3.18/1.69
% 3.18/1.69 % (30366)Memory used [KB]: 1407
% 3.18/1.69 % (30366)Time elapsed: 0.002 s
% 3.18/1.69 % (30366)Instructions burned: 3 (million)
% 3.18/1.69 % (30366)------------------------------
% 3.18/1.69 % (30366)------------------------------
% 3.26/1.70 % (30368)Instruction limit reached!
% 3.26/1.70 % (30368)------------------------------
% 3.26/1.70 % (30368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.70 % (30368)Termination reason: Unknown
% 3.26/1.70 % (30368)Termination phase: shuffling
% 3.26/1.70
% 3.26/1.70 % (30368)Memory used [KB]: 1791
% 3.26/1.70 % (30368)Time elapsed: 0.006 s
% 3.26/1.70 % (30368)Instructions burned: 16 (million)
% 3.26/1.70 % (30368)------------------------------
% 3.26/1.70 % (30368)------------------------------
% 3.26/1.70 % (30369)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_30214 for (2999ds/3Mi)
% 3.26/1.70 % (30369)Instruction limit reached!
% 3.26/1.70 % (30369)------------------------------
% 3.26/1.70 % (30369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.70 % (30369)Termination reason: Unknown
% 3.26/1.70 % (30369)Termination phase: shuffling
% 3.26/1.70
% 3.26/1.70 % (30369)Memory used [KB]: 1407
% 3.26/1.70 % (30369)Time elapsed: 0.002 s
% 3.26/1.70 % (30369)Instructions burned: 3 (million)
% 3.26/1.70 % (30369)------------------------------
% 3.26/1.70 % (30369)------------------------------
% 3.26/1.70 % (30370)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_30214 for (2999ds/1041Mi)
% 3.26/1.70 % (30371)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_30214 for (2999ds/7Mi)
% 3.26/1.70 % (30367)Instruction limit reached!
% 3.26/1.70 % (30367)------------------------------
% 3.26/1.70 % (30367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.70 % (30367)Termination reason: Unknown
% 3.26/1.70 % (30367)Termination phase: shuffling
% 3.26/1.70
% 3.26/1.70 % (30367)Memory used [KB]: 2174
% 3.26/1.70 % (30367)Time elapsed: 0.012 s
% 3.26/1.70 % (30367)Instructions burned: 39 (million)
% 3.26/1.70 % (30367)------------------------------
% 3.26/1.70 % (30367)------------------------------
% 3.26/1.71 % (30371)Instruction limit reached!
% 3.26/1.71 % (30371)------------------------------
% 3.26/1.71 % (30371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.71 % (30371)Termination reason: Unknown
% 3.26/1.71 % (30371)Termination phase: shuffling
% 3.26/1.71
% 3.26/1.71 % (30371)Memory used [KB]: 1535
% 3.26/1.71 % (30371)Time elapsed: 0.004 s
% 3.26/1.71 % (30371)Instructions burned: 9 (million)
% 3.26/1.71 % (30371)------------------------------
% 3.26/1.71 % (30371)------------------------------
% 3.26/1.71 % (30372)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_30214 for (2999ds/16Mi)
% 3.26/1.71 % (30364)First to succeed.
% 3.26/1.71 % (30364)Refutation found. Thanks to Tanya!
% 3.26/1.71 % SZS status Theorem for DTF2THF_30214
% 3.26/1.71 % SZS output start Proof for DTF2THF_30214
% 3.26/1.71 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.26/1.71 thf(type_def_6, type, set: ($tType) > $tType).
% 3.26/1.71 thf(type_def_8, type, labeled_graph: ($tType * $tType) > $tType).
% 3.26/1.71 thf(type_def_9, type, allegorical_term: ($tType) > $tType).
% 3.26/1.71 thf(type_def_10, type, itself: ($tType) > $tType).
% 3.26/1.71 thf(type_def_11, type, nat: $tType).
% 3.26/1.71 thf(type_def_12, type, a: $tType).
% 3.26/1.71 thf(type_def_13, type, b: $tType).
% 3.26/1.71 thf(func_def_0, type, allegorical_term: ($tType) > $tType).
% 3.26/1.71 thf(func_def_1, type, labeled_graph: ($tType * $tType) > $tType).
% 3.26/1.71 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 3.26/1.71 thf(func_def_3, type, set: ($tType) > $tType).
% 3.26/1.71 thf(func_def_4, type, nat: $tType).
% 3.26/1.71 thf(func_def_5, type, itself: ($tType) > $tType).
% 3.26/1.71 thf(func_def_6, type, b: $tType).
% 3.26/1.71 thf(func_def_7, type, a: $tType).
% 3.26/1.71 thf(func_def_8, type, type: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_9, type, bot: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_10, type, ord: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_11, type, order: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_12, type, linorder: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_13, type, preorder: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_14, type, order_bot: !>[X0: $tType]:($o)).
% 3.26/1.71 thf(func_def_15, type, bNF_Gr1317331620lImage: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > set @ product_prod @ X1 @ X1)).
% 3.26/1.71 thf(func_def_16, type, bNF_Greatest_univ: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 3.26/1.71 thf(func_def_17, type, equiv_congruent: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.26/1.71 thf(func_def_18, type, equiv_congruent2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1 > X2) > $o)).
% 3.26/1.71 thf(func_def_19, type, equiv_equiv: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_20, type, equiv_proj: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > set @ X1)).
% 3.26/1.71 thf(func_def_21, type, equiv_quotient: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > set @ set @ X0)).
% 3.26/1.71 thf(func_def_22, type, getRel: !>[X0: $tType, X1: $tType]:(X0 > labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X1)).
% 3.26/1.71 thf(func_def_23, type, semantics: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > allegorical_term @ X0 > set @ product_prod @ X1 @ X1)).
% 3.26/1.71 thf(func_def_24, type, graph_homomorphism: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.26/1.71 thf(func_def_25, type, graph_union: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.26/1.71 thf(func_def_26, type, labeled_LG: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > labeled_graph @ X0 @ X1)).
% 3.26/1.71 thf(func_def_27, type, labele1974067554_graph: !>[X0: $tType, X1: $tType, X2: $tType]:((set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > X2) > labeled_graph @ X0 @ X1 > X2)).
% 3.26/1.71 thf(func_def_28, type, labeled_edges: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X0 @ product_prod @ X1 @ X1)).
% 3.26/1.71 thf(func_def_29, type, labeled_vertices: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ X1)).
% 3.26/1.71 thf(func_def_30, type, on_triple: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ product_prod @ X2 @ product_prod @ X0 @ X0 @ product_prod @ X2 @ product_prod @ X1 @ X1)).
% 3.26/1.71 thf(func_def_31, type, restrict: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 3.26/1.71 thf(func_def_32, type, idempotent: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_33, type, order_1409979114der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_34, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_35, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.26/1.71 thf(func_def_36, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.26/1.71 thf(func_def_37, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.26/1.71 thf(func_def_38, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.26/1.71 thf(func_def_39, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.26/1.71 thf(func_def_40, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 3.26/1.71 thf(func_def_41, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_42, type, sym: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_43, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_44, type, trans: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 3.26/1.71 thf(func_def_45, type, agree_on: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 3.26/1.71 thf(func_def_46, 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)).
% 3.26/1.71 thf(func_def_47, 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)).
% 3.26/1.71 thf(func_def_48, type, fin_maintained: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.26/1.71 thf(func_def_49, 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)).
% 3.26/1.71 thf(func_def_50, 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)).
% 3.26/1.71 thf(func_def_51, type, maintained: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ labeled_graph @ X0 @ X1 @ labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > $o)).
% 3.26/1.71 thf(func_def_52, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.26/1.71 thf(func_def_53, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.26/1.71 thf(func_def_54, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 3.26/1.71 thf(func_def_55, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 3.26/1.71 thf(func_def_56, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.26/1.71 thf(func_def_57, type, standa1068660421e_rule: !>[X0: $tType]:(X0 > X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.26/1.71 thf(func_def_58, type, standa244753842y_rule: !>[X0: $tType]:(product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.26/1.71 thf(func_def_59, type, standa245363227y_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.26/1.71 thf(func_def_60, type, standa1805702094y_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.26/1.71 thf(func_def_61, type, standa2114124375e_rule: !>[X0: $tType]:(X0 > product_prod @ labeled_graph @ X0 @ nat @ labeled_graph @ X0 @ nat)).
% 3.26/1.71 thf(func_def_62, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 3.26/1.71 thf(func_def_63, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.26/1.71 thf(func_def_64, type, g: labeled_graph @ a @ b).
% 3.26/1.71 thf(func_def_65, type, i: a).
% 3.26/1.71 thf(func_def_66, type, l: a).
% 3.26/1.71 thf(func_def_67, type, y: b).
% 3.26/1.71 thf(func_def_68, type, z: b).
% 3.26/1.71 thf(func_def_84, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.26/1.71 thf(func_def_85, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.26/1.71 thf(func_def_86, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.26/1.71 thf(func_def_87, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.26/1.71 thf(func_def_88, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_89, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_90, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.26/1.71 thf(func_def_91, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.26/1.71 thf(func_def_92, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.26/1.71 thf(func_def_93, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_94, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_95, type, sK11: !>[X0: $tType]:(set @ X0 > product_prod @ X0 @ X0 > X0)).
% 3.26/1.71 thf(func_def_96, type, sK12: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 3.26/1.71 thf(func_def_97, type, sK13: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 3.26/1.71 thf(func_def_98, type, sK14: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 3.26/1.71 thf(func_def_99, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.26/1.71 thf(func_def_100, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.26/1.71 thf(func_def_101, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_102, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_103, type, sK19: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_104, type, sK20: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_105, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.26/1.71 thf(func_def_106, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_107, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_108, type, sK24: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.26/1.71 thf(func_def_109, type, sK25: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.26/1.71 thf(func_def_110, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 3.26/1.71 thf(func_def_111, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 3.26/1.71 thf(func_def_112, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 3.26/1.71 thf(func_def_113, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 3.26/1.71 thf(func_def_114, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 3.26/1.71 thf(func_def_115, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 3.26/1.71 thf(func_def_116, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 3.26/1.71 thf(func_def_117, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.26/1.71 thf(func_def_118, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.26/1.71 thf(func_def_119, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.26/1.71 thf(func_def_120, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 3.26/1.71 thf(func_def_121, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.26/1.71 thf(func_def_122, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.26/1.71 thf(func_def_123, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.26/1.71 thf(func_def_124, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.26/1.71 thf(func_def_125, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_126, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_127, type, sK43: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.26/1.71 thf(func_def_128, type, sK44: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.26/1.71 thf(func_def_129, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.26/1.71 thf(func_def_130, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.26/1.71 thf(func_def_131, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.26/1.71 thf(func_def_132, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.26/1.71 thf(func_def_133, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 3.26/1.71 thf(func_def_134, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 3.26/1.71 thf(func_def_135, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 3.26/1.71 thf(func_def_136, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 3.26/1.71 thf(func_def_137, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 3.26/1.71 thf(func_def_138, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 3.26/1.71 thf(func_def_139, type, sK55: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.26/1.71 thf(func_def_140, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.26/1.71 thf(func_def_141, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.26/1.71 thf(func_def_142, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.26/1.71 thf(func_def_143, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.26/1.71 thf(func_def_144, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.26/1.71 thf(func_def_145, type, sK61: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.26/1.71 thf(func_def_146, type, sK62: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.26/1.71 thf(func_def_148, type, ph64: !>[X0: $tType]:(X0)).
% 3.26/1.71 thf(f951,plain,(
% 3.26/1.71 $false),
% 3.26/1.71 inference(subsumption_resolution,[status(thm)],[f947,f909])).
% 3.26/1.71 thf(f909,plain,(
% 3.26/1.71 ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ y @ z) @ (getRel @ a @ b @ i @ g)) = $true)),
% 3.26/1.71 inference(cnf_transformation,[status(thm)],[f341])).
% 3.26/1.71 thf(f341,plain,(
% 3.26/1.71 ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ y @ z) @ (getRel @ a @ b @ i @ g)) = $true)),
% 3.26/1.71 inference(fool_elimination,[status(thm)],[f340])).
% 3.26/1.71 thf(f340,plain,(
% 3.26/1.71 (member @ product_prod @ b @ b @ (product_Pair @ b @ b @ y @ z) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 inference(rectify,[status(thm)],[f2])).
% 3.26/1.71 thf(f2,axiom,(
% 3.26/1.71 (member @ product_prod @ b @ b @ (product_Pair @ b @ b @ y @ z) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 file('/export/starexec/sandbox2/tmp/tmp.ANXxQNlu00/DTF2THF_30214.p',fact_1_aI)).
% 3.26/1.71 thf(f947,plain,(
% 3.26/1.71 ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ y @ z) @ (getRel @ a @ b @ i @ g)) != $true)),
% 3.26/1.71 inference(trivial_inequality_removal,[status(thm)],[f942])).
% 3.26/1.71 thf(f942,plain,(
% 3.26/1.71 ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ y @ z) @ (getRel @ a @ b @ i @ g)) != $true) | ($true != $true)),
% 3.26/1.71 inference(superposition,[status(thm)],[f867,f893])).
% 3.26/1.71 thf(f893,plain,(
% 3.26/1.71 ( ! [X0 : b,X1 : b] : (((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X1 @ X0) @ (getRel @ a @ b @ i @ g)) = $true) | ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X0 @ X1) @ (getRel @ a @ b @ i @ g)) != $true)) )),
% 3.26/1.71 inference(cnf_transformation,[status(thm)],[f786])).
% 3.26/1.71 thf(f786,plain,(
% 3.26/1.71 ((trans @ b @ (getRel @ a @ b @ i @ g)) = $true) & ! [X0 : b,X1 : b] : (((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X1 @ X0) @ (getRel @ a @ b @ i @ g)) = $true) | ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X0 @ X1) @ (getRel @ a @ b @ i @ g)) != $true)) & ((refl_on @ b @ (labeled_vertices @ a @ b @ g) @ (getRel @ a @ b @ i @ g)) = $true)),
% 3.26/1.71 inference(ennf_transformation,[status(thm)],[f283])).
% 3.26/1.71 thf(f283,plain,(
% 3.26/1.71 ! [X0 : b,X1 : b] : (((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X0 @ X1) @ (getRel @ a @ b @ i @ g)) = $true) => ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X1 @ X0) @ (getRel @ a @ b @ i @ g)) = $true)) & ((trans @ b @ (getRel @ a @ b @ i @ g)) = $true) & ((refl_on @ b @ (labeled_vertices @ a @ b @ g) @ (getRel @ a @ b @ i @ g)) = $true)),
% 3.26/1.71 inference(fool_elimination,[status(thm)],[f282])).
% 3.26/1.71 thf(f282,plain,(
% 3.26/1.71 ! [X0 : b,X1 : b] : ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X0 @ X1) @ (getRel @ a @ b @ i @ g)) => (member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X1 @ X0) @ (getRel @ a @ b @ i @ g))) & (trans @ b @ (getRel @ a @ b @ i @ g)) & (refl_on @ b @ (labeled_vertices @ a @ b @ g) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 inference(rectify,[status(thm)],[f1])).
% 3.26/1.71 thf(f1,axiom,(
% 3.26/1.71 ! [X7 : b,X8 : b] : ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X7 @ X8) @ (getRel @ a @ b @ i @ g)) => (member @ product_prod @ b @ b @ (product_Pair @ b @ b @ X8 @ X7) @ (getRel @ a @ b @ i @ g))) & (trans @ b @ (getRel @ a @ b @ i @ g)) & (refl_on @ b @ (labeled_vertices @ a @ b @ g) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 file('/export/starexec/sandbox2/tmp/tmp.ANXxQNlu00/DTF2THF_30214.p',fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062)).
% 3.26/1.71 thf(f867,plain,(
% 3.26/1.71 ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ z @ y) @ (getRel @ a @ b @ i @ g)) != $true)),
% 3.26/1.71 inference(cnf_transformation,[status(thm)],[f766])).
% 3.26/1.71 thf(f766,plain,(
% 3.26/1.71 ((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ z @ y) @ (getRel @ a @ b @ i @ g)) != $true)),
% 3.26/1.71 inference(flattening,[status(thm)],[f725])).
% 3.26/1.71 thf(f725,plain,(
% 3.26/1.71 ~((member @ product_prod @ b @ b @ (product_Pair @ b @ b @ z @ y) @ (getRel @ a @ b @ i @ g)) = $true)),
% 3.26/1.71 inference(fool_elimination,[status(thm)],[f724])).
% 3.26/1.71 thf(f724,plain,(
% 3.26/1.71 ~(member @ product_prod @ b @ b @ (product_Pair @ b @ b @ z @ y) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 inference(rectify,[status(thm)],[f280])).
% 3.26/1.71 thf(f280,negated_conjecture,(
% 3.26/1.71 ~(member @ product_prod @ b @ b @ (product_Pair @ b @ b @ z @ y) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 inference(negated_conjecture,[status(cth)],[f279])).
% 3.26/1.71 thf(f279,conjecture,(
% 3.26/1.71 (member @ product_prod @ b @ b @ (product_Pair @ b @ b @ z @ y) @ (getRel @ a @ b @ i @ g))),
% 3.26/1.71 file('/export/starexec/sandbox2/tmp/tmp.ANXxQNlu00/DTF2THF_30214.p',conj_0)).
% 3.26/1.71 % SZS output end Proof for DTF2THF_30214
% 3.26/1.71 % (30364)------------------------------
% 3.26/1.71 % (30364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.71 % (30364)Termination reason: Refutation
% 3.26/1.71
% 3.26/1.71 % (30364)Memory used [KB]: 6908
% 3.26/1.71 % (30364)Time elapsed: 0.028 s
% 3.26/1.71 % (30364)Instructions burned: 78 (million)
% 3.26/1.71 % (30364)------------------------------
% 3.26/1.71 % (30364)------------------------------
% 3.26/1.71 % (30358)Success in time 0.038 s
%------------------------------------------------------------------------------