%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP176^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n020.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 2.87s 1.63s
% Output : Refutation 2.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP176^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n020.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:28:57 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.47 thf(spec,logic,$$dhol).
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % File : ITP176^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.47 % Domain : Interactive Theorem Proving
% 0.28/0.47 % Problem : Sledgehammer StandardRules problem prob_305__5389912_1
% 0.28/0.47 % Version : Especial.
% 0.28/0.47 % English :
% 0.28/0.47
% 0.28/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.47 % Source : [Des21]
% 0.28/0.47 % Names : StandardRules/prob_305__5389912_1 [Des21]
% 0.28/0.47
% 0.28/0.47 % Status : Theorem
% 0.28/0.47 % Rating : 0.00 v7.5.0
% 0.28/0.47 % Syntax : Number of formulae : 324 ( 136 unt; 48 typ; 0 def)
% 0.28/0.47 % Number of atoms : 710 ( 268 equ; 0 cnn)
% 0.28/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.28/0.47 % Number of connectives : 4770 ( 66 ~; 10 |; 46 &;4346 @)
% 0.28/0.47 % ( 0 <=>; 302 =>; 0 <=; 0 <~>)
% 0.28/0.47 % Maximal formula depth : 28 ( 8 avg)
% 0.28/0.47 % Number of types : 3 ( 2 usr)
% 0.28/0.47 % Number of type conns : 199 ( 199 >; 0 *; 0 +; 0 <<)
% 0.28/0.47 % Number of symbols : 49 ( 46 usr; 5 con; 0-6 aty)
% 0.28/0.47 % Number of variables : 1206 ( 92 ^;1043 !; 14 ?;1206 :)
% 0.28/0.47 % ( 57 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.47 % SPC : TH1_THM_EQU_NAR
% 0.28/0.47
% 0.28/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:22:48.434
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % Could-be-implicit typings (6)
% 0.28/0.47 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 0.28/0.47 standard_Constant: $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 0.28/0.47 labeled_graph: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.28/0.47 product_prod: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Set_Oset,type,
% 0.28/0.47 set: $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_tf_b,type,
% 0.28/0.47 b: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_tf_a,type,
% 0.28/0.47 a: $tType ).
% 0.28/0.47
% 0.28/0.47 % Explicit typings (42)
% 0.28/0.47 thf(sy_cl_HOL_Otype,type,
% 0.28/0.47 type:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.28/0.47 bot:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.28/0.47 ord:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.28/0.47 order:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/0.47 linorder:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.28/0.47 preorder:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.28/0.47 order_bot:
% 0.28/0.47 !>[A: $tType] : $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
% 0.28/0.47 bNF_Greatest_univ:
% 0.28/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 0.28/0.47 equiv_congruent:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Equiv__Relations_Ocongruent2,type,
% 0.28/0.47 equiv_congruent2:
% 0.28/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Equiv__Relations_Oequiv,type,
% 0.28/0.47 equiv_equiv:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Equiv__Relations_Oproj,type,
% 0.28/0.47 equiv_proj:
% 0.28/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Hilbert__Choice_OEps,type,
% 0.28/0.47 hilbert_Eps:
% 0.28/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_If,type,
% 0.28/0.47 if:
% 0.28/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt,type,
% 0.28/0.47 standard_S_Idt:
% 0.28/0.47 !>[V: $tType] : ( standard_Constant @ V ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 0.28/0.47 getRel:
% 0.28/0.47 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 0.28/0.47 graph_homomorphism:
% 0.28/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 0.28/0.47 graph_union:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 0.28/0.47 labeled_LG:
% 0.28/0.47 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 0.28/0.47 labele1974067554_graph:
% 0.28/0.47 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 0.28/0.47 labeled_edges:
% 0.28/0.47 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 0.28/0.47 labeled_vertices:
% 0.28/0.47 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_LabeledGraphs_Orestrict,type,
% 0.28/0.47 restrict:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 0.28/0.47 order_1409979114der_on:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.28/0.47 bot_bot:
% 0.28/0.47 !>[A: $tType] : A ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.47 ord_less_eq:
% 0.28/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.28/0.47 product_Pair:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.28/0.47 product_rec_prod:
% 0.28/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Relation_OImage,type,
% 0.28/0.47 image:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Relation_Oconverse,type,
% 0.28/0.47 converse:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Relation_Orefl__on,type,
% 0.28/0.47 refl_on:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Relation_Ototal__on,type,
% 0.28/0.47 total_on:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_OCollect,type,
% 0.28/0.47 collect:
% 0.28/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_Oinsert,type,
% 0.28/0.47 insert:
% 0.28/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.28/0.47 is_singleton:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.28/0.47 the_elem:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_member,type,
% 0.28/0.47 member:
% 0.28/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_G,type,
% 0.28/0.47 g: labeled_graph @ ( standard_Constant @ a ) @ b ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_P____,type,
% 0.28/0.47 p: b > b > $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_f____,type,
% 0.28/0.47 f: b > b ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_x____,type,
% 0.28/0.47 x: b ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_y____,type,
% 0.28/0.47 y: b ).
% 0.28/0.47
% 0.28/0.47 % Relevant facts (256)
% 0.28/0.47 thf(fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062,axiom,
% 0.28/0.47 ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ x @ ( bot_bot @ ( set @ b ) ) ) )
% 0.28/0.47 = ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ y @ ( bot_bot @ ( set @ b ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % \<open>getRel S_Idt G `` {x} = getRel S_Idt G `` {y}\<close>
% 0.28/0.47 thf(fact_1_assms_I1_J,axiom,
% 0.28/0.47 ( g
% 0.28/0.47 = ( restrict @ ( standard_Constant @ a ) @ b @ g ) ) ).
% 0.28/0.47
% 0.28/0.47 % assms(1)
% 0.28/0.47 thf(fact_2_P__eq,axiom,
% 0.28/0.47 ! [X: b] :
% 0.28/0.47 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X @ ( bot_bot @ ( set @ b ) ) ) )
% 0.28/0.47 != ( bot_bot @ ( set @ b ) ) )
% 0.28/0.47 => ( ( hilbert_Eps @ b @ ( p @ ( hilbert_Eps @ b @ ( p @ X ) ) ) )
% 0.28/0.47 = ( hilbert_Eps @ b @ ( p @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % P_eq
% 0.28/0.47 thf(fact_3_P,axiom,
% 0.28/0.47 ( p
% 0.28/0.47 = ( ^ [X2: b,Y: b] : ( member @ b @ Y @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % P
% 0.28/0.47 thf(fact_4__092_060open_062_Ix_M_Ay_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 0.28/0.47 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ x @ y ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 0.28/0.47
% 0.28/0.47 % \<open>(x, y) \<in> getRel S_Idt G\<close>
% 0.28/0.47 thf(fact_5__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
% 0.28/0.47 ( f
% 0.28/0.47 = ( ^ [X2: b] :
% 0.28/0.47 ( if @ b
% 0.28/0.47 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ b ) ) )
% 0.28/0.47 @ X2
% 0.28/0.47 @ ( hilbert_Eps @ b @ ( p @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % \<open>f \<equiv> \<lambda>x. if getRel S_Idt G `` {x} = {} then x else Eps (P x)\<close>
% 0.28/0.47 thf(fact_6_singleton__conv,axiom,
% 0.28/0.47 ! [A: $tType,A2: A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [X2: A] : ( X2 = A2 ) )
% 0.28/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % singleton_conv
% 0.28/0.47 thf(fact_7_singleton__conv2,axiom,
% 0.28/0.47 ! [A: $tType,A2: A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.28/0.47 @ A2 ) )
% 0.28/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % singleton_conv2
% 0.28/0.47 thf(fact_8_tuple__disj,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,Z2: A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [Y: A] :
% 0.28/0.47 ( ( Y = X3 )
% 0.28/0.47 | ( Y = Z2 ) ) )
% 0.28/0.47 = ( insert @ A @ X3 @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % tuple_disj
% 0.28/0.47 thf(fact_9_Image__empty1,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,X4: set @ B] :
% 0.28/0.47 ( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X4 )
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Image_empty1
% 0.28/0.47 thf(fact_10_Image__empty2,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Image_empty2
% 0.28/0.47 thf(fact_11_singletonI,axiom,
% 0.28/0.47 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % singletonI
% 0.28/0.47 thf(fact_12_some__equality,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,A2: A] :
% 0.28/0.47 ( ( P @ A2 )
% 0.28/0.47 => ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 => ( X5 = A2 ) )
% 0.28/0.47 => ( ( hilbert_Eps @ A @ P )
% 0.28/0.47 = A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % some_equality
% 0.28/0.47 thf(fact_13_some__eq__trivial,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] :
% 0.28/0.47 ( ( hilbert_Eps @ A
% 0.28/0.47 @ ^ [Y: A] : ( Y = X3 ) )
% 0.28/0.47 = X3 ) ).
% 0.28/0.47
% 0.28/0.47 % some_eq_trivial
% 0.28/0.47 thf(fact_14_empty__Collect__eq,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o] :
% 0.28/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.28/0.47 = ( collect @ A @ P ) )
% 0.28/0.47 = ( ! [X2: A] :
% 0.28/0.47 ~ ( P @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % empty_Collect_eq
% 0.28/0.47 thf(fact_15_Collect__empty__eq,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o] :
% 0.28/0.47 ( ( ( collect @ A @ P )
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 = ( ! [X2: A] :
% 0.28/0.47 ~ ( P @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_empty_eq
% 0.28/0.47 thf(fact_16_all__not__in__conv,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( ! [X2: A] :
% 0.28/0.47 ~ ( member @ A @ X2 @ A3 ) )
% 0.28/0.47 = ( A3
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % all_not_in_conv
% 0.28/0.47 thf(fact_17_empty__iff,axiom,
% 0.28/0.47 ! [A: $tType,C2: A] :
% 0.28/0.47 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % empty_iff
% 0.28/0.47 thf(fact_18_insert__absorb2,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,A3: set @ A] :
% 0.28/0.47 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
% 0.28/0.47 = ( insert @ A @ X3 @ A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_absorb2
% 0.28/0.47 thf(fact_19_insert__iff,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 0.28/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 0.28/0.47 = ( ( A2 = B2 )
% 0.28/0.47 | ( member @ A @ A2 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_iff
% 0.28/0.47 thf(fact_20_insertCI,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 0.28/0.47 ( ( ~ ( member @ A @ A2 @ B3 )
% 0.28/0.47 => ( A2 = B2 ) )
% 0.28/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insertCI
% 0.28/0.47 thf(fact_21_some__sym__eq__trivial,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] :
% 0.28/0.47 ( ( hilbert_Eps @ A
% 0.28/0.47 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.28/0.47 @ X3 ) )
% 0.28/0.47 = X3 ) ).
% 0.28/0.47
% 0.28/0.47 % some_sym_eq_trivial
% 0.28/0.47 thf(fact_22_ImageI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),A3: set @ A] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.28/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ImageI
% 0.28/0.47 thf(fact_23_f,axiom,
% 0.28/0.47 ( f
% 0.28/0.47 = ( ^ [X2: b] :
% 0.28/0.47 ( if @ b
% 0.28/0.47 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ b ) ) )
% 0.28/0.47 @ X2
% 0.28/0.47 @ ( hilbert_Eps @ b @ ( p @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f
% 0.28/0.47 thf(fact_24_Image__singleton__iff,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A2: B] :
% 0.28/0.47 ( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.28/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B2 ) @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % Image_singleton_iff
% 0.28/0.47 thf(fact_25_pred__equals__eq2,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
% 0.28/0.47 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) ) )
% 0.28/0.47 = ( R = S ) ) ).
% 0.28/0.47
% 0.28/0.47 % pred_equals_eq2
% 0.28/0.47 thf(fact_26_rev__ImageI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,R2: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.28/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_ImageI
% 0.28/0.47 thf(fact_27_Image__iff,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 0.28/0.47 ( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A3 ) )
% 0.28/0.47 = ( ? [X2: B] :
% 0.28/0.47 ( ( member @ B @ X2 @ A3 )
% 0.28/0.47 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Image_iff
% 0.28/0.47 thf(fact_28_ImageE,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 0.28/0.47 ( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A3 ) )
% 0.28/0.47 => ~ ! [X5: B] :
% 0.28/0.47 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R2 )
% 0.28/0.47 => ~ ( member @ B @ X5 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ImageE
% 0.28/0.47 thf(fact_29_Image__singleton,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A2: B] :
% 0.28/0.47 ( ( image @ B @ A @ R2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
% 0.28/0.47 = ( collect @ A
% 0.28/0.47 @ ^ [B4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B4 ) @ R2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Image_singleton
% 0.28/0.47 thf(fact_30_ex__in__conv,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.28/0.47 = ( A3
% 0.28/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_in_conv
% 0.28/0.47 thf(fact_31_equals0I,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ! [Y3: A] :
% 0.28/0.47 ~ ( member @ A @ Y3 @ A3 )
% 0.28/0.47 => ( A3
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equals0I
% 0.28/0.47 thf(fact_32_equals0D,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,A2: A] :
% 0.28/0.47 ( ( A3
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % equals0D
% 0.28/0.47 thf(fact_33_emptyE,axiom,
% 0.28/0.47 ! [A: $tType,A2: A] :
% 0.28/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % emptyE
% 0.28/0.47 thf(fact_34_mk__disjoint__insert,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.28/0.47 ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ? [B5: set @ A] :
% 0.28/0.47 ( ( A3
% 0.28/0.47 = ( insert @ A @ A2 @ B5 ) )
% 0.28/0.47 & ~ ( member @ A @ A2 @ B5 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mk_disjoint_insert
% 0.28/0.47 thf(fact_35_insert__commute,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,Y4: A,A3: set @ A] :
% 0.28/0.47 ( ( insert @ A @ X3 @ ( insert @ A @ Y4 @ A3 ) )
% 0.28/0.47 = ( insert @ A @ Y4 @ ( insert @ A @ X3 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_commute
% 0.28/0.47 thf(fact_36_insert__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
% 0.28/0.47 ( ~ ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( ~ ( member @ A @ B2 @ B3 )
% 0.28/0.47 => ( ( ( insert @ A @ A2 @ A3 )
% 0.28/0.47 = ( insert @ A @ B2 @ B3 ) )
% 0.28/0.47 = ( ( ( A2 = B2 )
% 0.28/0.47 => ( A3 = B3 ) )
% 0.28/0.47 & ( ( A2 != B2 )
% 0.28/0.47 => ? [C3: set @ A] :
% 0.28/0.47 ( ( A3
% 0.28/0.47 = ( insert @ A @ B2 @ C3 ) )
% 0.28/0.47 & ~ ( member @ A @ B2 @ C3 )
% 0.28/0.47 & ( B3
% 0.28/0.47 = ( insert @ A @ A2 @ C3 ) )
% 0.28/0.47 & ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_eq_iff
% 0.28/0.47 thf(fact_37_insert__absorb,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.28/0.47 ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( ( insert @ A @ A2 @ A3 )
% 0.28/0.47 = A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_absorb
% 0.28/0.47 thf(fact_38_insert__ident,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ~ ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( ~ ( member @ A @ X3 @ B3 )
% 0.28/0.47 => ( ( ( insert @ A @ X3 @ A3 )
% 0.28/0.47 = ( insert @ A @ X3 @ B3 ) )
% 0.28/0.47 = ( A3 = B3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_ident
% 0.28/0.47 thf(fact_39_Set_Oset__insert,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,A3: set @ A] :
% 0.28/0.47 ( ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ~ ! [B5: set @ A] :
% 0.28/0.47 ( ( A3
% 0.28/0.47 = ( insert @ A @ X3 @ B5 ) )
% 0.28/0.47 => ( member @ A @ X3 @ B5 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Set.set_insert
% 0.28/0.47 thf(fact_40_insertI2,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 0.28/0.47 ( ( member @ A @ A2 @ B3 )
% 0.28/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insertI2
% 0.28/0.47 thf(fact_41_insertI1,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % insertI1
% 0.28/0.47 thf(fact_42_insertE,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 0.28/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 0.28/0.47 => ( ( A2 != B2 )
% 0.28/0.47 => ( member @ A @ A2 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insertE
% 0.28/0.47 thf(fact_43_some__eq__imp,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,A2: A,B2: A] :
% 0.28/0.47 ( ( ( hilbert_Eps @ A @ P )
% 0.28/0.47 = A2 )
% 0.28/0.47 => ( ( P @ B2 )
% 0.28/0.47 => ( P @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % some_eq_imp
% 0.28/0.47 thf(fact_44_tfl__some,axiom,
% 0.28/0.47 ! [A: $tType,P2: A > $o,X6: A] :
% 0.28/0.47 ( ( P2 @ X6 )
% 0.28/0.47 => ( P2 @ ( hilbert_Eps @ A @ P2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % tfl_some
% 0.28/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.47 = ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.28/0.47 = A3 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_47_Collect__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 = ( Q @ X5 ) )
% 0.28/0.47 => ( ( collect @ A @ P )
% 0.28/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_cong
% 0.28/0.47 thf(fact_48_ext,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ( ( F @ X5 )
% 0.28/0.47 = ( G @ X5 ) )
% 0.28/0.47 => ( F = G ) ) ).
% 0.28/0.47
% 0.28/0.47 % ext
% 0.28/0.47 thf(fact_49_Eps__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 = ( Q @ X5 ) )
% 0.28/0.47 => ( ( hilbert_Eps @ A @ P )
% 0.28/0.47 = ( hilbert_Eps @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Eps_cong
% 0.28/0.47 thf(fact_50_someI,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,X3: A] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % someI
% 0.28/0.47 thf(fact_51_empty__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.28/0.47 = ( collect @ A
% 0.28/0.47 @ ^ [X2: A] : $false ) ) ).
% 0.28/0.47
% 0.28/0.47 % empty_def
% 0.28/0.47 thf(fact_52_insert__Collect,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.47 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.47 = ( collect @ A
% 0.28/0.47 @ ^ [U: A] :
% 0.28/0.47 ( ( U != A2 )
% 0.28/0.47 => ( P @ U ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_Collect
% 0.28/0.47 thf(fact_53_insert__compr,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( insert @ A )
% 0.28/0.47 = ( ^ [A4: A,B6: set @ A] :
% 0.28/0.47 ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( X2 = A4 )
% 0.28/0.47 | ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_compr
% 0.28/0.47 thf(fact_54_some1__equality,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,A2: A] :
% 0.28/0.47 ( ? [X6: A] :
% 0.28/0.47 ( ( P @ X6 )
% 0.28/0.47 & ! [Y3: A] :
% 0.28/0.47 ( ( P @ Y3 )
% 0.28/0.47 => ( Y3 = X6 ) ) )
% 0.28/0.47 => ( ( P @ A2 )
% 0.28/0.47 => ( ( hilbert_Eps @ A @ P )
% 0.28/0.47 = A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % some1_equality
% 0.28/0.47 thf(fact_55_some__eq__ex,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o] :
% 0.28/0.47 ( ( P @ ( hilbert_Eps @ A @ P ) )
% 0.28/0.47 = ( ? [X7: A] : ( P @ X7 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % some_eq_ex
% 0.28/0.47 thf(fact_56_someI2__bex,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ? [X6: A] :
% 0.28/0.47 ( ( member @ A @ X6 @ A3 )
% 0.28/0.47 & ( P @ X6 ) )
% 0.28/0.47 => ( ! [X5: A] :
% 0.28/0.47 ( ( ( member @ A @ X5 @ A3 )
% 0.28/0.47 & ( P @ X5 ) )
% 0.28/0.47 => ( Q @ X5 ) )
% 0.28/0.47 => ( Q
% 0.28/0.47 @ ( hilbert_Eps @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ A3 )
% 0.28/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % someI2_bex
% 0.28/0.47 thf(fact_57_someI2__ex,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ? [X_1: A] : ( P @ X_1 )
% 0.28/0.47 => ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 => ( Q @ X5 ) )
% 0.28/0.47 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % someI2_ex
% 0.28/0.47 thf(fact_58_someI__ex,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o] :
% 0.28/0.47 ( ? [X_1: A] : ( P @ X_1 )
% 0.28/0.47 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % someI_ex
% 0.28/0.47 thf(fact_59_someI2,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.28/0.47 ( ( P @ A2 )
% 0.28/0.47 => ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 => ( Q @ X5 ) )
% 0.28/0.47 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % someI2
% 0.28/0.47 thf(fact_60_singleton__inject,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B2: A] :
% 0.28/0.47 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 => ( A2 = B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % singleton_inject
% 0.28/0.47 thf(fact_61_insert__not__empty,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.28/0.47 ( ( insert @ A @ A2 @ A3 )
% 0.28/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_not_empty
% 0.28/0.47 thf(fact_62_doubleton__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,B2: A,C2: A,D: A] :
% 0.28/0.47 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 = ( ( ( A2 = C2 )
% 0.28/0.47 & ( B2 = D ) )
% 0.28/0.47 | ( ( A2 = D )
% 0.28/0.47 & ( B2 = C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % doubleton_eq_iff
% 0.28/0.47 thf(fact_63_singleton__iff,axiom,
% 0.28/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.28/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( B2 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % singleton_iff
% 0.28/0.47 thf(fact_64_singletonD,axiom,
% 0.28/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.28/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 => ( B2 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % singletonD
% 0.28/0.47 thf(fact_65_Collect__conv__if2,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,A2: A] :
% 0.28/0.47 ( ( ( P @ A2 )
% 0.28/0.47 => ( ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( A2 = X2 )
% 0.28/0.47 & ( P @ X2 ) ) )
% 0.28/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 & ( ~ ( P @ A2 )
% 0.28/0.47 => ( ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( A2 = X2 )
% 0.28/0.47 & ( P @ X2 ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_conv_if2
% 0.28/0.47 thf(fact_66_Collect__conv__if,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,A2: A] :
% 0.28/0.47 ( ( ( P @ A2 )
% 0.28/0.47 => ( ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( X2 = A2 )
% 0.28/0.47 & ( P @ X2 ) ) )
% 0.28/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 & ( ~ ( P @ A2 )
% 0.28/0.47 => ( ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( X2 = A2 )
% 0.28/0.47 & ( P @ X2 ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_conv_if
% 0.28/0.47 thf(fact_67_some__in__eq,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( member @ A
% 0.28/0.47 @ ( hilbert_Eps @ A
% 0.28/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.28/0.47 @ A3 )
% 0.28/0.47 = ( A3
% 0.28/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % some_in_eq
% 0.28/0.47 thf(fact_68_congr_I1_J,axiom,
% 0.28/0.47 ! [X: b,Y5: b,L2: standard_Constant @ a] :
% 0.28/0.47 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X @ Y5 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) )
% 0.28/0.47 => ( equiv_congruent @ b @ ( set @ b ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g )
% 0.28/0.47 @ ^ [V2: b] : ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) @ ( insert @ b @ V2 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congr(1)
% 0.28/0.47 thf(fact_69_vert__P,axiom,
% 0.28/0.47 ! [X3: b] :
% 0.28/0.47 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.28/0.47 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ ( hilbert_Eps @ b @ ( p @ X3 ) ) ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % vert_P
% 0.28/0.47 thf(fact_70__092_060open_062_IS__Idt_M_Ax_M_Ay_J_A_092_060in_062_Aedges_AG_092_060close_062,axiom,
% 0.28/0.47 member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ ( standard_S_Idt @ a ) @ ( product_Pair @ b @ b @ x @ y ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) ).
% 0.28/0.47
% 0.28/0.47 % \<open>(S_Idt, x, y) \<in> edges G\<close>
% 0.28/0.47 thf(fact_71_restrict__idemp,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,X3: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X3 ) )
% 0.28/0.47 = ( restrict @ A @ B @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % restrict_idemp
% 0.28/0.47 thf(fact_72_r2,axiom,
% 0.28/0.47 ! [X3: b] :
% 0.28/0.47 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X3 @ ( bot_bot @ ( set @ b ) ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ b ) ) )
% 0.28/0.47 = ( ~ ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % r2
% 0.28/0.47 thf(fact_73_r1,axiom,
% 0.28/0.47 ! [X3: b] :
% 0.28/0.47 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.28/0.47 = ( p @ X3 @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % r1
% 0.28/0.47 thf(fact_74_congr_I2_J,axiom,
% 0.28/0.47 ! [X: b,Y5: b,L2: standard_Constant @ a] :
% 0.28/0.47 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X @ Y5 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) )
% 0.28/0.47 => ( equiv_congruent @ b @ ( set @ b ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g )
% 0.28/0.47 @ ^ [V2: b] : ( image @ b @ b @ ( converse @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) ) @ ( insert @ b @ V2 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congr(2)
% 0.28/0.47 thf(fact_75_the__elem__eq,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] :
% 0.28/0.47 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = X3 ) ).
% 0.28/0.47
% 0.28/0.47 % the_elem_eq
% 0.28/0.47 thf(fact_76_bot__apply,axiom,
% 0.28/0.47 ! [C: $tType,D2: $tType] :
% 0.28/0.47 ( ( bot @ C )
% 0.28/0.47 => ( ( bot_bot @ ( D2 > C ) )
% 0.28/0.47 = ( ^ [X2: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_apply
% 0.28/0.47 thf(fact_77_prod_Oinject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.28/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.28/0.47 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X22 = Y22 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_78_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B7: B] :
% 0.28/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.28/0.47 = ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.28/0.47 = ( ( A2 = A5 )
% 0.28/0.47 & ( B2 = B7 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_79_converse__inject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( ( converse @ B @ A @ R2 )
% 0.28/0.47 = ( converse @ B @ A @ S2 ) )
% 0.28/0.47 = ( R2 = S2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse_inject
% 0.28/0.47 thf(fact_80_converse__converse,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
% 0.28/0.47 = R2 ) ).
% 0.28/0.47
% 0.28/0.47 % converse_converse
% 0.28/0.47 thf(fact_81_converse__iff,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R2 ) )
% 0.28/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse_iff
% 0.28/0.47 thf(fact_82_vertices__restrict,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,G2: labeled_graph @ B @ A] :
% 0.28/0.47 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G2 ) )
% 0.28/0.47 = ( labeled_vertices @ B @ A @ G2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % vertices_restrict
% 0.28/0.47 thf(fact_83_converse__empty,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.28/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse_empty
% 0.28/0.47 thf(fact_84__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 0.28/0.47 ! [X3: b] :
% 0.28/0.47 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 0.28/0.47 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ X3 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % \<open>\<And>x. x \<in> vertices G \<Longrightarrow> (x, x) \<in> getRel S_Idt G\<close>
% 0.28/0.47 thf(fact_85_labeled__graph_Oexpand,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 0.28/0.47 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 0.28/0.47 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 0.28/0.47 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 0.28/0.47 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 0.28/0.47 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.expand
% 0.28/0.47 thf(fact_86_converse_Oinducts,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,X1: B,X22: A,R2: set @ ( product_prod @ A @ B ),P: B > A > $o] :
% 0.28/0.47 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R2 ) )
% 0.28/0.47 => ( ! [A6: A,B8: B] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B8 ) @ R2 )
% 0.28/0.47 => ( P @ B8 @ A6 ) )
% 0.28/0.47 => ( P @ X1 @ X22 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse.inducts
% 0.28/0.47 thf(fact_87_converse_Ointros,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.28/0.47 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse.intros
% 0.28/0.47 thf(fact_88_converse_Osimps,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 0.28/0.47 = ( ? [A4: A,B4: B] :
% 0.28/0.47 ( ( A1 = B4 )
% 0.28/0.47 & ( A22 = A4 )
% 0.28/0.47 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse.simps
% 0.28/0.47 thf(fact_89_converse_Ocases,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 0.28/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse.cases
% 0.28/0.47 thf(fact_90_converseE,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
% 0.28/0.47 => ~ ! [X5: B,Y3: A] :
% 0.28/0.47 ( ( Yx
% 0.28/0.47 = ( product_Pair @ A @ B @ Y3 @ X5 ) )
% 0.28/0.47 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ Y3 ) @ R2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % converseE
% 0.28/0.47 thf(fact_91_converseD,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R2 ) )
% 0.28/0.47 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % converseD
% 0.28/0.47 thf(fact_92_bot__set__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.28/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_set_def
% 0.28/0.47 thf(fact_93_bot__empty__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.28/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_empty_eq
% 0.28/0.47 thf(fact_94_bot__empty__eq2,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( bot_bot @ ( A > B > $o ) )
% 0.28/0.47 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_empty_eq2
% 0.28/0.47 thf(fact_95_old_Oprod_Oinducts,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.28/0.47 ( ! [A6: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B8 ) )
% 0.28/0.47 => ( P @ Prod ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inducts
% 0.28/0.47 thf(fact_96_old_Oprod_Oexhaust,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
% 0.28/0.47 ~ ! [A6: A,B8: B] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( product_Pair @ A @ B @ A6 @ B8 ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.exhaust
% 0.28/0.47 thf(fact_97_prod__induct7,axiom,
% 0.28/0.47 ! [G3: $tType,F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 0.28/0.47 ( ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
% 0.28/0.47 => ( P @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct7
% 0.28/0.47 thf(fact_98_prod__induct6,axiom,
% 0.28/0.47 ! [F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.28/0.47 ( ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.28/0.47 => ( P @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct6
% 0.28/0.47 thf(fact_99_prod__induct5,axiom,
% 0.28/0.47 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.28/0.47 ( ! [A6: A,B8: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 0.28/0.47 => ( P @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct5
% 0.28/0.47 thf(fact_100_prod__induct4,axiom,
% 0.28/0.47 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.28/0.47 ( ! [A6: A,B8: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B8 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.28/0.47 => ( P @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct4
% 0.28/0.47 thf(fact_101_prod__induct3,axiom,
% 0.28/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.28/0.47 ( ! [A6: A,B8: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) )
% 0.28/0.47 => ( P @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct3
% 0.28/0.47 thf(fact_102_prod__cases7,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,G3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 0.28/0.47 ~ ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2,G4: G3] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases7
% 0.28/0.47 thf(fact_103_prod__cases6,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.28/0.47 ~ ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases6
% 0.28/0.47 thf(fact_104_prod__cases5,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.28/0.47 ~ ! [A6: A,B8: B,C4: C,D3: D2,E2: E] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases5
% 0.28/0.47 thf(fact_105_prod__cases4,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.28/0.47 ~ ! [A6: A,B8: B,C4: C,D3: D2] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B8 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases4
% 0.28/0.47 thf(fact_106_prod__cases3,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.28/0.47 ~ ! [A6: A,B8: B,C4: C] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases3
% 0.28/0.47 thf(fact_107_Pair__inject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B7: B] :
% 0.28/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.28/0.47 = ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.28/0.47 => ~ ( ( A2 = A5 )
% 0.28/0.47 => ( B2 != B7 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_108_prod__cases,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
% 0.28/0.47 ( ! [A6: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B8 ) )
% 0.28/0.47 => ( P @ P3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_109_surj__pair,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
% 0.28/0.47 ? [X5: A,Y3: B] :
% 0.28/0.47 ( P3
% 0.28/0.47 = ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_110_bot__fun__def,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( bot @ B )
% 0.28/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.28/0.47 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_fun_def
% 0.28/0.47 thf(fact_111_local_Orefl,axiom,
% 0.28/0.47 refl_on @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 0.28/0.47
% 0.28/0.47 % local.refl
% 0.28/0.47 thf(fact_112_equiv,axiom,
% 0.28/0.47 equiv_equiv @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 0.28/0.47
% 0.28/0.47 % equiv
% 0.28/0.47 thf(fact_113_getRel__dom_I1_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: labeled_graph @ A @ B,A2: B,B2: B,L3: A] :
% 0.28/0.47 ( ( G2
% 0.28/0.47 = ( restrict @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L3 @ G2 ) )
% 0.28/0.47 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % getRel_dom(1)
% 0.28/0.47 thf(fact_114_getRel__dom_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: labeled_graph @ A @ B,A2: B,B2: B,L3: A] :
% 0.28/0.47 ( ( G2
% 0.28/0.47 = ( restrict @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L3 @ G2 ) )
% 0.28/0.47 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % getRel_dom(2)
% 0.28/0.47 thf(fact_115_old_Oprod_Orec,axiom,
% 0.28/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.28/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.28/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.rec
% 0.28/0.47 thf(fact_116_is__singleton__the__elem,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( is_singleton @ A )
% 0.28/0.47 = ( ^ [A7: set @ A] :
% 0.28/0.47 ( A7
% 0.28/0.47 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % is_singleton_the_elem
% 0.28/0.47 thf(fact_117_refl__on__converse,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.28/0.47 ( ( refl_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
% 0.28/0.47 = ( refl_on @ A @ A3 @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_on_converse
% 0.28/0.47 thf(fact_118_is__singletonI,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % is_singletonI
% 0.28/0.47 thf(fact_119_bot2E,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,X3: A,Y4: B] :
% 0.28/0.47 ~ ( bot_bot @ ( A > B > $o ) @ X3 @ Y4 ) ).
% 0.28/0.47
% 0.28/0.47 % bot2E
% 0.28/0.47 thf(fact_120_refl__onD,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
% 0.28/0.47 ( ( refl_on @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_onD
% 0.28/0.47 thf(fact_121_refl__onD1,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 0.28/0.47 ( ( refl_on @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
% 0.28/0.47 => ( member @ A @ X3 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_onD1
% 0.28/0.47 thf(fact_122_refl__onD2,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 0.28/0.47 ( ( refl_on @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
% 0.28/0.47 => ( member @ A @ Y4 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_onD2
% 0.28/0.47 thf(fact_123_refl__on__empty,axiom,
% 0.28/0.47 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_on_empty
% 0.28/0.47 thf(fact_124_is__singletonI_H,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( A3
% 0.28/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 => ( ! [X5: A,Y3: A] :
% 0.28/0.47 ( ( member @ A @ X5 @ A3 )
% 0.28/0.47 => ( ( member @ A @ Y3 @ A3 )
% 0.28/0.47 => ( X5 = Y3 ) ) )
% 0.28/0.47 => ( is_singleton @ A @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % is_singletonI'
% 0.28/0.47 thf(fact_125_refl__on__singleton,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] : ( refl_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_on_singleton
% 0.28/0.47 thf(fact_126_is__singleton__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( is_singleton @ A )
% 0.28/0.47 = ( ^ [A7: set @ A] :
% 0.28/0.47 ? [X2: A] :
% 0.28/0.47 ( A7
% 0.28/0.47 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % is_singleton_def
% 0.28/0.47 thf(fact_127_is__singletonE,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( is_singleton @ A @ A3 )
% 0.28/0.47 => ~ ! [X5: A] :
% 0.28/0.47 ( A3
% 0.28/0.47 != ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % is_singletonE
% 0.28/0.47 thf(fact_128_equiv__class__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
% 0.28/0.47 = ( ( ( image @ A @ A @ R2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( image @ A @ A @ R2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 & ( member @ A @ X3 @ A3 )
% 0.28/0.47 & ( member @ A @ Y4 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equiv_class_eq_iff
% 0.28/0.47 thf(fact_129_eq__equiv__class__iff,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( ( member @ A @ Y4 @ A3 )
% 0.28/0.47 => ( ( ( image @ A @ A @ R2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( image @ A @ A @ R2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_equiv_class_iff
% 0.28/0.47 thf(fact_130_equiv__class__eq,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 0.28/0.47 => ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equiv_class_eq
% 0.28/0.47 thf(fact_131_eq__equiv__class,axiom,
% 0.28/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A,A3: set @ A] :
% 0.28/0.47 ( ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 => ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ A @ B2 @ A3 )
% 0.28/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_equiv_class
% 0.28/0.47 thf(fact_132_congruentI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.28/0.47 ( ! [Y3: A,Z3: A] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 0.28/0.47 => ( ( F @ Y3 )
% 0.28/0.47 = ( F @ Z3 ) ) )
% 0.28/0.47 => ( equiv_congruent @ A @ B @ R2 @ F ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruentI
% 0.28/0.47 thf(fact_133_congruentD,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F: A > B,Y4: A,Z2: A] :
% 0.28/0.47 ( ( equiv_congruent @ A @ B @ R2 @ F )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R2 )
% 0.28/0.47 => ( ( F @ Y4 )
% 0.28/0.47 = ( F @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruentD
% 0.28/0.47 thf(fact_134_equiv__class__self,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( member @ A @ A2 @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equiv_class_self
% 0.28/0.47 thf(fact_135_Collect__empty__eq__bot,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o] :
% 0.28/0.47 ( ( ( collect @ A @ P )
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 = ( P
% 0.28/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_empty_eq_bot
% 0.28/0.47 thf(fact_136_graph__single,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,C2: B] :
% 0.28/0.47 ( ( 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 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.28/0.47 = ( 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 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_single
% 0.28/0.47 thf(fact_137_refl__on__domain,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.28/0.47 ( ( refl_on @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 0.28/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 & ( member @ A @ B2 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refl_on_domain
% 0.28/0.47 thf(fact_138_total__on__singleton,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] : ( total_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % total_on_singleton
% 0.28/0.47 thf(fact_139_labeled__graph_Oinject,axiom,
% 0.28/0.47 ! [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 ) ),Y22: set @ V] :
% 0.28/0.47 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 0.28/0.47 = ( labeled_LG @ L @ V @ Y1 @ Y22 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X22 = Y22 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.inject
% 0.28/0.47 thf(fact_140_total__on__converse,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.28/0.47 ( ( total_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
% 0.28/0.47 = ( total_on @ A @ A3 @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % total_on_converse
% 0.28/0.47 thf(fact_141_labeled__graph_Ocollapse,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.28/0.47 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 0.28/0.47 = Labeled_graph ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.collapse
% 0.28/0.47 thf(fact_142_graph__empty__e,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,V3: set @ B] :
% 0.28/0.47 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
% 0.28/0.47 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_empty_e
% 0.28/0.47 thf(fact_143_total__on__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( total_on @ A )
% 0.28/0.47 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.28/0.47 ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ A7 )
% 0.28/0.47 => ! [Y: A] :
% 0.28/0.47 ( ( member @ A @ Y @ A7 )
% 0.28/0.47 => ( ( X2 != Y )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 )
% 0.28/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % total_on_def
% 0.28/0.47 thf(fact_144_total__onI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.28/0.47 ( ! [X5: A,Y3: A] :
% 0.28/0.47 ( ( member @ A @ X5 @ A3 )
% 0.28/0.47 => ( ( member @ A @ Y3 @ A3 )
% 0.28/0.47 => ( ( X5 != Y3 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ R2 )
% 0.28/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X5 ) @ R2 ) ) ) ) )
% 0.28/0.47 => ( total_on @ A @ A3 @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % total_onI
% 0.28/0.47 thf(fact_145_total__on__empty,axiom,
% 0.28/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% 0.28/0.47
% 0.28/0.47 % total_on_empty
% 0.28/0.47 thf(fact_146_labeled__graph_Osel_I2_J,axiom,
% 0.28/0.47 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.28/0.47 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.28/0.47 = X22 ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.sel(2)
% 0.28/0.47 thf(fact_147_labeled__graph_Oexhaust,axiom,
% 0.28/0.47 ! [L: $tType,V: $tType,Y4: labeled_graph @ L @ V] :
% 0.28/0.47 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 0.28/0.47 ( Y4
% 0.28/0.47 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.exhaust
% 0.28/0.47 thf(fact_148_labeled__graph_Oinduct,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType,P: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 0.28/0.47 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 0.28/0.47 => ( P @ Labeled_graph ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.induct
% 0.28/0.47 thf(fact_149_labeled__graph_Osel_I1_J,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.28/0.47 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.28/0.47 = X1 ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.sel(1)
% 0.28/0.47 thf(fact_150_labeled__graph_Oexhaust__sel,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 0.28/0.47 ( Labeled_graph
% 0.28/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.exhaust_sel
% 0.28/0.47 thf(fact_151_linear__order__on__singleton,axiom,
% 0.28/0.47 ! [A: $tType,X3: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linear_order_on_singleton
% 0.28/0.47 thf(fact_152_labeled__graph_Osplit__sel,axiom,
% 0.28/0.47 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.28/0.47 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 0.28/0.47 = ( ( Labeled_graph
% 0.28/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.28/0.47 => ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.split_sel
% 0.28/0.47 thf(fact_153_labeled__graph_Osplit__sel__asm,axiom,
% 0.28/0.47 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.28/0.47 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 0.28/0.47 = ( ~ ( ( Labeled_graph
% 0.28/0.47 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 0.28/0.47 & ~ ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.split_sel_asm
% 0.28/0.47 thf(fact_154_graph__homomorphism__empty,axiom,
% 0.28/0.47 ! [B: $tType,C: $tType,A: $tType,G2: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 0.28/0.47 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G2 @ F )
% 0.28/0.47 = ( ( F
% 0.28/0.47 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 0.28/0.47 & ( G2
% 0.28/0.47 = ( restrict @ A @ C @ G2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_homomorphism_empty
% 0.28/0.47 thf(fact_155_linear__order__on__converse,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.28/0.47 ( ( order_1409979114der_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
% 0.28/0.47 = ( order_1409979114der_on @ A @ A3 @ R2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % linear_order_on_converse
% 0.28/0.47 thf(fact_156_labeled__graph_Ocase__distrib,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,V: $tType,L: $tType,H: A > B,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 0.28/0.47 ( ( H @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 0.28/0.47 = ( labele1974067554_graph @ L @ V @ B
% 0.28/0.47 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : ( H @ ( F @ X13 @ X24 ) )
% 0.28/0.47 @ Labeled_graph ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.case_distrib
% 0.28/0.47 thf(fact_157_labeled__graph_Ocase,axiom,
% 0.28/0.47 ! [L: $tType,A: $tType,V: $tType,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 0.28/0.47 ( ( labele1974067554_graph @ L @ V @ A @ F @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 0.28/0.47 = ( F @ X1 @ X22 ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.case
% 0.28/0.47 thf(fact_158_edges__def,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType] :
% 0.28/0.47 ( ( labeled_edges @ L @ V )
% 0.28/0.47 = ( labele1974067554_graph @ L @ V @ ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) )
% 0.28/0.47 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X13 ) ) ).
% 0.28/0.47
% 0.28/0.47 % edges_def
% 0.28/0.47 thf(fact_159_vertices__def,axiom,
% 0.28/0.47 ! [V: $tType,L: $tType] :
% 0.28/0.47 ( ( labeled_vertices @ L @ V )
% 0.28/0.47 = ( labele1974067554_graph @ L @ V @ ( set @ V )
% 0.28/0.47 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X24 ) ) ).
% 0.28/0.47
% 0.28/0.47 % vertices_def
% 0.28/0.47 thf(fact_160_lnear__order__on__empty,axiom,
% 0.28/0.47 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % lnear_order_on_empty
% 0.28/0.47 thf(fact_161_labeled__graph_Ocase__eq__if,axiom,
% 0.28/0.47 ! [A: $tType,V: $tType,L: $tType] :
% 0.28/0.47 ( ( labele1974067554_graph @ L @ V @ A )
% 0.28/0.47 = ( ^ [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 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % labeled_graph.case_eq_if
% 0.28/0.47 thf(fact_162_congruent2__implies__congruent,axiom,
% 0.28/0.47 ! [B: $tType,C: $tType,A: $tType,A3: set @ A,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,A2: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R1 )
% 0.28/0.47 => ( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
% 0.28/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.28/0.47 => ( equiv_congruent @ B @ C @ R22 @ ( F @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruent2_implies_congruent
% 0.28/0.47 thf(fact_163_proj__def,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( equiv_proj @ B @ A )
% 0.28/0.47 = ( ^ [R3: set @ ( product_prod @ B @ A ),X2: B] : ( image @ B @ A @ R3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % proj_def
% 0.28/0.47 thf(fact_164_equiv__class__subset,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equiv_class_subset
% 0.28/0.47 thf(fact_165_order__refl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_refl
% 0.28/0.47 thf(fact_166_subset__antisym,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.28/0.47 => ( A3 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_antisym
% 0.28/0.47 thf(fact_167_subsetI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ( ( member @ A @ X5 @ A3 )
% 0.28/0.47 => ( member @ A @ X5 @ B3 ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % subsetI
% 0.28/0.47 thf(fact_168_converse__mono,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S2 ) )
% 0.28/0.47 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse_mono
% 0.28/0.47 thf(fact_169_empty__subsetI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.28/0.47
% 0.28/0.47 % empty_subsetI
% 0.28/0.47 thf(fact_170_subset__empty,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 = ( A3
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_empty
% 0.28/0.47 thf(fact_171_insert__subset,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 0.28/0.47 = ( ( member @ A @ X3 @ B3 )
% 0.28/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_subset
% 0.28/0.47 thf(fact_172_singleton__insert__inj__eq_H,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,A3: set @ A,B2: A] :
% 0.28/0.47 ( ( ( insert @ A @ A2 @ A3 )
% 0.28/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( ( A2 = B2 )
% 0.28/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % singleton_insert_inj_eq'
% 0.28/0.47 thf(fact_173_singleton__insert__inj__eq,axiom,
% 0.28/0.47 ! [A: $tType,B2: A,A2: A,A3: set @ A] :
% 0.28/0.47 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 = ( insert @ A @ A2 @ A3 ) )
% 0.28/0.47 = ( ( A2 = B2 )
% 0.28/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % singleton_insert_inj_eq
% 0.28/0.47 thf(fact_174_dual__order_Oantisym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B2: A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.antisym
% 0.28/0.47 thf(fact_175_dual__order_Oeq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.28/0.47 = ( ^ [A4: A,B4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.47 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.eq_iff
% 0.28/0.47 thf(fact_176_dual__order_Otrans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.28/0.47 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.trans
% 0.28/0.47 thf(fact_177_linorder__wlog,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.28/0.47 ( ! [A6: A,B8: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A6 @ B8 )
% 0.28/0.47 => ( P @ A6 @ B8 ) )
% 0.28/0.47 => ( ! [A6: A,B8: A] :
% 0.28/0.47 ( ( P @ B8 @ A6 )
% 0.28/0.47 => ( P @ A6 @ B8 ) )
% 0.28/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_wlog
% 0.28/0.47 thf(fact_178_dual__order_Orefl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.refl
% 0.28/0.47 thf(fact_179_order__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X3: A,Y4: A,Z2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.28/0.47 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_trans
% 0.28/0.47 thf(fact_180_order__class_Oorder_Oantisym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_class.order.antisym
% 0.28/0.47 thf(fact_181_ord__le__eq__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord @ A )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( B2 = C2 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_le_eq_trans
% 0.28/0.47 thf(fact_182_ord__eq__le__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord @ A )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( A2 = B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_eq_le_trans
% 0.28/0.47 thf(fact_183_order__class_Oorder_Oeq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.28/0.47 = ( ^ [A4: A,B4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.47 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_class.order.eq_iff
% 0.28/0.47 thf(fact_184_antisym__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [Y4: A,X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ Y4 @ X3 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 = ( X3 = Y4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym_conv
% 0.28/0.47 thf(fact_185_le__cases3,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X3: A,Y4: A,Z2: A] :
% 0.28/0.47 ( ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ X3 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Y4 @ X3 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.28/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_cases3
% 0.28/0.47 thf(fact_186_order_Otrans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.trans
% 0.28/0.47 thf(fact_187_le__cases,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X3: A,Y4: A] :
% 0.28/0.47 ( ~ ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_cases
% 0.28/0.47 thf(fact_188_eq__refl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( X3 = Y4 )
% 0.28/0.47 => ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_refl
% 0.28/0.47 thf(fact_189_linear,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 | ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linear
% 0.28/0.47 thf(fact_190_antisym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y4 @ X3 )
% 0.28/0.47 => ( X3 = Y4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym
% 0.28/0.47 thf(fact_191_eq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.28/0.47 = ( ^ [X2: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.47 & ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_iff
% 0.28/0.47 thf(fact_192_ord__le__eq__subst,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( ord @ B )
% 0.28/0.47 & ( ord @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ( F @ B2 )
% 0.28/0.47 = C2 )
% 0.28/0.47 => ( ! [X5: A,Y3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_le_eq_subst
% 0.28/0.47 thf(fact_193_ord__eq__le__subst,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( ord @ B )
% 0.28/0.47 & ( ord @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 = ( F @ B2 ) )
% 0.28/0.47 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.28/0.47 => ( ! [X5: B,Y3: B] :
% 0.28/0.47 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_eq_le_subst
% 0.28/0.47 thf(fact_194_order__subst2,axiom,
% 0.28/0.47 ! [A: $tType,C: $tType] :
% 0.28/0.47 ( ( ( order @ C )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.28/0.47 => ( ! [X5: A,Y3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.28/0.47 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_subst2
% 0.28/0.47 thf(fact_195_order__subst1,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( order @ B )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.28/0.47 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.28/0.47 => ( ! [X5: B,Y3: B] :
% 0.28/0.47 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_subst1
% 0.28/0.47 thf(fact_196_le__fun__def,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.28/0.47 = ( ^ [F4: A > B,G5: A > B] :
% 0.28/0.47 ! [X2: A] : ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( G5 @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_fun_def
% 0.28/0.47 thf(fact_197_le__funI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ! [F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.28/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_funI
% 0.28/0.47 thf(fact_198_le__funE,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ! [F: A > B,G: A > B,X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_funE
% 0.28/0.47 thf(fact_199_le__funD,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ! [F: A > B,G: A > B,X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_funD
% 0.28/0.47 thf(fact_200_Image__mono,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A8: set @ A,A3: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R2 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ A3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A8 ) @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Image_mono
% 0.28/0.47 thf(fact_201_Collect__subset,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,P: A > $o] :
% 0.28/0.47 ( ord_less_eq @ ( set @ A )
% 0.28/0.47 @ ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ A3 )
% 0.28/0.47 & ( P @ X2 ) ) )
% 0.28/0.47 @ A3 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_subset
% 0.28/0.47 thf(fact_202_Collect__mono__iff,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.28/0.47 = ( ! [X2: A] :
% 0.28/0.47 ( ( P @ X2 )
% 0.28/0.47 => ( Q @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mono_iff
% 0.28/0.47 thf(fact_203_set__eq__subset,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.28/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % set_eq_subset
% 0.28/0.47 thf(fact_204_subset__trans,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_trans
% 0.28/0.47 thf(fact_205_Collect__mono,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 => ( Q @ X5 ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mono
% 0.28/0.47 thf(fact_206_subset__refl,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.28/0.47
% 0.28/0.47 % subset_refl
% 0.28/0.47 thf(fact_207_subset__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ! [T2: A] :
% 0.28/0.47 ( ( member @ A @ T2 @ A7 )
% 0.28/0.47 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_iff
% 0.28/0.47 thf(fact_208_equalityD2,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ( A3 = B3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % equalityD2
% 0.28/0.47 thf(fact_209_equalityD1,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ( A3 = B3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % equalityD1
% 0.28/0.47 thf(fact_210_subset__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ A7 )
% 0.28/0.47 => ( member @ A @ X2 @ B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_eq
% 0.28/0.47 thf(fact_211_equalityE,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ( A3 = B3 )
% 0.28/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equalityE
% 0.28/0.47 thf(fact_212_subsetD,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.28/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subsetD
% 0.28/0.47 thf(fact_213_in__mono,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.47 => ( ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( member @ A @ X3 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % in_mono
% 0.28/0.47 thf(fact_214_converse__subset__swap,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ A )] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S2 ) )
% 0.28/0.47 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % converse_subset_swap
% 0.28/0.47 thf(fact_215_subrelI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ! [X5: A,Y3: B] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R2 )
% 0.28/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S2 ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % subrelI
% 0.28/0.47 thf(fact_216_bot_Oextremum__uniqueI,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_bot @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.28/0.47 => ( A2
% 0.28/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot.extremum_uniqueI
% 0.28/0.47 thf(fact_217_bot_Oextremum__unique,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_bot @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot.extremum_unique
% 0.28/0.47 thf(fact_218_bot_Oextremum,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_bot @ A )
% 0.28/0.47 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot.extremum
% 0.28/0.47 thf(fact_219_insert__mono,axiom,
% 0.28/0.47 ! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_mono
% 0.28/0.47 thf(fact_220_subset__insert,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.28/0.47 ( ~ ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
% 0.28/0.47 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_insert
% 0.28/0.47 thf(fact_221_subset__insertI,axiom,
% 0.28/0.47 ! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_insertI
% 0.28/0.47 thf(fact_222_subset__insertI2,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_insertI2
% 0.28/0.47 thf(fact_223_subset__singleton__iff,axiom,
% 0.28/0.47 ! [A: $tType,X4: set @ A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 = ( ( X4
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 | ( X4
% 0.28/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_singleton_iff
% 0.28/0.47 thf(fact_224_subset__singletonD,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.47 => ( ( A3
% 0.28/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.47 | ( A3
% 0.28/0.47 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_singletonD
% 0.28/0.47 thf(fact_225_restrict__subsD,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G2 ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G2 ) ) )
% 0.28/0.47 => ( G2
% 0.28/0.47 = ( restrict @ A @ B @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % restrict_subsD
% 0.28/0.47 thf(fact_226_congruent2I_H,axiom,
% 0.28/0.47 ! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 0.28/0.47 ( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R1 )
% 0.28/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z22 ) @ R22 )
% 0.28/0.47 => ( ( F @ Y12 @ Y23 )
% 0.28/0.47 = ( F @ Z1 @ Z22 ) ) ) )
% 0.28/0.47 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruent2I'
% 0.28/0.47 thf(fact_227_congruent2D,axiom,
% 0.28/0.47 ! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z12: A,Y22: B,Z23: B] :
% 0.28/0.47 ( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R1 )
% 0.28/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R22 )
% 0.28/0.47 => ( ( F @ Y1 @ Y22 )
% 0.28/0.47 = ( F @ Z12 @ Z23 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruent2D
% 0.28/0.47 thf(fact_228_proj__iff,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 )
% 0.28/0.47 => ( ( ( equiv_proj @ A @ A @ R2 @ X3 )
% 0.28/0.47 = ( equiv_proj @ A @ A @ R2 @ Y4 ) )
% 0.28/0.47 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % proj_iff
% 0.28/0.47 thf(fact_229_refines__equiv__class__eq2,axiom,
% 0.28/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 0.28/0.47 => ( ( equiv_equiv @ A @ A3 @ R )
% 0.28/0.47 => ( ( equiv_equiv @ A @ A3 @ S )
% 0.28/0.47 => ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 = ( image @ A @ A @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refines_equiv_class_eq2
% 0.28/0.47 thf(fact_230_refines__equiv__class__eq,axiom,
% 0.28/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 0.28/0.47 => ( ( equiv_equiv @ A @ A3 @ R )
% 0.28/0.47 => ( ( equiv_equiv @ A @ A3 @ S )
% 0.28/0.47 => ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 = ( image @ A @ A @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % refines_equiv_class_eq
% 0.28/0.47 thf(fact_231_congruent2I,axiom,
% 0.28/0.47 ! [C: $tType,B: $tType,A: $tType,A12: set @ A,R1: set @ ( product_prod @ A @ A ),A23: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A12 @ R1 )
% 0.28/0.47 => ( ( equiv_equiv @ B @ A23 @ R22 )
% 0.28/0.47 => ( ! [Y3: A,Z3: A,W: B] :
% 0.28/0.47 ( ( member @ B @ W @ A23 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R1 )
% 0.28/0.47 => ( ( F @ Y3 @ W )
% 0.28/0.47 = ( F @ Z3 @ W ) ) ) )
% 0.28/0.47 => ( ! [Y3: B,Z3: B,W: A] :
% 0.28/0.47 ( ( member @ A @ W @ A12 )
% 0.28/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z3 ) @ R22 )
% 0.28/0.47 => ( ( F @ W @ Y3 )
% 0.28/0.47 = ( F @ W @ Z3 ) ) ) )
% 0.28/0.47 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruent2I
% 0.28/0.47 thf(fact_232_congruent2__commuteI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F: A > A > B] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ! [Y3: A,Z3: A] :
% 0.28/0.47 ( ( member @ A @ Y3 @ A3 )
% 0.28/0.47 => ( ( member @ A @ Z3 @ A3 )
% 0.28/0.47 => ( ( F @ Y3 @ Z3 )
% 0.28/0.47 = ( F @ Z3 @ Y3 ) ) ) )
% 0.28/0.47 => ( ! [Y3: A,Z3: A,W: A] :
% 0.28/0.47 ( ( member @ A @ W @ A3 )
% 0.28/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 0.28/0.47 => ( ( F @ W @ Y3 )
% 0.28/0.47 = ( F @ W @ Z3 ) ) ) )
% 0.28/0.47 => ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % congruent2_commuteI
% 0.28/0.47 thf(fact_233_subset__equiv__class,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.47 => ( ( member @ A @ B2 @ A3 )
% 0.28/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_equiv_class
% 0.28/0.47 thf(fact_234_insert__subsetI,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,A3: set @ A,X4: set @ A] :
% 0.28/0.47 ( ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ A3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X4 ) @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % insert_subsetI
% 0.28/0.47 thf(fact_235_subset__emptyI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ~ ( member @ A @ X5 @ A3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_emptyI
% 0.28/0.47 thf(fact_236_univ__commute,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F: A > B,X3: A] :
% 0.28/0.47 ( ( equiv_equiv @ A @ A3 @ R2 )
% 0.28/0.47 => ( ( equiv_congruent @ A @ B @ R2 @ F )
% 0.28/0.47 => ( ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( ( bNF_Greatest_univ @ A @ B @ F @ ( equiv_proj @ A @ A @ R2 @ X3 ) )
% 0.28/0.47 = ( F @ X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % univ_commute
% 0.28/0.47 thf(fact_237_pred__subset__eq,axiom,
% 0.28/0.47 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.28/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ R )
% 0.28/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ S ) )
% 0.28/0.47 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.28/0.47
% 0.28/0.47 % pred_subset_eq
% 0.28/0.47 thf(fact_238_less__eq__set__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ( ord_less_eq @ ( A > $o )
% 0.28/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A7 )
% 0.28/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_eq_set_def
% 0.28/0.47 thf(fact_239_pred__subset__eq2,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > B > $o )
% 0.28/0.47 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
% 0.28/0.47 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) )
% 0.28/0.47 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.28/0.47
% 0.28/0.47 % pred_subset_eq2
% 0.28/0.47 thf(fact_240_univ__def,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( bNF_Greatest_univ @ B @ A )
% 0.28/0.47 = ( ^ [F4: B > A,X8: set @ B] :
% 0.28/0.47 ( F4
% 0.28/0.47 @ ( hilbert_Eps @ B
% 0.28/0.47 @ ^ [X2: B] : ( member @ B @ X2 @ X8 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % univ_def
% 0.28/0.47 thf(fact_241_ssubst__Pair__rhs,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 0.28/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
% 0.28/0.47 => ( ( S3 = S2 )
% 0.28/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ssubst_Pair_rhs
% 0.28/0.47 thf(fact_242_subset__Collect__iff,axiom,
% 0.28/0.47 ! [A: $tType,B3: set @ A,A3: set @ A,P: A > $o] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3
% 0.28/0.47 @ ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ A3 )
% 0.28/0.47 & ( P @ X2 ) ) ) )
% 0.28/0.47 = ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ B3 )
% 0.28/0.47 => ( P @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_Collect_iff
% 0.28/0.47 thf(fact_243_subset__CollectI,axiom,
% 0.28/0.47 ! [A: $tType,B3: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.28/0.47 => ( ! [X5: A] :
% 0.28/0.47 ( ( member @ A @ X5 @ B3 )
% 0.28/0.47 => ( ( Q @ X5 )
% 0.28/0.47 => ( P @ X5 ) ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A )
% 0.28/0.47 @ ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ B3 )
% 0.28/0.47 & ( Q @ X2 ) ) )
% 0.28/0.47 @ ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ A3 )
% 0.28/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_CollectI
% 0.28/0.47 thf(fact_244_Collect__restrict,axiom,
% 0.28/0.47 ! [A: $tType,X4: set @ A,P: A > $o] :
% 0.28/0.47 ( ord_less_eq @ ( set @ A )
% 0.28/0.47 @ ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ X4 )
% 0.28/0.47 & ( P @ X2 ) ) )
% 0.28/0.47 @ X4 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_restrict
% 0.28/0.47 thf(fact_245_prop__restrict,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,Z4: set @ A,X4: set @ A,P: A > $o] :
% 0.28/0.47 ( ( member @ A @ X3 @ Z4 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 0.28/0.47 @ ( collect @ A
% 0.28/0.47 @ ^ [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ X4 )
% 0.28/0.47 & ( P @ X2 ) ) ) )
% 0.28/0.47 => ( P @ X3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prop_restrict
% 0.28/0.47 thf(fact_246_graph__unionI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 0.28/0.47 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.28/0.47 = G_2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_unionI
% 0.28/0.47 thf(fact_247_less__by__empty,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
% 0.28/0.47 ( ( A3
% 0.28/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_by_empty
% 0.28/0.47 thf(fact_248_predicate2I,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.28/0.47 ( ! [X5: A,Y3: B] :
% 0.28/0.47 ( ( P @ X5 @ Y3 )
% 0.28/0.47 => ( Q @ X5 @ Y3 ) )
% 0.28/0.47 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.28/0.47
% 0.28/0.47 % predicate2I
% 0.28/0.47 thf(fact_249_predicate1I,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X5: A] :
% 0.28/0.47 ( ( P @ X5 )
% 0.28/0.47 => ( Q @ X5 ) )
% 0.28/0.47 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.28/0.47
% 0.28/0.47 % predicate1I
% 0.28/0.47 thf(fact_250_graph__union__idemp_I1_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( graph_union @ A @ B @ A3 @ A3 )
% 0.28/0.47 = A3 ) ).
% 0.28/0.47
% 0.28/0.47 % graph_union_idemp(1)
% 0.28/0.47 thf(fact_251_graph__union__idemp_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( graph_union @ A @ B @ A3 @ ( graph_union @ A @ B @ A3 @ B3 ) )
% 0.28/0.47 = ( graph_union @ A @ B @ A3 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_union_idemp(2)
% 0.28/0.47 thf(fact_252_graph__union__idemp_I3_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( graph_union @ A @ B @ A3 @ ( graph_union @ A @ B @ B3 @ A3 ) )
% 0.28/0.47 = ( graph_union @ A @ B @ B3 @ A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_union_idemp(3)
% 0.28/0.47 thf(fact_253_graph__union__preserves__restrict,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 0.28/0.47 ( ( G_1
% 0.28/0.47 = ( restrict @ A @ B @ G_1 ) )
% 0.28/0.47 => ( ( G_2
% 0.28/0.47 = ( restrict @ A @ B @ G_2 ) )
% 0.28/0.47 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 0.28/0.47 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % graph_union_preserves_restrict
% 0.28/0.47 thf(fact_254_rev__predicate2D,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,P: A > B > $o,X3: A,Y4: B,Q: A > B > $o] :
% 0.28/0.47 ( ( P @ X3 @ Y4 )
% 0.28/0.47 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.28/0.47 => ( Q @ X3 @ Y4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_predicate2D
% 0.28/0.47 thf(fact_255_rev__predicate1D,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,X3: A,Q: A > $o] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.28/0.47 => ( Q @ X3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_predicate1D
% 0.28/0.47
% 0.28/0.47 % Type constructors (16)
% 0.28/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.28/0.47 ! [A9: $tType,A10: $tType] :
% 0.28/0.47 ( ( order_bot @ A10 )
% 0.28/0.47 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.28/0.47 ! [A9: $tType,A10: $tType] :
% 0.28/0.47 ( ( preorder @ A10 )
% 0.28/0.47 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/0.47 ! [A9: $tType,A10: $tType] :
% 0.28/0.47 ( ( order @ A10 )
% 0.28/0.47 => ( order @ ( A9 > A10 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.28/0.47 ! [A9: $tType,A10: $tType] :
% 0.28/0.47 ( ( ord @ A10 )
% 0.28/0.47 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.28/0.47 ! [A9: $tType,A10: $tType] :
% 0.28/0.47 ( ( bot @ A10 )
% 0.28/0.47 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
% 0.28/0.47 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.28/0.47 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.28/0.47 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 0.28/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
% 0.28/0.47 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
% 0.28/0.47 order_bot @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.28/0.47 preorder @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.28/0.47 linorder @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 0.28/0.47 order @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
% 0.28/0.47 ord @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
% 0.28/0.47 bot @ $o ).
% 0.28/0.47
% 0.28/0.47 % Helper facts (3)
% 0.28/0.47 thf(help_If_3_1_T,axiom,
% 0.28/0.47 ! [P: $o] :
% 0.28/0.47 ( ( P = $true )
% 0.28/0.47 | ( P = $false ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(help_If_2_1_T,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,Y4: A] :
% 0.28/0.47 ( ( if @ A @ $false @ X3 @ Y4 )
% 0.28/0.47 = Y4 ) ).
% 0.28/0.47
% 0.28/0.47 thf(help_If_1_1_T,axiom,
% 0.28/0.47 ! [A: $tType,X3: A,Y4: A] :
% 0.28/0.47 ( ( if @ A @ $true @ X3 @ Y4 )
% 0.28/0.47 = X3 ) ).
% 0.28/0.47
% 0.28/0.47 % Conjectures (1)
% 0.28/0.47 thf(conj_0,conjecture,
% 0.28/0.47 ( ( hilbert_Eps @ b
% 0.28/0.47 @ ^ [Y: b] : ( member @ b @ Y @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ x @ ( bot_bot @ ( set @ b ) ) ) ) ) )
% 0.28/0.47 = ( hilbert_Eps @ b
% 0.28/0.47 @ ^ [Y: b] : ( member @ b @ Y @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ y @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 2.70/1.55 ---- Embedded in THF ---
% 2.70/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.70/1.55 %%% Generated on Tue Feb 24 08:28:58 EST 2026
% 2.70/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.70/1.55 %%% Logic specification used:
% 2.70/1.55 %%% thf(spec, logic, $$dhol).
% 2.70/1.55
% 2.70/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.27O21xWGGm/DTF2DTF16637.p
% 2.70/1.55 thf(ty_t_LabeledGraphSemantics_OStandard__Constant, type, standard_Constant: ($tType > $tType)).
% 2.70/1.55 thf(ty_t_LabeledGraphs_Olabeled__graph, type, labeled_graph: ($tType > ($tType > $tType))).
% 2.70/1.55 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.70/1.55 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.70/1.55 thf(ty_tf_b, type, b: $tType).
% 2.70/1.55 thf(ty_tf_a, type, a: $tType).
% 2.70/1.55 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.70/1.55 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv, type, bNF_Greatest_univ: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 2.70/1.55 thf(sy_c_Equiv__Relations_Ocongruent, type, equiv_congruent: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.70/1.55 thf(sy_c_Equiv__Relations_Ocongruent2, type, equiv_congruent2: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > ((A > (B > C)) > $o)))))).
% 2.70/1.55 thf(sy_c_Equiv__Relations_Oequiv, type, equiv_equiv: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.70/1.55 thf(sy_c_Equiv__Relations_Oproj, type, equiv_proj: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > (set @ A)))))).
% 2.70/1.55 thf(sy_c_Hilbert__Choice_OEps, type, hilbert_Eps: (!> [A:$tType]: (((A > $o) > A)))).
% 2.70/1.55 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.70/1.55 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt, type, standard_S_Idt: (!> [V:$tType]: ((standard_Constant @ V)))).
% 2.70/1.55 thf(sy_c_LabeledGraphSemantics_OgetRel, type, getRel: (!> [B:$tType,A:$tType]: ((B > (((labeled_graph @ B) @ A) > (set @ ((product_prod @ A) @ A))))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Ograph__homomorphism, type, graph_homomorphism: (!> [A:$tType,B:$tType,C:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ C) > ((set @ ((product_prod @ B) @ C)) > $o)))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Ograph__union, type, graph_union: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > (((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B)))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG, type, labeled_LG: (!> [L:$tType,V:$tType]: (((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > ((labeled_graph @ L) @ V)))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph, type, labele1974067554_graph: (!> [L:$tType,V:$tType,A:$tType]: ((((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)) > (((labeled_graph @ L) @ V) > A))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges, type, labeled_edges: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ ((product_prod @ L) @ ((product_prod @ V) @ V))))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices, type, labeled_vertices: (!> [L:$tType,V:$tType]: ((((labeled_graph @ L) @ V) > (set @ V))))).
% 2.70/1.55 thf(sy_c_LabeledGraphs_Orestrict, type, restrict: (!> [A:$tType,B:$tType]: ((((labeled_graph @ A) @ B) > ((labeled_graph @ A) @ B))))).
% 2.70/1.55 thf(sy_c_Order__Relation_Olinear__order__on, type, order_1409979114der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.70/1.55 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.70/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.70/1.55 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.70/1.55 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.70/1.55 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 2.70/1.55 thf(sy_c_Relation_Oconverse, type, converse: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ B) @ A)))))).
% 2.70/1.55 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.70/1.55 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.70/1.55 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.70/1.55 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.70/1.55 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.70/1.55 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.70/1.55 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.70/1.55 thf(sy_v_G, type, g: ((labeled_graph @ (standard_Constant @ a)) @ b)).
% 2.70/1.55 thf(sy_v_P____, type, p: (b > (b > $o))).
% 2.70/1.55 thf(sy_v_f____, type, f: (b > b)).
% 2.70/1.55 thf(sy_v_x____, type, x: b).
% 2.70/1.55 thf(sy_v_y____, type, y: b).
% 2.70/1.55 thf(fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062, axiom, (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ x) @ (bot_bot @ (set @ b)))) = ((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ y) @ (bot_bot @ (set @ b)))))).
% 2.70/1.55 thf(fact_1_assms_I1_J, axiom, (g = (((restrict @ (standard_Constant @ a)) @ b) @ g))).
% 2.70/1.55 thf(fact_2_P__eq, axiom, (! [X:b]: (((((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X) @ (bot_bot @ (set @ b)))) != (bot_bot @ (set @ b))) => (((hilbert_Eps @ b) @ (p @ ((hilbert_Eps @ b) @ (p @ X)))) = ((hilbert_Eps @ b) @ (p @ X))))))).
% 2.70/1.55 thf(fact_3_P, axiom, (p = (^ [X2:b,Y:b]: ((((member @ b) @ Y) @ ((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X2) @ (bot_bot @ (set @ b))))))))).
% 2.70/1.55 thf(fact_4__092_060open_062_Ix_M_Ay_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062, axiom, (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ x) @ y)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g))).
% 2.70/1.55 thf(fact_5__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062, axiom, (f = (^ [X2:b]: (((((if @ b) @ (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X2) @ (bot_bot @ (set @ b)))) = (bot_bot @ (set @ b)))) @ X2) @ ((hilbert_Eps @ b) @ (p @ X2))))))).
% 2.70/1.55 thf(fact_6_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X2:A]: ((X2 = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_7_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_8_tuple__disj, axiom, (! [A:$tType,X3:A,Z2:A]: ((((collect @ A) @ (^ [Y:A]: (((Y = X3) | (Y = Z2))))) = (((insert @ A) @ X3) @ (((insert @ A) @ Z2) @ (bot_bot @ (set @ A)))))))).
% 2.70/1.55 thf(fact_9_Image__empty1, axiom, (! [B:$tType,A:$tType,X4:(set @ B)]: ((((((image @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) @ X4) = (bot_bot @ (set @ A)))))).
% 2.70/1.55 thf(fact_10_Image__empty2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((image @ B) @ A) @ R) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.70/1.55 thf(fact_11_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_12_some__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X5:A]: (((P @ X5) => (X5 = A2)))) => (((hilbert_Eps @ A) @ P) = A2)))))).
% 2.70/1.55 thf(fact_13_some__eq__trivial, axiom, (! [A:$tType,X3:A]: ((((hilbert_Eps @ A) @ (^ [Y:A]: ((Y = X3)))) = X3)))).
% 2.70/1.55 thf(fact_14_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.70/1.55 thf(fact_15_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.70/1.55 thf(fact_16_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_17_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.70/1.55 thf(fact_18_insert__absorb2, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ X3) @ A3)) = (((insert @ A) @ X3) @ A3))))).
% 2.70/1.55 thf(fact_19_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) = ((A2 = B2) | (((member @ A) @ A2) @ A3)))))).
% 2.70/1.55 thf(fact_20_insertCI, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B3))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))).
% 2.70/1.55 thf(fact_21_some__sym__eq__trivial, axiom, (! [A:$tType,X3:A]: ((((hilbert_Eps @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ X3)) = X3)))).
% 2.70/1.55 thf(fact_22_ImageI, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B)),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => ((((member @ A) @ A2) @ A3) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R2) @ A3))))))).
% 2.70/1.55 thf(fact_23_f, axiom, (f = (^ [X2:b]: (((((if @ b) @ (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X2) @ (bot_bot @ (set @ b)))) = (bot_bot @ (set @ b)))) @ X2) @ ((hilbert_Eps @ b) @ (p @ X2))))))).
% 2.70/1.55 thf(fact_24_Image__singleton__iff, axiom, (! [A:$tType,B:$tType,B2:A,R2:(set @ ((product_prod @ B) @ A)),A2:B]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R2) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A2) @ B2)) @ R2))))).
% 2.70/1.55 thf(fact_25_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R))) = (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ S)))) = (R = S))))).
% 2.70/1.55 thf(fact_26_rev__ImageI, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B2:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ A3) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R2) @ A3))))))).
% 2.70/1.55 thf(fact_27_Image__iff, axiom, (! [A:$tType,B:$tType,B2:A,R2:(set @ ((product_prod @ B) @ A)),A3:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R2) @ A3)) = (? [X2:B]: (((((member @ B) @ X2) @ A3) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X2) @ B2)) @ R2)))))))).
% 2.70/1.55 thf(fact_28_ImageE, axiom, (! [A:$tType,B:$tType,B2:A,R2:(set @ ((product_prod @ B) @ A)),A3:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R2) @ A3)) => (~ ((! [X5:B]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X5) @ B2)) @ R2) => (~ ((((member @ B) @ X5) @ A3)))))))))))).
% 2.70/1.55 thf(fact_29_Image__singleton, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ B) @ A)),A2:B]: ((((((image @ B) @ A) @ R2) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B)))) = ((collect @ A) @ (^ [B4:A]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A2) @ B4)) @ R2)))))))).
% 2.70/1.55 thf(fact_30_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_31_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_32_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.70/1.55 thf(fact_33_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.70/1.55 thf(fact_34_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (? [B5:(set @ A)]: (((A3 = (((insert @ A) @ A2) @ B5)) & (~ ((((member @ A) @ A2) @ B5)))))))))).
% 2.70/1.55 thf(fact_35_insert__commute, axiom, (! [A:$tType,X3:A,Y4:A,A3:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ Y4) @ A3)) = (((insert @ A) @ Y4) @ (((insert @ A) @ X3) @ A3)))))).
% 2.70/1.55 thf(fact_36_insert__eq__iff, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A,B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((~ ((((member @ A) @ B2) @ B3))) => (((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ B3)) = (((A2 = B2) => (A3 = B3)) & ((A2 != B2) => (? [C3:(set @ A)]: (((A3 = (((insert @ A) @ B2) @ C3)) & ((~ ((((member @ A) @ B2) @ C3))) & ((B3 = (((insert @ A) @ A2) @ C3)) & (~ ((((member @ A) @ A2) @ C3)))))))))))))))).
% 2.70/1.55 thf(fact_37_insert__absorb, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ A3) = A3))))).
% 2.70/1.55 thf(fact_38_insert__ident, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((~ ((((member @ A) @ X3) @ B3))) => (((((insert @ A) @ X3) @ A3) = (((insert @ A) @ X3) @ B3)) = (A3 = B3))))))).
% 2.70/1.55 thf(fact_39_Set_Oset__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (~ ((! [B5:(set @ A)]: (((A3 = (((insert @ A) @ X3) @ B5)) => (((member @ A) @ X3) @ B5)))))))))).
% 2.70/1.55 thf(fact_40_insertI2, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: (((((member @ A) @ A2) @ B3) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))).
% 2.70/1.55 thf(fact_41_insertI1, axiom, (! [A:$tType,A2:A,B3:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B3))))).
% 2.70/1.55 thf(fact_42_insertE, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) => ((A2 != B2) => (((member @ A) @ A2) @ A3)))))).
% 2.70/1.55 thf(fact_43_some__eq__imp, axiom, (! [A:$tType,P:(A > $o),A2:A,B2:A]: (((((hilbert_Eps @ A) @ P) = A2) => ((P @ B2) => (P @ A2)))))).
% 2.70/1.55 thf(fact_44_tfl__some, axiom, (! [A:$tType,P2:(A > $o),X6:A]: (((P2 @ X6) => (P2 @ ((hilbert_Eps @ A) @ P2)))))).
% 2.70/1.55 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.70/1.55 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 2.70/1.55 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.70/1.55 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 2.70/1.55 thf(fact_49_Eps__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((hilbert_Eps @ A) @ P) = ((hilbert_Eps @ A) @ Q)))))).
% 2.70/1.55 thf(fact_50_someI, axiom, (! [A:$tType,P:(A > $o),X3:A]: (((P @ X3) => (P @ ((hilbert_Eps @ A) @ P)))))).
% 2.70/1.55 thf(fact_51_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X2:A]: ($false))))))).
% 2.70/1.55 thf(fact_52_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U:A]: (((U != A2) => (P @ U))))))))).
% 2.70/1.55 thf(fact_53_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A,B6:(set @ A)]: (((collect @ A) @ (^ [X2:A]: (((X2 = A4) | (((member @ A) @ X2) @ B6))))))))))).
% 2.70/1.55 thf(fact_54_some1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X6:A]: (((P @ X6) & (! [Y3:A]: (((P @ Y3) => (Y3 = X6))))))) => ((P @ A2) => (((hilbert_Eps @ A) @ P) = A2)))))).
% 2.70/1.55 thf(fact_55_some__eq__ex, axiom, (! [A:$tType,P:(A > $o)]: (((P @ ((hilbert_Eps @ A) @ P)) = (? [X7:A]: ((P @ X7))))))).
% 2.70/1.55 thf(fact_56_someI2__bex, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o),Q:(A > $o)]: (((? [X6:A]: (((((member @ A) @ X6) @ A3) & (P @ X6)))) => ((! [X5:A]: ((((((member @ A) @ X5) @ A3) & (P @ X5)) => (Q @ X5)))) => (Q @ ((hilbert_Eps @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A3) & (P @ X2))))))))))).
% 2.70/1.55 thf(fact_57_someI2__ex, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => ((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (Q @ ((hilbert_Eps @ A) @ P))))))).
% 2.70/1.55 thf(fact_58_someI__ex, axiom, (! [A:$tType,P:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => (P @ ((hilbert_Eps @ A) @ P)))))).
% 2.70/1.55 thf(fact_59_someI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (Q @ ((hilbert_Eps @ A) @ P))))))).
% 2.70/1.55 thf(fact_60_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 2.70/1.55 thf(fact_61_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))).
% 2.70/1.55 thf(fact_62_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C2:A,D:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C2) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A2 = C2) & (B2 = D)) | ((A2 = D) & (B2 = C2))))))).
% 2.70/1.55 thf(fact_63_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 2.70/1.55 thf(fact_64_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 2.70/1.55 thf(fact_65_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X2:A]: (((A2 = X2) & (P @ X2))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X2:A]: (((A2 = X2) & (P @ X2))))) = (bot_bot @ (set @ A)))))))).
% 2.70/1.55 thf(fact_66_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X2:A]: (((X2 = A2) & (P @ X2))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X2:A]: (((X2 = A2) & (P @ X2))))) = (bot_bot @ (set @ A)))))))).
% 2.70/1.55 thf(fact_67_some__in__eq, axiom, (! [A:$tType,A3:(set @ A)]: (((((member @ A) @ ((hilbert_Eps @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3))))) @ A3) = (A3 != (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_68_congr_I1_J, axiom, (! [X:b,Y5:b,L2:(standard_Constant @ a)]: (((((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X) @ Y5)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ L2) @ g)) => ((((equiv_congruent @ b) @ (set @ b)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (^ [V2:b]: (((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ L2) @ g)) @ (((insert @ b) @ V2) @ (bot_bot @ (set @ b))))))))))).
% 2.70/1.55 thf(fact_69_vert__P, axiom, (! [X3:b]: (((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X3) @ ((hilbert_Eps @ b) @ (p @ X3)))) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)))))).
% 2.70/1.55 thf(fact_70__092_060open_062_IS__Idt_M_Ax_M_Ay_J_A_092_060in_062_Aedges_AG_092_060close_062, axiom, (((member @ ((product_prod @ (standard_Constant @ a)) @ ((product_prod @ b) @ b))) @ ((((product_Pair @ (standard_Constant @ a)) @ ((product_prod @ b) @ b)) @ (standard_S_Idt @ a)) @ ((((product_Pair @ b) @ b) @ x) @ y))) @ (((labeled_edges @ (standard_Constant @ a)) @ b) @ g))).
% 2.70/1.55 thf(fact_71_restrict__idemp, axiom, (! [B:$tType,A:$tType,X3:((labeled_graph @ A) @ B)]: (((((restrict @ A) @ B) @ (((restrict @ A) @ B) @ X3)) = (((restrict @ A) @ B) @ X3))))).
% 2.70/1.55 thf(fact_72_r2, axiom, (! [X3:b]: (((((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ X3) @ (bot_bot @ (set @ b)))) = (bot_bot @ (set @ b))) = (~ ((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)))))))).
% 2.70/1.55 thf(fact_73_r1, axiom, (! [X3:b]: (((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) = ((p @ X3) @ X3))))).
% 2.70/1.55 thf(fact_74_congr_I2_J, axiom, (! [X:b,Y5:b,L2:(standard_Constant @ a)]: (((((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X) @ Y5)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ L2) @ g)) => ((((equiv_congruent @ b) @ (set @ b)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (^ [V2:b]: (((((image @ b) @ b) @ (((converse @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ L2) @ g))) @ (((insert @ b) @ V2) @ (bot_bot @ (set @ b))))))))))).
% 2.70/1.55 thf(fact_75_the__elem__eq, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3)))).
% 2.70/1.55 thf(fact_76_bot__apply, axiom, (! [C:$tType,D2:$tType]: (((bot @ C) => ((bot_bot @ (D2 > C)) = (^ [X2:D2]: ((bot_bot @ C)))))))).
% 2.70/1.55 thf(fact_77_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.70/1.55 thf(fact_78_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A5:A,B7:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B7)) = ((A2 = A5) & (B2 = B7)))))).
% 2.70/1.55 thf(fact_79_converse__inject, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ B) @ A)),S2:(set @ ((product_prod @ B) @ A))]: ((((((converse @ B) @ A) @ R2) = (((converse @ B) @ A) @ S2)) = (R2 = S2))))).
% 2.70/1.55 thf(fact_80_converse__converse, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B))]: (((((converse @ B) @ A) @ (((converse @ A) @ B) @ R2)) = R2)))).
% 2.70/1.55 thf(fact_81_converse__iff, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ (((converse @ B) @ A) @ R2)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A2)) @ R2))))).
% 2.70/1.55 thf(fact_82_vertices__restrict, axiom, (! [A:$tType,B:$tType,G2:((labeled_graph @ B) @ A)]: (((((labeled_vertices @ B) @ A) @ (((restrict @ B) @ A) @ G2)) = (((labeled_vertices @ B) @ A) @ G2))))).
% 2.70/1.55 thf(fact_83_converse__empty, axiom, (! [B:$tType,A:$tType]: (((((converse @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.70/1.55 thf(fact_84__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062, axiom, (! [X3:b]: (((((member @ b) @ X3) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) => (((member @ ((product_prod @ b) @ b)) @ ((((product_Pair @ b) @ b) @ X3) @ X3)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)))))).
% 2.70/1.55 thf(fact_85_labeled__graph_Oexpand, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V),Labeled_graph2:((labeled_graph @ L) @ V)]: (((((((labeled_edges @ L) @ V) @ Labeled_graph) = (((labeled_edges @ L) @ V) @ Labeled_graph2)) & ((((labeled_vertices @ L) @ V) @ Labeled_graph) = (((labeled_vertices @ L) @ V) @ Labeled_graph2))) => (Labeled_graph = Labeled_graph2))))).
% 2.70/1.55 thf(fact_86_converse_Oinducts, axiom, (! [B:$tType,A:$tType,X1:B,X22:A,R2:(set @ ((product_prod @ A) @ B)),P:(B > (A > $o))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X1) @ X22)) @ (((converse @ A) @ B) @ R2)) => ((! [A6:A,B8:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B8)) @ R2) => ((P @ B8) @ A6)))) => ((P @ X1) @ X22)))))).
% 2.70/1.55 thf(fact_87_converse_Ointros, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A2)) @ (((converse @ A) @ B) @ R2)))))).
% 2.70/1.55 thf(fact_88_converse_Osimps, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R2)) = (? [A4:A,B4:B]: (((A1 = B4) & ((A22 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R2))))))))).
% 2.70/1.55 thf(fact_89_converse_Ocases, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R2)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A22) @ A1)) @ R2))))).
% 2.70/1.55 thf(fact_90_converseE, axiom, (! [A:$tType,B:$tType,Yx:((product_prod @ A) @ B),R2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ Yx) @ (((converse @ B) @ A) @ R2)) => (~ ((! [X5:B,Y3:A]: (((Yx = ((((product_Pair @ A) @ B) @ Y3) @ X5)) => (~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X5) @ Y3)) @ R2)))))))))))).
% 2.70/1.55 thf(fact_91_converseD, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ (((converse @ B) @ A) @ R2)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A2)) @ R2))))).
% 2.70/1.55 thf(fact_92_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.70/1.55 thf(fact_93_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.70/1.55 thf(fact_94_bot__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((bot_bot @ (A > (B > $o))) = (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))))))))).
% 2.70/1.55 thf(fact_95_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B8:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B8)))) => (P @ Prod))))).
% 2.70/1.55 thf(fact_96_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y4:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B8:B]: ((Y4 != ((((product_Pair @ A) @ B) @ A6) @ B8))))))))).
% 2.70/1.55 thf(fact_97_prod__induct7, axiom, (! [G3:$tType,F2:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))))]: (((! [A6:A,B8:B,C4:C,D3:D2,E2:E,F3:F2,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G3)) @ E2) @ ((((product_Pair @ F2) @ G3) @ F3) @ G4))))))))) => (P @ X3))))).
% 2.70/1.55 thf(fact_98_prod__induct6, axiom, (! [F2:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))))]: (((! [A6:A,B8:B,C4:C,D3:D2,E2:E,F3:F2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F2)) @ D3) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P @ X3))))).
% 2.70/1.55 thf(fact_99_prod__induct5, axiom, (! [E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: (((! [A6:A,B8:B,C4:C,D3:D2,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C4) @ ((((product_Pair @ D2) @ E) @ D3) @ E2))))))) => (P @ X3))))).
% 2.70/1.55 thf(fact_100_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A6:A,B8:B,C4:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B8) @ ((((product_Pair @ C) @ D2) @ C4) @ D3)))))) => (P @ X3))))).
% 2.70/1.55 thf(fact_101_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A6:A,B8:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B8) @ C4))))) => (P @ X3))))).
% 2.70/1.55 thf(fact_102_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F2:$tType,G3:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))))]: ((~ ((! [A6:A,B8:B,C4:C,D3:D2,E2:E,F3:F2,G4:G3]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G3)) @ E2) @ ((((product_Pair @ F2) @ G3) @ F3) @ G4)))))))))))))).
% 2.70/1.55 thf(fact_103_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F2:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A6:A,B8:B,C4:C,D3:D2,E2:E,F3:F2]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F2)) @ D3) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 2.70/1.55 thf(fact_104_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: ((~ ((! [A6:A,B8:B,C4:C,D3:D2,E2:E]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C4) @ ((((product_Pair @ D2) @ E) @ D3) @ E2)))))))))))).
% 2.70/1.55 thf(fact_105_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A6:A,B8:B,C4:C,D3:D2]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B8) @ ((((product_Pair @ C) @ D2) @ C4) @ D3))))))))))).
% 2.70/1.55 thf(fact_106_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A6:A,B8:B,C4:C]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B8) @ C4)))))))))).
% 2.70/1.55 thf(fact_107_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A5:A,B7:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B7)) => (~ (((A2 = A5) => (B2 != B7)))))))).
% 2.70/1.55 thf(fact_108_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P3:((product_prod @ A) @ B)]: (((! [A6:A,B8:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B8)))) => (P @ P3))))).
% 2.70/1.55 thf(fact_109_surj__pair, axiom, (! [A:$tType,B:$tType,P3:((product_prod @ A) @ B)]: ((? [X5:A,Y3:B]: ((P3 = ((((product_Pair @ A) @ B) @ X5) @ Y3))))))).
% 2.70/1.55 thf(fact_110_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 2.70/1.55 thf(fact_111_local_Orefl, axiom, (((refl_on @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g))).
% 2.70/1.55 thf(fact_112_equiv, axiom, (((equiv_equiv @ b) @ (((labeled_vertices @ (standard_Constant @ a)) @ b) @ g)) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g))).
% 2.70/1.55 thf(fact_113_getRel__dom_I1_J, axiom, (! [B:$tType,A:$tType,G2:((labeled_graph @ A) @ B),A2:B,B2:B,L3:A]: (((G2 = (((restrict @ A) @ B) @ G2)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((getRel @ A) @ B) @ L3) @ G2)) => (((member @ B) @ A2) @ (((labeled_vertices @ A) @ B) @ G2))))))).
% 2.70/1.55 thf(fact_114_getRel__dom_I2_J, axiom, (! [B:$tType,A:$tType,G2:((labeled_graph @ A) @ B),A2:B,B2:B,L3:A]: (((G2 = (((restrict @ A) @ B) @ G2)) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A2) @ B2)) @ ((((getRel @ A) @ B) @ L3) @ G2)) => (((member @ B) @ B2) @ (((labeled_vertices @ A) @ B) @ G2))))))).
% 2.70/1.55 thf(fact_115_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.70/1.55 thf(fact_116_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((A7 = (((insert @ A) @ ((the_elem @ A) @ A7)) @ (bot_bot @ (set @ A)))))))))).
% 2.70/1.55 thf(fact_117_refl__on__converse, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A3) @ (((converse @ A) @ A) @ R2)) = (((refl_on @ A) @ A3) @ R2))))).
% 2.70/1.55 thf(fact_118_is__singletonI, axiom, (! [A:$tType,X3:A]: (((is_singleton @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_119_bot2E, axiom, (! [A:$tType,B:$tType,X3:A,Y4:B]: ((~ ((((bot_bot @ (A > (B > $o))) @ X3) @ Y4)))))).
% 2.70/1.55 thf(fact_120_refl__onD, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R2)))))).
% 2.70/1.55 thf(fact_121_refl__onD1, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y4:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2) => (((member @ A) @ X3) @ A3)))))).
% 2.70/1.55 thf(fact_122_refl__onD2, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y4:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2) => (((member @ A) @ Y4) @ A3)))))).
% 2.70/1.55 thf(fact_123_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.70/1.55 thf(fact_124_is__singletonI_H, axiom, (! [A:$tType,A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y3:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y3) @ A3) => (X5 = Y3))))) => ((is_singleton @ A) @ A3)))))).
% 2.70/1.55 thf(fact_125_refl__on__singleton, axiom, (! [A:$tType,X3:A]: ((((refl_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.70/1.55 thf(fact_126_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((? [X2:A]: ((A7 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 2.70/1.55 thf(fact_127_is__singletonE, axiom, (! [A:$tType,A3:(set @ A)]: ((((is_singleton @ A) @ A3) => (~ ((! [X5:A]: ((A3 != (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))))).
% 2.70/1.55 thf(fact_128_equiv__class__eq__iff, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y4:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2) = ((((((image @ A) @ A) @ R2) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R2) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) & ((((member @ A) @ X3) @ A3) & (((member @ A) @ Y4) @ A3)))))))).
% 2.70/1.55 thf(fact_129_eq__equiv__class__iff, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y4:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((member @ A) @ X3) @ A3) => ((((member @ A) @ Y4) @ A3) => ((((((image @ A) @ A) @ R2) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R2) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2)))))))).
% 2.70/1.55 thf(fact_130_equiv__class__eq, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => (((((image @ A) @ A) @ R2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))))))))).
% 2.70/1.55 thf(fact_131_eq__equiv__class, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A,A3:(set @ A)]: (((((((image @ A) @ A) @ R2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) => ((((equiv_equiv @ A) @ A3) @ R2) => ((((member @ A) @ B2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2))))))).
% 2.70/1.55 thf(fact_132_congruentI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),F:(A > B)]: (((! [Y3:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R2) => ((F @ Y3) = (F @ Z3))))) => ((((equiv_congruent @ A) @ B) @ R2) @ F))))).
% 2.70/1.55 thf(fact_133_congruentD, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),F:(A > B),Y4:A,Z2:A]: ((((((equiv_congruent @ A) @ B) @ R2) @ F) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ R2) => ((F @ Y4) = (F @ Z2))))))).
% 2.70/1.55 thf(fact_134_equiv__class__self, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((member @ A) @ A2) @ A3) => (((member @ A) @ A2) @ ((((image @ A) @ A) @ R2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))))).
% 2.70/1.55 thf(fact_135_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.70/1.55 thf(fact_136_graph__single, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,C2:B]: ((((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A2) @ ((((product_Pair @ B) @ B) @ B2) @ C2))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B2) @ (((insert @ B) @ C2) @ (bot_bot @ (set @ B))))) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (((insert @ ((product_prod @ A) @ ((product_prod @ B) @ B))) @ ((((product_Pair @ A) @ ((product_prod @ B) @ B)) @ A2) @ ((((product_Pair @ B) @ B) @ B2) @ C2))) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))))) @ (((insert @ B) @ B2) @ (((insert @ B) @ C2) @ (bot_bot @ (set @ B)))))))))).
% 2.70/1.55 thf(fact_137_refl__on__domain, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ A) @ A2) @ A3) & (((member @ A) @ B2) @ A3))))))).
% 2.70/1.55 thf(fact_138_total__on__singleton, axiom, (! [A:$tType,X3:A]: ((((total_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.70/1.55 thf(fact_139_labeled__graph_Oinject, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V),Y1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),Y22:(set @ V)]: (((((((labeled_LG @ L) @ V) @ X1) @ X22) = ((((labeled_LG @ L) @ V) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.70/1.55 thf(fact_140_total__on__converse, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ A3) @ (((converse @ A) @ A) @ R2)) = (((total_on @ A) @ A3) @ R2))))).
% 2.70/1.55 thf(fact_141_labeled__graph_Ocollapse, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V)]: ((((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)) = Labeled_graph)))).
% 2.70/1.55 thf(fact_142_graph__empty__e, axiom, (! [A:$tType,B:$tType,V3:(set @ B)]: ((((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V3) = (((restrict @ A) @ B) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ V3)))))).
% 2.70/1.55 thf(fact_143_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A7:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (! [Y:A]: (((((member @ A) @ Y) @ A7) => ((X2 != Y) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ R3))))))))))))))).
% 2.70/1.55 thf(fact_144_total__onI, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((! [X5:A,Y3:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y3) @ A3) => ((X5 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y3)) @ R2) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X5)) @ R2))))))) => (((total_on @ A) @ A3) @ R2))))).
% 2.70/1.55 thf(fact_145_total__on__empty, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R2)))).
% 2.70/1.55 thf(fact_146_labeled__graph_Osel_I2_J, axiom, (! [L:$tType,V:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((labeled_vertices @ L) @ V) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = X22)))).
% 2.70/1.55 thf(fact_147_labeled__graph_Oexhaust, axiom, (! [L:$tType,V:$tType,Y4:((labeled_graph @ L) @ V)]: ((~ ((! [X12:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X23:(set @ V)]: ((Y4 != ((((labeled_LG @ L) @ V) @ X12) @ X23))))))))).
% 2.70/1.55 thf(fact_148_labeled__graph_Oinduct, axiom, (! [V:$tType,L:$tType,P:(((labeled_graph @ L) @ V) > $o),Labeled_graph:((labeled_graph @ L) @ V)]: (((! [X1a:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X2a:(set @ V)]: ((P @ ((((labeled_LG @ L) @ V) @ X1a) @ X2a)))) => (P @ Labeled_graph))))).
% 2.70/1.55 thf(fact_149_labeled__graph_Osel_I1_J, axiom, (! [V:$tType,L:$tType,X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((labeled_edges @ L) @ V) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = X1)))).
% 2.70/1.55 thf(fact_150_labeled__graph_Oexhaust__sel, axiom, (! [V:$tType,L:$tType,Labeled_graph:((labeled_graph @ L) @ V)]: ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))).
% 2.70/1.55 thf(fact_151_linear__order__on__singleton, axiom, (! [A:$tType,X3:A]: ((((order_1409979114der_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.70/1.55 thf(fact_152_labeled__graph_Osplit__sel, axiom, (! [A:$tType,V:$tType,L:$tType,P:(A > $o),F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P @ (((((labele1974067554_graph @ L) @ V) @ A) @ F) @ Labeled_graph)) = ((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) => (P @ ((F @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))).
% 2.70/1.55 thf(fact_153_labeled__graph_Osplit__sel__asm, axiom, (! [A:$tType,V:$tType,L:$tType,P:(A > $o),F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((P @ (((((labele1974067554_graph @ L) @ V) @ A) @ F) @ Labeled_graph)) = (~ (((Labeled_graph = ((((labeled_LG @ L) @ V) @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph))) & (~ ((P @ ((F @ (((labeled_edges @ L) @ V) @ Labeled_graph)) @ (((labeled_vertices @ L) @ V) @ Labeled_graph)))))))))))).
% 2.70/1.55 thf(fact_154_graph__homomorphism__empty, axiom, (! [B:$tType,C:$tType,A:$tType,G2:((labeled_graph @ A) @ C),F:(set @ ((product_prod @ B) @ C))]: ((((((((graph_homomorphism @ A) @ B) @ C) @ ((((labeled_LG @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B))))) @ (bot_bot @ (set @ B)))) @ G2) @ F) = ((F = (bot_bot @ (set @ ((product_prod @ B) @ C)))) & (G2 = (((restrict @ A) @ C) @ G2))))))).
% 2.70/1.55 thf(fact_155_linear__order__on__converse, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((order_1409979114der_on @ A) @ A3) @ (((converse @ A) @ A) @ R2)) = (((order_1409979114der_on @ A) @ A3) @ R2))))).
% 2.70/1.55 thf(fact_156_labeled__graph_Ocase__distrib, axiom, (! [A:$tType,B:$tType,V:$tType,L:$tType,H:(A > B),F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),Labeled_graph:((labeled_graph @ L) @ V)]: (((H @ (((((labele1974067554_graph @ L) @ V) @ A) @ F) @ Labeled_graph)) = (((((labele1974067554_graph @ L) @ V) @ B) @ (^ [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X24:(set @ V)]: ((H @ ((F @ X13) @ X24))))) @ Labeled_graph))))).
% 2.70/1.55 thf(fact_157_labeled__graph_Ocase, axiom, (! [L:$tType,A:$tType,V:$tType,F:((set @ ((product_prod @ L) @ ((product_prod @ V) @ V))) > ((set @ V) > A)),X1:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X22:(set @ V)]: (((((((labele1974067554_graph @ L) @ V) @ A) @ F) @ ((((labeled_LG @ L) @ V) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 2.70/1.55 thf(fact_158_edges__def, axiom, (! [V:$tType,L:$tType]: ((((labeled_edges @ L) @ V) = ((((labele1974067554_graph @ L) @ V) @ (set @ ((product_prod @ L) @ ((product_prod @ V) @ V)))) @ (^ [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X24:(set @ V)]: (X13))))))).
% 2.70/1.55 thf(fact_159_vertices__def, axiom, (! [V:$tType,L:$tType]: ((((labeled_vertices @ L) @ V) = ((((labele1974067554_graph @ L) @ V) @ (set @ V)) @ (^ [X13:(set @ ((product_prod @ L) @ ((product_prod @ V) @ V))),X24:(set @ V)]: (X24))))))).
% 2.70/1.55 thf(fact_160_lnear__order__on__empty, axiom, (! [A:$tType]: ((((order_1409979114der_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.70/1.55 thf(fact_161_labeled__graph_Ocase__eq__if, axiom, (! [A:$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)))))))).
% 2.70/1.55 thf(fact_162_congruent2__implies__congruent, axiom, (! [B:$tType,C:$tType,A:$tType,A3:(set @ A),R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C)),A2:A]: (((((equiv_equiv @ A) @ A3) @ R1) => (((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F) => ((((member @ A) @ A2) @ A3) => ((((equiv_congruent @ B) @ C) @ R22) @ (F @ A2)))))))).
% 2.70/1.55 thf(fact_163_proj__def, axiom, (! [A:$tType,B:$tType]: ((((equiv_proj @ B) @ A) = (^ [R3:(set @ ((product_prod @ B) @ A)),X2:B]: (((((image @ B) @ A) @ R3) @ (((insert @ B) @ X2) @ (bot_bot @ (set @ B)))))))))).
% 2.70/1.55 thf(fact_164_equiv__class__subset, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))))))))).
% 2.70/1.55 thf(fact_165_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.70/1.55 thf(fact_166_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 2.70/1.55 thf(fact_167_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ A) @ X5) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.70/1.55 thf(fact_168_converse__mono, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ B) @ A)),S2:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ (((converse @ B) @ A) @ R2)) @ (((converse @ B) @ A) @ S2)) = (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ R2) @ S2))))).
% 2.70/1.55 thf(fact_169_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 2.70/1.55 thf(fact_170_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_171_insert__subset, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = ((((member @ A) @ X3) @ B3) & (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))).
% 2.70/1.55 thf(fact_172_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.70/1.55 thf(fact_173_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A3:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A3)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.70/1.55 thf(fact_174_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.70/1.55 thf(fact_175_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.70/1.55 thf(fact_176_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.70/1.55 thf(fact_177_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B8:A]: (((((ord_less_eq @ A) @ A6) @ B8) => ((P @ A6) @ B8)))) => ((! [A6:A,B8:A]: ((((P @ B8) @ A6) => ((P @ A6) @ B8)))) => ((P @ A2) @ B2))))))))).
% 2.70/1.55 thf(fact_178_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.70/1.55 thf(fact_179_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 2.70/1.55 thf(fact_180_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.70/1.55 thf(fact_181_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.70/1.55 thf(fact_182_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.70/1.55 thf(fact_183_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.70/1.55 thf(fact_184_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y4:A,X3:A]: (((((ord_less_eq @ A) @ Y4) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y4) = (X3 = Y4))))))))).
% 2.70/1.55 thf(fact_185_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y4:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ Z2)))) => (((((ord_less_eq @ A) @ Y4) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y4)))) => (((((ord_less_eq @ A) @ Z2) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ X3)))) => (((((ord_less_eq @ A) @ Y4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y4))))))))))))))))).
% 2.70/1.55 thf(fact_186_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.70/1.55 thf(fact_187_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X3)))))))).
% 2.70/1.55 thf(fact_188_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y4:A]: (((X3 = Y4) => (((ord_less_eq @ A) @ X3) @ Y4)))))))).
% 2.70/1.55 thf(fact_189_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X3)))))))).
% 2.70/1.55 thf(fact_190_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X3) => (X3 = Y4))))))))).
% 2.70/1.55 thf(fact_191_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) & (((ord_less_eq @ A) @ Y) @ X2))))))))).
% 2.70/1.55 thf(fact_192_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.70/1.55 thf(fact_193_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y3:B]: (((((ord_less_eq @ B) @ X5) @ Y3) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.70/1.55 thf(fact_194_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.70/1.55 thf(fact_195_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y3:B]: (((((ord_less_eq @ B) @ X5) @ Y3) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.70/1.55 thf(fact_196_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G5:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F4 @ X2)) @ (G5 @ X2))))))))))).
% 2.70/1.55 thf(fact_197_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.70/1.55 thf(fact_198_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.70/1.55 thf(fact_199_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.70/1.55 thf(fact_200_Image__mono, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),R2:(set @ ((product_prod @ A) @ B)),A8:(set @ A),A3:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ R2) => ((((ord_less_eq @ (set @ A)) @ A8) @ A3) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R4) @ A8)) @ ((((image @ A) @ B) @ R2) @ A3))))))).
% 2.70/1.55 thf(fact_201_Collect__subset, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A3) & (P @ X2)))))) @ A3)))).
% 2.70/1.55 thf(fact_202_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.70/1.55 thf(fact_203_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z:(set @ A)]: ((Y2 = Z))) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))).
% 2.70/1.55 thf(fact_204_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => (((ord_less_eq @ (set @ A)) @ A3) @ C5)))))).
% 2.70/1.55 thf(fact_205_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.70/1.55 thf(fact_206_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.70/1.55 thf(fact_207_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A7) => (((member @ A) @ T2) @ B6)))))))))).
% 2.70/1.55 thf(fact_208_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3))))).
% 2.70/1.55 thf(fact_209_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.70/1.55 thf(fact_210_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((member @ A) @ X2) @ B6)))))))))).
% 2.70/1.55 thf(fact_211_equalityE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))).
% 2.70/1.55 thf(fact_212_subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 2.70/1.55 thf(fact_213_in__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B3)))))).
% 2.70/1.55 thf(fact_214_converse__subset__swap, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ (((converse @ B) @ A) @ S2)) = (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ (((converse @ A) @ B) @ R2)) @ S2))))).
% 2.70/1.55 thf(fact_215_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X5:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y3)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y3)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S2))))).
% 2.70/1.55 thf(fact_216_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.70/1.55 thf(fact_217_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.70/1.55 thf(fact_218_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.70/1.55 thf(fact_219_insert__mono, axiom, (! [A:$tType,C5:(set @ A),D4:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C5) @ D4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C5)) @ (((insert @ A) @ A2) @ D4)))))).
% 2.70/1.55 thf(fact_220_subset__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))).
% 2.70/1.55 thf(fact_221_subset__insertI, axiom, (! [A:$tType,B3:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((insert @ A) @ A2) @ B3))))).
% 2.70/1.55 thf(fact_222_subset__insertI2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ B3)))))).
% 2.70/1.55 thf(fact_223_subset__singleton__iff, axiom, (! [A:$tType,X4:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X4 = (bot_bot @ (set @ A))) | (X4 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 2.70/1.55 thf(fact_224_subset__singletonD, axiom, (! [A:$tType,A3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => ((A3 = (bot_bot @ (set @ A))) | (A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.70/1.55 thf(fact_225_restrict__subsD, axiom, (! [B:$tType,A:$tType,G2:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G2)) @ (((labeled_edges @ A) @ B) @ (((restrict @ A) @ B) @ G2))) => (G2 = (((restrict @ A) @ B) @ G2)))))).
% 2.70/1.55 thf(fact_226_congruent2I_H, axiom, (! [C:$tType,B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C))]: (((! [Y12:A,Z1:A,Y23:B,Z22:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y12) @ Z1)) @ R1) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y23) @ Z22)) @ R22) => (((F @ Y12) @ Y23) = ((F @ Z1) @ Z22)))))) => ((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F))))).
% 2.70/1.55 thf(fact_227_congruent2D, axiom, (! [A:$tType,C:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C)),Y1:A,Z12:A,Y22:B,Z23:B]: ((((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y1) @ Z12)) @ R1) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y22) @ Z23)) @ R22) => (((F @ Y1) @ Y22) = ((F @ Z12) @ Z23)))))))).
% 2.70/1.55 thf(fact_228_proj__iff, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y4:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) @ A3) => ((((((equiv_proj @ A) @ A) @ R2) @ X3) = ((((equiv_proj @ A) @ A) @ R2) @ Y4)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2))))))).
% 2.70/1.55 thf(fact_229_refines__equiv__class__eq2, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A)),A3:(set @ A),A2:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ S) => ((((equiv_equiv @ A) @ A3) @ R) => ((((equiv_equiv @ A) @ A3) @ S) => (((((image @ A) @ A) @ S) @ ((((image @ A) @ A) @ R) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = ((((image @ A) @ A) @ S) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))).
% 2.70/1.55 thf(fact_230_refines__equiv__class__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A)),A3:(set @ A),A2:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ S) => ((((equiv_equiv @ A) @ A3) @ R) => ((((equiv_equiv @ A) @ A3) @ S) => (((((image @ A) @ A) @ R) @ ((((image @ A) @ A) @ S) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = ((((image @ A) @ A) @ S) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))).
% 2.70/1.55 thf(fact_231_congruent2I, axiom, (! [C:$tType,B:$tType,A:$tType,A12:(set @ A),R1:(set @ ((product_prod @ A) @ A)),A23:(set @ B),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C))]: (((((equiv_equiv @ A) @ A12) @ R1) => ((((equiv_equiv @ B) @ A23) @ R22) => ((! [Y3:A,Z3:A,W:B]: (((((member @ B) @ W) @ A23) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R1) => (((F @ Y3) @ W) = ((F @ Z3) @ W)))))) => ((! [Y3:B,Z3:B,W:A]: (((((member @ A) @ W) @ A12) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y3) @ Z3)) @ R22) => (((F @ W) @ Y3) = ((F @ W) @ Z3)))))) => ((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F)))))))).
% 2.70/1.55 thf(fact_232_congruent2__commuteI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),F:(A > (A > B))]: (((((equiv_equiv @ A) @ A3) @ R2) => ((! [Y3:A,Z3:A]: (((((member @ A) @ Y3) @ A3) => ((((member @ A) @ Z3) @ A3) => (((F @ Y3) @ Z3) = ((F @ Z3) @ Y3)))))) => ((! [Y3:A,Z3:A,W:A]: (((((member @ A) @ W) @ A3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R2) => (((F @ W) @ Y3) = ((F @ W) @ Z3)))))) => ((((((equiv_congruent2 @ A) @ A) @ B) @ R2) @ R2) @ F))))))).
% 2.70/1.55 thf(fact_233_subset__equiv__class, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),B2:A,A2:A]: (((((equiv_equiv @ A) @ A3) @ R2) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) => ((((member @ A) @ B2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2))))))).
% 2.70/1.55 thf(fact_234_insert__subsetI, axiom, (! [A:$tType,X3:A,A3:(set @ A),X4:(set @ A)]: (((((member @ A) @ X3) @ A3) => ((((ord_less_eq @ (set @ A)) @ X4) @ A3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ X4)) @ A3)))))).
% 2.70/1.55 thf(fact_235_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.70/1.55 thf(fact_236_univ__commute, axiom, (! [B:$tType,A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),F:(A > B),X3:A]: (((((equiv_equiv @ A) @ A3) @ R2) => (((((equiv_congruent @ A) @ B) @ R2) @ F) => ((((member @ A) @ X3) @ A3) => (((((bNF_Greatest_univ @ A) @ B) @ F) @ ((((equiv_proj @ A) @ A) @ R2) @ X3)) = (F @ X3)))))))).
% 2.70/1.55 thf(fact_237_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ R)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 2.70/1.55 thf(fact_238_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A7)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B6)))))))))).
% 2.70/1.55 thf(fact_239_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R)))) @ (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ S)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 2.70/1.55 thf(fact_240_univ__def, axiom, (! [A:$tType,B:$tType]: ((((bNF_Greatest_univ @ B) @ A) = (^ [F4:(B > A),X8:(set @ B)]: ((F4 @ ((hilbert_Eps @ B) @ (^ [X2:B]: ((((member @ B) @ X2) @ X8))))))))))).
% 2.70/1.55 thf(fact_241_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S2:B,R:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S2)) @ R) => ((S3 = S2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S3)) @ R)))))).
% 2.70/1.55 thf(fact_242_subset__Collect__iff, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((ord_less_eq @ (set @ A)) @ B3) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A3) & (P @ X2)))))) = (! [X2:A]: (((((member @ A) @ X2) @ B3) => (P @ X2))))))))).
% 2.70/1.55 thf(fact_243_subset__CollectI, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ B3) => ((Q @ X5) => (P @ X5))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ B3) & (Q @ X2)))))) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A3) & (P @ X2))))))))))).
% 2.70/1.55 thf(fact_244_Collect__restrict, axiom, (! [A:$tType,X4:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ X4) & (P @ X2)))))) @ X4)))).
% 2.70/1.55 thf(fact_245_prop__restrict, axiom, (! [A:$tType,X3:A,Z4:(set @ A),X4:(set @ A),P:(A > $o)]: (((((member @ A) @ X3) @ Z4) => ((((ord_less_eq @ (set @ A)) @ Z4) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ X4) & (P @ X2)))))) => (P @ X3)))))).
% 2.70/1.55 thf(fact_246_graph__unionI, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ ((product_prod @ B) @ B)))) @ (((labeled_edges @ A) @ B) @ G_1)) @ (((labeled_edges @ A) @ B) @ G_2)) => ((((ord_less_eq @ (set @ B)) @ (((labeled_vertices @ A) @ B) @ G_1)) @ (((labeled_vertices @ A) @ B) @ G_2)) => (((((graph_union @ A) @ B) @ G_1) @ G_2) = G_2)))))).
% 2.70/1.55 thf(fact_247_less__by__empty, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A)),B3:(set @ ((product_prod @ A) @ A))]: (((A3 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A3) @ B3))))).
% 2.70/1.55 thf(fact_248_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X5:A,Y3:B]: ((((P @ X5) @ Y3) => ((Q @ X5) @ Y3)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 2.70/1.55 thf(fact_249_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 2.70/1.55 thf(fact_250_graph__union__idemp_I1_J, axiom, (! [B:$tType,A:$tType,A3:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A3) @ A3) = A3)))).
% 2.70/1.55 thf(fact_251_graph__union__idemp_I2_J, axiom, (! [B:$tType,A:$tType,A3:((labeled_graph @ A) @ B),B3:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A3) @ ((((graph_union @ A) @ B) @ A3) @ B3)) = ((((graph_union @ A) @ B) @ A3) @ B3))))).
% 2.70/1.55 thf(fact_252_graph__union__idemp_I3_J, axiom, (! [B:$tType,A:$tType,A3:((labeled_graph @ A) @ B),B3:((labeled_graph @ A) @ B)]: ((((((graph_union @ A) @ B) @ A3) @ ((((graph_union @ A) @ B) @ B3) @ A3)) = ((((graph_union @ A) @ B) @ B3) @ A3))))).
% 2.70/1.55 thf(fact_253_graph__union__preserves__restrict, axiom, (! [B:$tType,A:$tType,G_1:((labeled_graph @ A) @ B),G_2:((labeled_graph @ A) @ B)]: (((G_1 = (((restrict @ A) @ B) @ G_1)) => ((G_2 = (((restrict @ A) @ B) @ G_2)) => (((((graph_union @ A) @ B) @ G_1) @ G_2) = (((restrict @ A) @ B) @ ((((graph_union @ A) @ B) @ G_1) @ G_2)))))))).
% 2.70/1.55 thf(fact_254_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X3:A,Y4:B,Q:(A > (B > $o))]: ((((P @ X3) @ Y4) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X3) @ Y4)))))).
% 2.70/1.55 thf(fact_255_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X3:A,Q:(A > $o)]: (((P @ X3) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X3)))))).
% 2.70/1.55 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A9:$tType,A10:$tType]: (((order_bot @ A10) => (order_bot @ (A9 > A10)))))).
% 2.70/1.55 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.70/1.55 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.70/1.55 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.70/1.55 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 2.70/1.55 thf(tcon_Set_Oset___Orderings_Oorder__bot_1, axiom, (! [A9:$tType]: ((order_bot @ (set @ A9))))).
% 2.70/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.70/1.55 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.70/1.55 thf(tcon_Set_Oset___Orderings_Oord_4, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.70/1.55 thf(tcon_Set_Oset___Orderings_Obot_5, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 2.70/1.55 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6, axiom, (order_bot @ $o)).
% 2.70/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, (preorder @ $o)).
% 2.70/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.70/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_8, axiom, (order @ $o)).
% 2.70/1.55 thf(tcon_HOL_Obool___Orderings_Oord_9, axiom, (ord @ $o)).
% 2.70/1.55 thf(tcon_HOL_Obool___Orderings_Obot_10, axiom, (bot @ $o)).
% 2.70/1.55 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.70/1.55 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y4:A]: ((((((if @ A) @ $false) @ X3) @ Y4) = Y4)))).
% 2.70/1.55 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y4:A]: ((((((if @ A) @ $true) @ X3) @ Y4) = X3)))).
% 2.70/1.55 thf(conj_0, conjecture, (((hilbert_Eps @ b) @ (^ [Y:b]: ((((member @ b) @ Y) @ ((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ x) @ (bot_bot @ (set @ b)))))))) = ((hilbert_Eps @ b) @ (^ [Y:b]: ((((member @ b) @ Y) @ ((((image @ b) @ b) @ ((((getRel @ (standard_Constant @ a)) @ b) @ (standard_S_Idt @ a)) @ g)) @ (((insert @ b) @ y) @ (bot_bot @ (set @ b)))))))))).
% 2.70/1.55 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.27O21xWGGm/DTF2DTF16637.p
% 2.70/1.55 ------------------------
% 2.70/1.58 ---- Cleaned THF ---
% 2.70/1.58 thf(ty_t_LabeledGraphSemantics_OStandard__Constant,type,
% 2.70/1.58 standard_Constant: $tType > $tType ).
% 2.70/1.58
% 2.70/1.58 thf(ty_t_LabeledGraphs_Olabeled__graph,type,
% 2.70/1.58 labeled_graph: $tType > $tType > $tType ).
% 2.70/1.58
% 2.70/1.58 thf(ty_t_Product__Type_Oprod,type,
% 2.70/1.58 product_prod: $tType > $tType > $tType ).
% 2.70/1.58
% 2.70/1.58 thf(ty_t_Set_Oset,type,
% 2.70/1.58 set: $tType > $tType ).
% 2.70/1.58
% 2.70/1.58 thf(ty_tf_b,type,
% 2.70/1.58 b: $tType ).
% 2.70/1.58
% 2.70/1.58 thf(ty_tf_a,type,
% 2.70/1.58 a: $tType ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_HOL_Otype,type,
% 2.70/1.58 type:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_Orderings_Obot,type,
% 2.70/1.58 bot:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_Orderings_Oord,type,
% 2.70/1.58 ord:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_Orderings_Oorder,type,
% 2.70/1.58 order:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_Orderings_Olinorder,type,
% 2.70/1.58 linorder:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_Orderings_Opreorder,type,
% 2.70/1.58 preorder:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.70/1.58 order_bot:
% 2.70/1.58 !>[A: $tType] : $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
% 2.70/1.58 bNF_Greatest_univ:
% 2.70/1.58 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 2.70/1.58 equiv_congruent:
% 2.70/1.58 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Equiv__Relations_Ocongruent2,type,
% 2.70/1.58 equiv_congruent2:
% 2.70/1.58 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Equiv__Relations_Oequiv,type,
% 2.70/1.58 equiv_equiv:
% 2.70/1.58 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Equiv__Relations_Oproj,type,
% 2.70/1.58 equiv_proj:
% 2.70/1.58 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Hilbert__Choice_OEps,type,
% 2.70/1.58 hilbert_Eps:
% 2.70/1.58 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_If,type,
% 2.70/1.58 if:
% 2.70/1.58 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt,type,
% 2.70/1.58 standard_S_Idt:
% 2.70/1.58 !>[V: $tType] : ( standard_Constant @ V ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphSemantics_OgetRel,type,
% 2.70/1.58 getRel:
% 2.70/1.58 !>[B: $tType,A: $tType] : ( B > ( labeled_graph @ B @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Ograph__homomorphism,type,
% 2.70/1.58 graph_homomorphism:
% 2.70/1.58 !>[A: $tType,B: $tType,C: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ C ) > ( set @ ( product_prod @ B @ C ) ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Ograph__union,type,
% 2.70/1.58 graph_union:
% 2.70/1.58 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Olabeled__graph_OLG,type,
% 2.70/1.58 labeled_LG:
% 2.70/1.58 !>[L: $tType,V: $tType] : ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > ( labeled_graph @ L @ V ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Olabeled__graph_Ocase__labeled__graph,type,
% 2.70/1.58 labele1974067554_graph:
% 2.70/1.58 !>[L: $tType,V: $tType,A: $tType] : ( ( ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A ) > ( labeled_graph @ L @ V ) > A ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges,type,
% 2.70/1.58 labeled_edges:
% 2.70/1.58 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices,type,
% 2.70/1.58 labeled_vertices:
% 2.70/1.58 !>[L: $tType,V: $tType] : ( ( labeled_graph @ L @ V ) > ( set @ V ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_LabeledGraphs_Orestrict,type,
% 2.70/1.58 restrict:
% 2.70/1.58 !>[A: $tType,B: $tType] : ( ( labeled_graph @ A @ B ) > ( labeled_graph @ A @ B ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 2.70/1.58 order_1409979114der_on:
% 2.70/1.58 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.70/1.58 bot_bot:
% 2.70/1.58 !>[A: $tType] : A ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.70/1.58 ord_less_eq:
% 2.70/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Product__Type_OPair,type,
% 2.70/1.58 product_Pair:
% 2.70/1.58 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.70/1.58 product_rec_prod:
% 2.70/1.58 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Relation_OImage,type,
% 2.70/1.58 image:
% 2.70/1.58 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Relation_Oconverse,type,
% 2.70/1.58 converse:
% 2.70/1.58 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Relation_Orefl__on,type,
% 2.70/1.58 refl_on:
% 2.70/1.58 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Relation_Ototal__on,type,
% 2.70/1.58 total_on:
% 2.70/1.58 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Set_OCollect,type,
% 2.70/1.58 collect:
% 2.70/1.58 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Set_Oinsert,type,
% 2.70/1.58 insert:
% 2.70/1.58 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Set_Ois__singleton,type,
% 2.70/1.58 is_singleton:
% 2.70/1.58 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_Set_Othe__elem,type,
% 2.70/1.58 the_elem:
% 2.70/1.58 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_c_member,type,
% 2.70/1.58 member:
% 2.70/1.58 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.70/1.58
% 2.70/1.58 thf(sy_v_G,type,
% 2.70/1.58 g: labeled_graph @ ( standard_Constant @ a ) @ b ).
% 2.70/1.58
% 2.70/1.58 thf(sy_v_P____,type,
% 2.70/1.58 p: b > b > $o ).
% 2.70/1.58
% 2.70/1.58 thf(sy_v_f____,type,
% 2.70/1.58 f: b > b ).
% 2.70/1.58
% 2.70/1.58 thf(sy_v_x____,type,
% 2.70/1.58 x: b ).
% 2.70/1.58
% 2.70/1.58 thf(sy_v_y____,type,
% 2.70/1.58 y: b ).
% 2.70/1.58
% 2.70/1.58 thf(fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062,axiom,
% 2.70/1.58 ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ x @ ( bot_bot @ ( set @ b ) ) ) )
% 2.70/1.58 = ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ y @ ( bot_bot @ ( set @ b ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_1_assms_I1_J,axiom,
% 2.70/1.58 ( g
% 2.70/1.58 = ( restrict @ ( standard_Constant @ a ) @ b @ g ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_2_P__eq,axiom,
% 2.70/1.58 ! [X: b] :
% 2.70/1.58 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X @ ( bot_bot @ ( set @ b ) ) ) )
% 2.70/1.58 != ( bot_bot @ ( set @ b ) ) )
% 2.70/1.58 => ( ( hilbert_Eps @ b @ ( p @ ( hilbert_Eps @ b @ ( p @ X ) ) ) )
% 2.70/1.58 = ( hilbert_Eps @ b @ ( p @ X ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_3_P,axiom,
% 2.70/1.58 ( p
% 2.70/1.58 = ( ^ [X2: b,Y: b] : ( member @ b @ Y @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_4__092_060open_062_Ix_M_Ay_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 2.70/1.58 member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ x @ y ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_5__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
% 2.70/1.58 ( f
% 2.70/1.58 = ( ^ [X2: b] :
% 2.70/1.58 ( if @ b
% 2.70/1.58 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ b ) ) )
% 2.70/1.58 @ X2
% 2.70/1.58 @ ( hilbert_Eps @ b @ ( p @ X2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_6_singleton__conv,axiom,
% 2.70/1.58 ! [A: $tType,A2: A] :
% 2.70/1.58 ( ( collect @ A
% 2.70/1.58 @ ^ [X2: A] : ( X2 = A2 ) )
% 2.70/1.58 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_7_singleton__conv2,axiom,
% 2.70/1.58 ! [A: $tType,A2: A] :
% 2.70/1.58 ( ( collect @ A
% 2.70/1.58 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.70/1.58 @ A2 ) )
% 2.70/1.58 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_8_tuple__disj,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,Z2: A] :
% 2.70/1.58 ( ( collect @ A
% 2.70/1.58 @ ^ [Y: A] :
% 2.70/1.58 ( ( Y = X3 )
% 2.70/1.58 | ( Y = Z2 ) ) )
% 2.70/1.58 = ( insert @ A @ X3 @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_9_Image__empty1,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,X4: set @ B] :
% 2.70/1.58 ( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X4 )
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_10_Image__empty2,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_11_singletonI,axiom,
% 2.70/1.58 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_12_some__equality,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,A2: A] :
% 2.70/1.58 ( ( P @ A2 )
% 2.70/1.58 => ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 => ( X5 = A2 ) )
% 2.70/1.58 => ( ( hilbert_Eps @ A @ P )
% 2.70/1.58 = A2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_13_some__eq__trivial,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] :
% 2.70/1.58 ( ( hilbert_Eps @ A
% 2.70/1.58 @ ^ [Y: A] : ( Y = X3 ) )
% 2.70/1.58 = X3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_14_empty__Collect__eq,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o] :
% 2.70/1.58 ( ( ( bot_bot @ ( set @ A ) )
% 2.70/1.58 = ( collect @ A @ P ) )
% 2.70/1.58 = ( ! [X2: A] :
% 2.70/1.58 ~ ( P @ X2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_15_Collect__empty__eq,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o] :
% 2.70/1.58 ( ( ( collect @ A @ P )
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 = ( ! [X2: A] :
% 2.70/1.58 ~ ( P @ X2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_16_all__not__in__conv,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( ! [X2: A] :
% 2.70/1.58 ~ ( member @ A @ X2 @ A3 ) )
% 2.70/1.58 = ( A3
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_17_empty__iff,axiom,
% 2.70/1.58 ! [A: $tType,C2: A] :
% 2.70/1.58 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_18_insert__absorb2,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,A3: set @ A] :
% 2.70/1.58 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
% 2.70/1.58 = ( insert @ A @ X3 @ A3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_19_insert__iff,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 2.70/1.58 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 2.70/1.58 = ( ( A2 = B2 )
% 2.70/1.58 | ( member @ A @ A2 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_20_insertCI,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 2.70/1.58 ( ( ~ ( member @ A @ A2 @ B3 )
% 2.70/1.58 => ( A2 = B2 ) )
% 2.70/1.58 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_21_some__sym__eq__trivial,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] :
% 2.70/1.58 ( ( hilbert_Eps @ A
% 2.70/1.58 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.70/1.58 @ X3 ) )
% 2.70/1.58 = X3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_22_ImageI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),A3: set @ A] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.70/1.58 => ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_23_f,axiom,
% 2.70/1.58 ( f
% 2.70/1.58 = ( ^ [X2: b] :
% 2.70/1.58 ( if @ b
% 2.70/1.58 @ ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X2 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ b ) ) )
% 2.70/1.58 @ X2
% 2.70/1.58 @ ( hilbert_Eps @ b @ ( p @ X2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_24_Image__singleton__iff,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A2: B] :
% 2.70/1.58 ( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.70/1.58 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B2 ) @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_25_pred__equals__eq2,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
% 2.70/1.58 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) ) )
% 2.70/1.58 = ( R = S ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_26_rev__ImageI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,R2: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.70/1.58 => ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_27_Image__iff,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 2.70/1.58 ( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A3 ) )
% 2.70/1.58 = ( ? [X2: B] :
% 2.70/1.58 ( ( member @ B @ X2 @ A3 )
% 2.70/1.58 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_28_ImageE,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 2.70/1.58 ( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A3 ) )
% 2.70/1.58 => ~ ! [X5: B] :
% 2.70/1.58 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R2 )
% 2.70/1.58 => ~ ( member @ B @ X5 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_29_Image__singleton,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A2: B] :
% 2.70/1.58 ( ( image @ B @ A @ R2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
% 2.70/1.58 = ( collect @ A
% 2.70/1.58 @ ^ [B4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B4 ) @ R2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_30_ex__in__conv,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.70/1.58 = ( A3
% 2.70/1.58 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_31_equals0I,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ! [Y3: A] :
% 2.70/1.58 ~ ( member @ A @ Y3 @ A3 )
% 2.70/1.58 => ( A3
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_32_equals0D,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,A2: A] :
% 2.70/1.58 ( ( A3
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_33_emptyE,axiom,
% 2.70/1.58 ! [A: $tType,A2: A] :
% 2.70/1.58 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_34_mk__disjoint__insert,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,A3: set @ A] :
% 2.70/1.58 ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ? [B5: set @ A] :
% 2.70/1.58 ( ( A3
% 2.70/1.58 = ( insert @ A @ A2 @ B5 ) )
% 2.70/1.58 & ~ ( member @ A @ A2 @ B5 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_35_insert__commute,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,Y4: A,A3: set @ A] :
% 2.70/1.58 ( ( insert @ A @ X3 @ ( insert @ A @ Y4 @ A3 ) )
% 2.70/1.58 = ( insert @ A @ Y4 @ ( insert @ A @ X3 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_36_insert__eq__iff,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
% 2.70/1.58 ( ~ ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( ~ ( member @ A @ B2 @ B3 )
% 2.70/1.58 => ( ( ( insert @ A @ A2 @ A3 )
% 2.70/1.58 = ( insert @ A @ B2 @ B3 ) )
% 2.70/1.58 = ( ( ( A2 = B2 )
% 2.70/1.58 => ( A3 = B3 ) )
% 2.70/1.58 & ( ( A2 != B2 )
% 2.70/1.58 => ? [C3: set @ A] :
% 2.70/1.58 ( ( A3
% 2.70/1.58 = ( insert @ A @ B2 @ C3 ) )
% 2.70/1.58 & ~ ( member @ A @ B2 @ C3 )
% 2.70/1.58 & ( B3
% 2.70/1.58 = ( insert @ A @ A2 @ C3 ) )
% 2.70/1.58 & ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_37_insert__absorb,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,A3: set @ A] :
% 2.70/1.58 ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( ( insert @ A @ A2 @ A3 )
% 2.70/1.58 = A3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_38_insert__ident,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ~ ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ( ~ ( member @ A @ X3 @ B3 )
% 2.70/1.58 => ( ( ( insert @ A @ X3 @ A3 )
% 2.70/1.58 = ( insert @ A @ X3 @ B3 ) )
% 2.70/1.58 = ( A3 = B3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_39_Set_Oset__insert,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,A3: set @ A] :
% 2.70/1.58 ( ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ~ ! [B5: set @ A] :
% 2.70/1.58 ( ( A3
% 2.70/1.58 = ( insert @ A @ X3 @ B5 ) )
% 2.70/1.58 => ( member @ A @ X3 @ B5 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_40_insertI2,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 2.70/1.58 ( ( member @ A @ A2 @ B3 )
% 2.70/1.58 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_41_insertI1,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_42_insertE,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 2.70/1.58 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 2.70/1.58 => ( ( A2 != B2 )
% 2.70/1.58 => ( member @ A @ A2 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_43_some__eq__imp,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,A2: A,B2: A] :
% 2.70/1.58 ( ( ( hilbert_Eps @ A @ P )
% 2.70/1.58 = A2 )
% 2.70/1.58 => ( ( P @ B2 )
% 2.70/1.58 => ( P @ A2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_44_tfl__some,axiom,
% 2.70/1.58 ! [A: $tType,P2: A > $o,X6: A] :
% 2.70/1.58 ( ( P2 @ X6 )
% 2.70/1.58 => ( P2 @ ( hilbert_Eps @ A @ P2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_45_mem__Collect__eq,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,P: A > $o] :
% 2.70/1.58 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.70/1.58 = ( P @ A2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_46_Collect__mem__eq,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( collect @ A
% 2.70/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.70/1.58 = A3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_47_Collect__cong,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 = ( Q @ X5 ) )
% 2.70/1.58 => ( ( collect @ A @ P )
% 2.70/1.58 = ( collect @ A @ Q ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_48_ext,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ( ( F @ X5 )
% 2.70/1.58 = ( G @ X5 ) )
% 2.70/1.58 => ( F = G ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_49_Eps__cong,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 = ( Q @ X5 ) )
% 2.70/1.58 => ( ( hilbert_Eps @ A @ P )
% 2.70/1.58 = ( hilbert_Eps @ A @ Q ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_50_someI,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,X3: A] :
% 2.70/1.58 ( ( P @ X3 )
% 2.70/1.58 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_51_empty__def,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( bot_bot @ ( set @ A ) )
% 2.70/1.58 = ( collect @ A
% 2.70/1.58 @ ^ [X2: A] : $false ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_52_insert__Collect,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,P: A > $o] :
% 2.70/1.58 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 2.70/1.58 = ( collect @ A
% 2.70/1.58 @ ^ [U: A] :
% 2.70/1.58 ( ( U != A2 )
% 2.70/1.58 => ( P @ U ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_53_insert__compr,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( insert @ A )
% 2.70/1.58 = ( ^ [A4: A,B6: set @ A] :
% 2.70/1.58 ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( X2 = A4 )
% 2.70/1.58 | ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_54_some1__equality,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,A2: A] :
% 2.70/1.58 ( ? [X6: A] :
% 2.70/1.58 ( ( P @ X6 )
% 2.70/1.58 & ! [Y3: A] :
% 2.70/1.58 ( ( P @ Y3 )
% 2.70/1.58 => ( Y3 = X6 ) ) )
% 2.70/1.58 => ( ( P @ A2 )
% 2.70/1.58 => ( ( hilbert_Eps @ A @ P )
% 2.70/1.58 = A2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_55_some__eq__ex,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o] :
% 2.70/1.58 ( ( P @ ( hilbert_Eps @ A @ P ) )
% 2.70/1.58 = ( ? [X7: A] : ( P @ X7 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_56_someI2__bex,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ? [X6: A] :
% 2.70/1.58 ( ( member @ A @ X6 @ A3 )
% 2.70/1.58 & ( P @ X6 ) )
% 2.70/1.58 => ( ! [X5: A] :
% 2.70/1.58 ( ( ( member @ A @ X5 @ A3 )
% 2.70/1.58 & ( P @ X5 ) )
% 2.70/1.58 => ( Q @ X5 ) )
% 2.70/1.58 => ( Q
% 2.70/1.58 @ ( hilbert_Eps @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ A3 )
% 2.70/1.58 & ( P @ X2 ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_57_someI2__ex,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ? [X_1: A] : ( P @ X_1 )
% 2.70/1.58 => ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 => ( Q @ X5 ) )
% 2.70/1.58 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_58_someI__ex,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o] :
% 2.70/1.58 ( ? [X_1: A] : ( P @ X_1 )
% 2.70/1.58 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_59_someI2,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 2.70/1.58 ( ( P @ A2 )
% 2.70/1.58 => ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 => ( Q @ X5 ) )
% 2.70/1.58 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_60_singleton__inject,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B2: A] :
% 2.70/1.58 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 => ( A2 = B2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_61_insert__not__empty,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,A3: set @ A] :
% 2.70/1.58 ( ( insert @ A @ A2 @ A3 )
% 2.70/1.58 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_62_doubleton__eq__iff,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,B2: A,C2: A,D: A] :
% 2.70/1.58 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 = ( ( ( A2 = C2 )
% 2.70/1.58 & ( B2 = D ) )
% 2.70/1.58 | ( ( A2 = D )
% 2.70/1.58 & ( B2 = C2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_63_singleton__iff,axiom,
% 2.70/1.58 ! [A: $tType,B2: A,A2: A] :
% 2.70/1.58 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( B2 = A2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_64_singletonD,axiom,
% 2.70/1.58 ! [A: $tType,B2: A,A2: A] :
% 2.70/1.58 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 => ( B2 = A2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_65_Collect__conv__if2,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,A2: A] :
% 2.70/1.58 ( ( ( P @ A2 )
% 2.70/1.58 => ( ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( A2 = X2 )
% 2.70/1.58 & ( P @ X2 ) ) )
% 2.70/1.58 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 & ( ~ ( P @ A2 )
% 2.70/1.58 => ( ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( A2 = X2 )
% 2.70/1.58 & ( P @ X2 ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_66_Collect__conv__if,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,A2: A] :
% 2.70/1.58 ( ( ( P @ A2 )
% 2.70/1.58 => ( ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( X2 = A2 )
% 2.70/1.58 & ( P @ X2 ) ) )
% 2.70/1.58 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 & ( ~ ( P @ A2 )
% 2.70/1.58 => ( ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( X2 = A2 )
% 2.70/1.58 & ( P @ X2 ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_67_some__in__eq,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( member @ A
% 2.70/1.58 @ ( hilbert_Eps @ A
% 2.70/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.70/1.58 @ A3 )
% 2.70/1.58 = ( A3
% 2.70/1.58 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_68_congr_I1_J,axiom,
% 2.70/1.58 ! [X: b,Y5: b,L2: standard_Constant @ a] :
% 2.70/1.58 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X @ Y5 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) )
% 2.70/1.58 => ( equiv_congruent @ b @ ( set @ b ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g )
% 2.70/1.58 @ ^ [V2: b] : ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) @ ( insert @ b @ V2 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_69_vert__P,axiom,
% 2.70/1.58 ! [X3: b] :
% 2.70/1.58 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.70/1.58 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ ( hilbert_Eps @ b @ ( p @ X3 ) ) ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_70__092_060open_062_IS__Idt_M_Ax_M_Ay_J_A_092_060in_062_Aedges_AG_092_060close_062,axiom,
% 2.70/1.58 member @ ( product_prod @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) ) @ ( product_Pair @ ( standard_Constant @ a ) @ ( product_prod @ b @ b ) @ ( standard_S_Idt @ a ) @ ( product_Pair @ b @ b @ x @ y ) ) @ ( labeled_edges @ ( standard_Constant @ a ) @ b @ g ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_71_restrict__idemp,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,X3: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( restrict @ A @ B @ ( restrict @ A @ B @ X3 ) )
% 2.70/1.58 = ( restrict @ A @ B @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_72_r2,axiom,
% 2.70/1.58 ! [X3: b] :
% 2.70/1.58 ( ( ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ X3 @ ( bot_bot @ ( set @ b ) ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ b ) ) )
% 2.70/1.58 = ( ~ ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_73_r1,axiom,
% 2.70/1.58 ! [X3: b] :
% 2.70/1.58 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.70/1.58 = ( p @ X3 @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_74_congr_I2_J,axiom,
% 2.70/1.58 ! [X: b,Y5: b,L2: standard_Constant @ a] :
% 2.70/1.58 ( ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X @ Y5 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) )
% 2.70/1.58 => ( equiv_congruent @ b @ ( set @ b ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g )
% 2.70/1.58 @ ^ [V2: b] : ( image @ b @ b @ ( converse @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ L2 @ g ) ) @ ( insert @ b @ V2 @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_75_the__elem__eq,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] :
% 2.70/1.58 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = X3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_76_bot__apply,axiom,
% 2.70/1.58 ! [C: $tType,D2: $tType] :
% 2.70/1.58 ( ( bot @ C )
% 2.70/1.58 => ( ( bot_bot @ ( D2 > C ) )
% 2.70/1.58 = ( ^ [X2: D2] : ( bot_bot @ C ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_77_prod_Oinject,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.70/1.58 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.70/1.58 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.70/1.58 = ( ( X1 = Y1 )
% 2.70/1.58 & ( X22 = Y22 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_78_old_Oprod_Oinject,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B7: B] :
% 2.70/1.58 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.70/1.58 = ( product_Pair @ A @ B @ A5 @ B7 ) )
% 2.70/1.58 = ( ( A2 = A5 )
% 2.70/1.58 & ( B2 = B7 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_79_converse__inject,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( ( converse @ B @ A @ R2 )
% 2.70/1.58 = ( converse @ B @ A @ S2 ) )
% 2.70/1.58 = ( R2 = S2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_80_converse__converse,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
% 2.70/1.58 = R2 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_81_converse__iff,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R2 ) )
% 2.70/1.58 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_82_vertices__restrict,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,G2: labeled_graph @ B @ A] :
% 2.70/1.58 ( ( labeled_vertices @ B @ A @ ( restrict @ B @ A @ G2 ) )
% 2.70/1.58 = ( labeled_vertices @ B @ A @ G2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_83_converse__empty,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.70/1.58 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_84__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 2.70/1.58 ! [X3: b] :
% 2.70/1.58 ( ( member @ b @ X3 @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) )
% 2.70/1.58 => ( member @ ( product_prod @ b @ b ) @ ( product_Pair @ b @ b @ X3 @ X3 ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_85_labeled__graph_Oexpand,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V,Labeled_graph2: labeled_graph @ L @ V] :
% 2.70/1.58 ( ( ( ( labeled_edges @ L @ V @ Labeled_graph )
% 2.70/1.58 = ( labeled_edges @ L @ V @ Labeled_graph2 ) )
% 2.70/1.58 & ( ( labeled_vertices @ L @ V @ Labeled_graph )
% 2.70/1.58 = ( labeled_vertices @ L @ V @ Labeled_graph2 ) ) )
% 2.70/1.58 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_86_converse_Oinducts,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,X1: B,X22: A,R2: set @ ( product_prod @ A @ B ),P: B > A > $o] :
% 2.70/1.58 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R2 ) )
% 2.70/1.58 => ( ! [A6: A,B8: B] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B8 ) @ R2 )
% 2.70/1.58 => ( P @ B8 @ A6 ) )
% 2.70/1.58 => ( P @ X1 @ X22 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_87_converse_Ointros,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.70/1.58 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_88_converse_Osimps,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 2.70/1.58 = ( ? [A4: A,B4: B] :
% 2.70/1.58 ( ( A1 = B4 )
% 2.70/1.58 & ( A22 = A4 )
% 2.70/1.58 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_89_converse_Ocases,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 2.70/1.58 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_90_converseE,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
% 2.70/1.58 => ~ ! [X5: B,Y3: A] :
% 2.70/1.58 ( ( Yx
% 2.70/1.58 = ( product_Pair @ A @ B @ Y3 @ X5 ) )
% 2.70/1.58 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ Y3 ) @ R2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_91_converseD,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R2 ) )
% 2.70/1.58 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_92_bot__set__def,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( bot_bot @ ( set @ A ) )
% 2.70/1.58 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_93_bot__empty__eq,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( bot_bot @ ( A > $o ) )
% 2.70/1.58 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_94_bot__empty__eq2,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( bot_bot @ ( A > B > $o ) )
% 2.70/1.58 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_95_old_Oprod_Oinducts,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.70/1.58 ( ! [A6: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B8 ) )
% 2.70/1.58 => ( P @ Prod ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_96_old_Oprod_Oexhaust,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
% 2.70/1.58 ~ ! [A6: A,B8: B] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( product_Pair @ A @ B @ A6 @ B8 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_97_prod__induct7,axiom,
% 2.70/1.58 ! [G3: $tType,F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 2.70/1.58 ( ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
% 2.70/1.58 => ( P @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_98_prod__induct6,axiom,
% 2.70/1.58 ! [F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.70/1.58 ( ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 2.70/1.58 => ( P @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_99_prod__induct5,axiom,
% 2.70/1.58 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 2.70/1.58 ( ! [A6: A,B8: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 2.70/1.58 => ( P @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_100_prod__induct4,axiom,
% 2.70/1.58 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 2.70/1.58 ( ! [A6: A,B8: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B8 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 2.70/1.58 => ( P @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_101_prod__induct3,axiom,
% 2.70/1.58 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.70/1.58 ( ! [A6: A,B8: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) )
% 2.70/1.58 => ( P @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_102_prod__cases7,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,G3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 2.70/1.58 ~ ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2,G4: G3] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_103_prod__cases6,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.70/1.58 ~ ! [A6: A,B8: B,C4: C,D3: D2,E2: E,F3: F2] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_104_prod__cases5,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 2.70/1.58 ~ ! [A6: A,B8: B,C4: C,D3: D2,E2: E] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_105_prod__cases4,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 2.70/1.58 ~ ! [A6: A,B8: B,C4: C,D3: D2] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B8 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_106_prod__cases3,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.70/1.58 ~ ! [A6: A,B8: B,C4: C] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_107_Pair__inject,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B7: B] :
% 2.70/1.58 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.70/1.58 = ( product_Pair @ A @ B @ A5 @ B7 ) )
% 2.70/1.58 => ~ ( ( A2 = A5 )
% 2.70/1.58 => ( B2 != B7 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_108_prod__cases,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
% 2.70/1.58 ( ! [A6: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B8 ) )
% 2.70/1.58 => ( P @ P3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_109_surj__pair,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
% 2.70/1.58 ? [X5: A,Y3: B] :
% 2.70/1.58 ( P3
% 2.70/1.58 = ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_110_bot__fun__def,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( bot @ B )
% 2.70/1.58 => ( ( bot_bot @ ( A > B ) )
% 2.70/1.58 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_111_local_Orefl,axiom,
% 2.70/1.58 refl_on @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_112_equiv,axiom,
% 2.70/1.58 equiv_equiv @ b @ ( labeled_vertices @ ( standard_Constant @ a ) @ b @ g ) @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_113_getRel__dom_I1_J,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,G2: labeled_graph @ A @ B,A2: B,B2: B,L3: A] :
% 2.70/1.58 ( ( G2
% 2.70/1.58 = ( restrict @ A @ B @ G2 ) )
% 2.70/1.58 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L3 @ G2 ) )
% 2.70/1.58 => ( member @ B @ A2 @ ( labeled_vertices @ A @ B @ G2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_114_getRel__dom_I2_J,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,G2: labeled_graph @ A @ B,A2: B,B2: B,L3: A] :
% 2.70/1.58 ( ( G2
% 2.70/1.58 = ( restrict @ A @ B @ G2 ) )
% 2.70/1.58 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A2 @ B2 ) @ ( getRel @ A @ B @ L3 @ G2 ) )
% 2.70/1.58 => ( member @ B @ B2 @ ( labeled_vertices @ A @ B @ G2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_115_old_Oprod_Orec,axiom,
% 2.70/1.58 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.70/1.58 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.70/1.58 = ( F1 @ A2 @ B2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_116_is__singleton__the__elem,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( is_singleton @ A )
% 2.70/1.58 = ( ^ [A7: set @ A] :
% 2.70/1.58 ( A7
% 2.70/1.58 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_117_refl__on__converse,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.70/1.58 ( ( refl_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
% 2.70/1.58 = ( refl_on @ A @ A3 @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_118_is__singletonI,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_119_bot2E,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,X3: A,Y4: B] :
% 2.70/1.58 ~ ( bot_bot @ ( A > B > $o ) @ X3 @ Y4 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_120_refl__onD,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
% 2.70/1.58 ( ( refl_on @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_121_refl__onD1,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 2.70/1.58 ( ( refl_on @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
% 2.70/1.58 => ( member @ A @ X3 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_122_refl__onD2,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 2.70/1.58 ( ( refl_on @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
% 2.70/1.58 => ( member @ A @ Y4 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_123_refl__on__empty,axiom,
% 2.70/1.58 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_124_is__singletonI_H,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( A3
% 2.70/1.58 != ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 => ( ! [X5: A,Y3: A] :
% 2.70/1.58 ( ( member @ A @ X5 @ A3 )
% 2.70/1.58 => ( ( member @ A @ Y3 @ A3 )
% 2.70/1.58 => ( X5 = Y3 ) ) )
% 2.70/1.58 => ( is_singleton @ A @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_125_refl__on__singleton,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] : ( refl_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_126_is__singleton__def,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( is_singleton @ A )
% 2.70/1.58 = ( ^ [A7: set @ A] :
% 2.70/1.58 ? [X2: A] :
% 2.70/1.58 ( A7
% 2.70/1.58 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_127_is__singletonE,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( is_singleton @ A @ A3 )
% 2.70/1.58 => ~ ! [X5: A] :
% 2.70/1.58 ( A3
% 2.70/1.58 != ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_128_equiv__class__eq__iff,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
% 2.70/1.58 = ( ( ( image @ A @ A @ R2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( image @ A @ A @ R2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 & ( member @ A @ X3 @ A3 )
% 2.70/1.58 & ( member @ A @ Y4 @ A3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_129_eq__equiv__class__iff,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ( ( member @ A @ Y4 @ A3 )
% 2.70/1.58 => ( ( ( image @ A @ A @ R2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( image @ A @ A @ R2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_130_equiv__class__eq,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 2.70/1.58 => ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_131_eq__equiv__class,axiom,
% 2.70/1.58 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A,A3: set @ A] :
% 2.70/1.58 ( ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 => ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ A @ B2 @ A3 )
% 2.70/1.58 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_132_congruentI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F: A > B] :
% 2.70/1.58 ( ! [Y3: A,Z3: A] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 2.70/1.58 => ( ( F @ Y3 )
% 2.70/1.58 = ( F @ Z3 ) ) )
% 2.70/1.58 => ( equiv_congruent @ A @ B @ R2 @ F ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_133_congruentD,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F: A > B,Y4: A,Z2: A] :
% 2.70/1.58 ( ( equiv_congruent @ A @ B @ R2 @ F )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R2 )
% 2.70/1.58 => ( ( F @ Y4 )
% 2.70/1.58 = ( F @ Z2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_134_equiv__class__self,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( member @ A @ A2 @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_135_Collect__empty__eq__bot,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o] :
% 2.70/1.58 ( ( ( collect @ A @ P )
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 = ( P
% 2.70/1.58 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_136_graph__single,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,A2: A,B2: B,C2: B] :
% 2.70/1.58 ( ( 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 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.70/1.58 = ( 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 @ C2 ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) ) @ ( insert @ B @ B2 @ ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_137_refl__on__domain,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 2.70/1.58 ( ( refl_on @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 2.70/1.58 => ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 & ( member @ A @ B2 @ A3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_138_total__on__singleton,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] : ( total_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_139_labeled__graph_Oinject,axiom,
% 2.70/1.58 ! [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 ) ),Y22: set @ V] :
% 2.70/1.58 ( ( ( labeled_LG @ L @ V @ X1 @ X22 )
% 2.70/1.58 = ( labeled_LG @ L @ V @ Y1 @ Y22 ) )
% 2.70/1.58 = ( ( X1 = Y1 )
% 2.70/1.58 & ( X22 = Y22 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_140_total__on__converse,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.70/1.58 ( ( total_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
% 2.70/1.58 = ( total_on @ A @ A3 @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_141_labeled__graph_Ocollapse,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.70/1.58 ( ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) )
% 2.70/1.58 = Labeled_graph ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_142_graph__empty__e,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,V3: set @ B] :
% 2.70/1.58 ( ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 )
% 2.70/1.58 = ( restrict @ A @ B @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ V3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_143_total__on__def,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( total_on @ A )
% 2.70/1.58 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 2.70/1.58 ! [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ A7 )
% 2.70/1.58 => ! [Y: A] :
% 2.70/1.58 ( ( member @ A @ Y @ A7 )
% 2.70/1.58 => ( ( X2 != Y )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 )
% 2.70/1.58 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_144_total__onI,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.70/1.58 ( ! [X5: A,Y3: A] :
% 2.70/1.58 ( ( member @ A @ X5 @ A3 )
% 2.70/1.58 => ( ( member @ A @ Y3 @ A3 )
% 2.70/1.58 => ( ( X5 != Y3 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ R2 )
% 2.70/1.58 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X5 ) @ R2 ) ) ) ) )
% 2.70/1.58 => ( total_on @ A @ A3 @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_145_total__on__empty,axiom,
% 2.70/1.58 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_146_labeled__graph_Osel_I2_J,axiom,
% 2.70/1.58 ! [L: $tType,V: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.70/1.58 ( ( labeled_vertices @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.70/1.58 = X22 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_147_labeled__graph_Oexhaust,axiom,
% 2.70/1.58 ! [L: $tType,V: $tType,Y4: labeled_graph @ L @ V] :
% 2.70/1.58 ~ ! [X12: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X23: set @ V] :
% 2.70/1.58 ( Y4
% 2.70/1.58 != ( labeled_LG @ L @ V @ X12 @ X23 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_148_labeled__graph_Oinduct,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType,P: ( labeled_graph @ L @ V ) > $o,Labeled_graph: labeled_graph @ L @ V] :
% 2.70/1.58 ( ! [X1a: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X2a: set @ V] : ( P @ ( labeled_LG @ L @ V @ X1a @ X2a ) )
% 2.70/1.58 => ( P @ Labeled_graph ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_149_labeled__graph_Osel_I1_J,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.70/1.58 ( ( labeled_edges @ L @ V @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.70/1.58 = X1 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_150_labeled__graph_Oexhaust__sel,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType,Labeled_graph: labeled_graph @ L @ V] :
% 2.70/1.58 ( Labeled_graph
% 2.70/1.58 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_151_linear__order__on__singleton,axiom,
% 2.70/1.58 ! [A: $tType,X3: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_152_labeled__graph_Osplit__sel,axiom,
% 2.70/1.58 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.70/1.58 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 2.70/1.58 = ( ( Labeled_graph
% 2.70/1.58 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.70/1.58 => ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_153_labeled__graph_Osplit__sel__asm,axiom,
% 2.70/1.58 ! [A: $tType,V: $tType,L: $tType,P: A > $o,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.70/1.58 ( ( P @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 2.70/1.58 = ( ~ ( ( Labeled_graph
% 2.70/1.58 = ( labeled_LG @ L @ V @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) )
% 2.70/1.58 & ~ ( P @ ( F @ ( labeled_edges @ L @ V @ Labeled_graph ) @ ( labeled_vertices @ L @ V @ Labeled_graph ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_154_graph__homomorphism__empty,axiom,
% 2.70/1.58 ! [B: $tType,C: $tType,A: $tType,G2: labeled_graph @ A @ C,F: set @ ( product_prod @ B @ C )] :
% 2.70/1.58 ( ( graph_homomorphism @ A @ B @ C @ ( labeled_LG @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) ) @ ( bot_bot @ ( set @ B ) ) ) @ G2 @ F )
% 2.70/1.58 = ( ( F
% 2.70/1.58 = ( bot_bot @ ( set @ ( product_prod @ B @ C ) ) ) )
% 2.70/1.58 & ( G2
% 2.70/1.58 = ( restrict @ A @ C @ G2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_155_linear__order__on__converse,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.70/1.58 ( ( order_1409979114der_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
% 2.70/1.58 = ( order_1409979114der_on @ A @ A3 @ R2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_156_labeled__graph_Ocase__distrib,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,V: $tType,L: $tType,H: A > B,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,Labeled_graph: labeled_graph @ L @ V] :
% 2.70/1.58 ( ( H @ ( labele1974067554_graph @ L @ V @ A @ F @ Labeled_graph ) )
% 2.70/1.58 = ( labele1974067554_graph @ L @ V @ B
% 2.70/1.58 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : ( H @ ( F @ X13 @ X24 ) )
% 2.70/1.58 @ Labeled_graph ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_157_labeled__graph_Ocase,axiom,
% 2.70/1.58 ! [L: $tType,A: $tType,V: $tType,F: ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) ) > ( set @ V ) > A,X1: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X22: set @ V] :
% 2.70/1.58 ( ( labele1974067554_graph @ L @ V @ A @ F @ ( labeled_LG @ L @ V @ X1 @ X22 ) )
% 2.70/1.58 = ( F @ X1 @ X22 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_158_edges__def,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType] :
% 2.70/1.58 ( ( labeled_edges @ L @ V )
% 2.70/1.58 = ( labele1974067554_graph @ L @ V @ ( set @ ( product_prod @ L @ ( product_prod @ V @ V ) ) )
% 2.70/1.58 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X13 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_159_vertices__def,axiom,
% 2.70/1.58 ! [V: $tType,L: $tType] :
% 2.70/1.58 ( ( labeled_vertices @ L @ V )
% 2.70/1.58 = ( labele1974067554_graph @ L @ V @ ( set @ V )
% 2.70/1.58 @ ^ [X13: set @ ( product_prod @ L @ ( product_prod @ V @ V ) ),X24: set @ V] : X24 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_160_lnear__order__on__empty,axiom,
% 2.70/1.58 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_161_labeled__graph_Ocase__eq__if,axiom,
% 2.70/1.58 ! [A: $tType,V: $tType,L: $tType] :
% 2.70/1.58 ( ( labele1974067554_graph @ L @ V @ A )
% 2.70/1.58 = ( ^ [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 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_162_congruent2__implies__congruent,axiom,
% 2.70/1.58 ! [B: $tType,C: $tType,A: $tType,A3: set @ A,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,A2: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R1 )
% 2.70/1.58 => ( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
% 2.70/1.58 => ( ( member @ A @ A2 @ A3 )
% 2.70/1.58 => ( equiv_congruent @ B @ C @ R22 @ ( F @ A2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_163_proj__def,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType] :
% 2.70/1.58 ( ( equiv_proj @ B @ A )
% 2.70/1.58 = ( ^ [R3: set @ ( product_prod @ B @ A ),X2: B] : ( image @ B @ A @ R3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_164_equiv__class__subset,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_165_order__refl,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( preorder @ A )
% 2.70/1.58 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_166_subset__antisym,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 2.70/1.58 => ( A3 = B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_167_subsetI,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ( ( member @ A @ X5 @ A3 )
% 2.70/1.58 => ( member @ A @ X5 @ B3 ) )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_168_converse__mono,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S2 ) )
% 2.70/1.58 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_169_empty__subsetI,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_170_subset__empty,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 = ( A3
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_171_insert__subset,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 2.70/1.58 = ( ( member @ A @ X3 @ B3 )
% 2.70/1.58 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_172_singleton__insert__inj__eq_H,axiom,
% 2.70/1.58 ! [A: $tType,A2: A,A3: set @ A,B2: A] :
% 2.70/1.58 ( ( ( insert @ A @ A2 @ A3 )
% 2.70/1.58 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( ( A2 = B2 )
% 2.70/1.58 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_173_singleton__insert__inj__eq,axiom,
% 2.70/1.58 ! [A: $tType,B2: A,A2: A,A3: set @ A] :
% 2.70/1.58 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 = ( insert @ A @ A2 @ A3 ) )
% 2.70/1.58 = ( ( A2 = B2 )
% 2.70/1.58 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_174_dual__order_Oantisym,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [B2: A,A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.58 => ( A2 = B2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_175_dual__order_Oeq__iff,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.70/1.58 = ( ^ [A4: A,B4: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.70/1.58 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_176_dual__order_Otrans,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [B2: A,A2: A,C2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.70/1.58 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_177_linorder__wlog,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( linorder @ A )
% 2.70/1.58 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.70/1.58 ( ! [A6: A,B8: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A6 @ B8 )
% 2.70/1.58 => ( P @ A6 @ B8 ) )
% 2.70/1.58 => ( ! [A6: A,B8: A] :
% 2.70/1.58 ( ( P @ B8 @ A6 )
% 2.70/1.58 => ( P @ A6 @ B8 ) )
% 2.70/1.58 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_178_dual__order_Orefl,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_179_order__trans,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( preorder @ A )
% 2.70/1.58 => ! [X3: A,Y4: A,Z2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.70/1.58 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_180_order__class_Oorder_Oantisym,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [A2: A,B2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.70/1.58 => ( A2 = B2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_181_ord__le__eq__trans,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( ord @ A )
% 2.70/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.58 => ( ( B2 = C2 )
% 2.70/1.58 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_182_ord__eq__le__trans,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( ord @ A )
% 2.70/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.70/1.58 ( ( A2 = B2 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.70/1.58 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_183_order__class_Oorder_Oeq__iff,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.70/1.58 = ( ^ [A4: A,B4: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.70/1.58 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_184_antisym__conv,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [Y4: A,X3: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ Y4 @ X3 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.70/1.58 = ( X3 = Y4 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_185_le__cases3,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( linorder @ A )
% 2.70/1.58 => ! [X3: A,Y4: A,Z2: A] :
% 2.70/1.58 ( ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.70/1.58 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 2.70/1.58 => ( ( ( ord_less_eq @ A @ Y4 @ X3 )
% 2.70/1.58 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.70/1.58 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 2.70/1.58 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 2.70/1.58 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 2.70/1.58 => ~ ( ord_less_eq @ A @ Y4 @ X3 ) )
% 2.70/1.58 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.70/1.58 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 2.70/1.58 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 2.70/1.58 => ~ ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_186_order_Otrans,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.70/1.58 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_187_le__cases,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( linorder @ A )
% 2.70/1.58 => ! [X3: A,Y4: A] :
% 2.70/1.58 ( ~ ( ord_less_eq @ A @ X3 @ Y4 )
% 2.70/1.58 => ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_188_eq__refl,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( preorder @ A )
% 2.70/1.58 => ! [X3: A,Y4: A] :
% 2.70/1.58 ( ( X3 = Y4 )
% 2.70/1.58 => ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_189_linear,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( linorder @ A )
% 2.70/1.58 => ! [X3: A,Y4: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.70/1.58 | ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_190_antisym,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ! [X3: A,Y4: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.70/1.58 => ( ( ord_less_eq @ A @ Y4 @ X3 )
% 2.70/1.58 => ( X3 = Y4 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_191_eq__iff,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order @ A )
% 2.70/1.58 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.70/1.58 = ( ^ [X2: A,Y: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.70/1.58 & ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_192_ord__le__eq__subst,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType] :
% 2.70/1.58 ( ( ( ord @ B )
% 2.70/1.58 & ( ord @ A ) )
% 2.70/1.58 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.58 => ( ( ( F @ B2 )
% 2.70/1.58 = C2 )
% 2.70/1.58 => ( ! [X5: A,Y3: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.70/1.58 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.70/1.58 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_193_ord__eq__le__subst,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType] :
% 2.70/1.58 ( ( ( ord @ B )
% 2.70/1.58 & ( ord @ A ) )
% 2.70/1.58 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.70/1.58 ( ( A2
% 2.70/1.58 = ( F @ B2 ) )
% 2.70/1.58 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.70/1.58 => ( ! [X5: B,Y3: B] :
% 2.70/1.58 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 2.70/1.58 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.70/1.58 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_194_order__subst2,axiom,
% 2.70/1.58 ! [A: $tType,C: $tType] :
% 2.70/1.58 ( ( ( order @ C )
% 2.70/1.58 & ( order @ A ) )
% 2.70/1.58 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.58 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.70/1.58 => ( ! [X5: A,Y3: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.70/1.58 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.70/1.58 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_195_order__subst1,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType] :
% 2.70/1.58 ( ( ( order @ B )
% 2.70/1.58 & ( order @ A ) )
% 2.70/1.58 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.70/1.58 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.70/1.58 => ( ! [X5: B,Y3: B] :
% 2.70/1.58 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 2.70/1.58 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.70/1.58 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_196_le__fun__def,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( ord @ B )
% 2.70/1.58 => ( ( ord_less_eq @ ( A > B ) )
% 2.70/1.58 = ( ^ [F4: A > B,G5: A > B] :
% 2.70/1.58 ! [X2: A] : ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( G5 @ X2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_197_le__funI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( ord @ B )
% 2.70/1.58 => ! [F: A > B,G: A > B] :
% 2.70/1.58 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 2.70/1.58 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_198_le__funE,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( ord @ B )
% 2.70/1.58 => ! [F: A > B,G: A > B,X3: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.70/1.58 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_199_le__funD,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType] :
% 2.70/1.58 ( ( ord @ B )
% 2.70/1.58 => ! [F: A > B,G: A > B,X3: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.70/1.58 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_200_Image__mono,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A8: set @ A,A3: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R2 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ A3 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A8 ) @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_201_Collect__subset,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,P: A > $o] :
% 2.70/1.58 ( ord_less_eq @ ( set @ A )
% 2.70/1.58 @ ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ A3 )
% 2.70/1.58 & ( P @ X2 ) ) )
% 2.70/1.58 @ A3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_202_Collect__mono__iff,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.70/1.58 = ( ! [X2: A] :
% 2.70/1.58 ( ( P @ X2 )
% 2.70/1.58 => ( Q @ X2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_203_set__eq__subset,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 2.70/1.58 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.70/1.58 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_204_subset__trans,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_205_Collect__mono,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 => ( Q @ X5 ) )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_206_subset__refl,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_207_subset__iff,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) )
% 2.70/1.58 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.70/1.58 ! [T2: A] :
% 2.70/1.58 ( ( member @ A @ T2 @ A7 )
% 2.70/1.58 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_208_equalityD2,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ( A3 = B3 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_209_equalityD1,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ( A3 = B3 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_210_subset__eq,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) )
% 2.70/1.58 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.70/1.58 ! [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ A7 )
% 2.70/1.58 => ( member @ A @ X2 @ B6 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_211_equalityE,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ( A3 = B3 )
% 2.70/1.58 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.70/1.58 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_212_subsetD,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.70/1.58 => ( ( member @ A @ C2 @ A3 )
% 2.70/1.58 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_213_in__mono,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.70/1.58 => ( ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ( member @ A @ X3 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_214_converse__subset__swap,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ A )] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S2 ) )
% 2.70/1.58 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_215_subrelI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ! [X5: A,Y3: B] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R2 )
% 2.70/1.58 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S2 ) )
% 2.70/1.58 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_216_bot_Oextremum__uniqueI,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order_bot @ A )
% 2.70/1.58 => ! [A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.70/1.58 => ( A2
% 2.70/1.58 = ( bot_bot @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_217_bot_Oextremum__unique,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order_bot @ A )
% 2.70/1.58 => ! [A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.70/1.58 = ( A2
% 2.70/1.58 = ( bot_bot @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_218_bot_Oextremum,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( order_bot @ A )
% 2.70/1.58 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_219_insert__mono,axiom,
% 2.70/1.58 ! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_220_subset__insert,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.70/1.58 ( ~ ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
% 2.70/1.58 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_221_subset__insertI,axiom,
% 2.70/1.58 ! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_222_subset__insertI2,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,B3: set @ A,B2: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_223_subset__singleton__iff,axiom,
% 2.70/1.58 ! [A: $tType,X4: set @ A,A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 = ( ( X4
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 | ( X4
% 2.70/1.58 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_224_subset__singletonD,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,X3: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.70/1.58 => ( ( A3
% 2.70/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.70/1.58 | ( A3
% 2.70/1.58 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_225_restrict__subsD,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,G2: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G2 ) @ ( labeled_edges @ A @ B @ ( restrict @ A @ B @ G2 ) ) )
% 2.70/1.58 => ( G2
% 2.70/1.58 = ( restrict @ A @ B @ G2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_226_congruent2I_H,axiom,
% 2.70/1.58 ! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 2.70/1.58 ( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R1 )
% 2.70/1.58 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z22 ) @ R22 )
% 2.70/1.58 => ( ( F @ Y12 @ Y23 )
% 2.70/1.58 = ( F @ Z1 @ Z22 ) ) ) )
% 2.70/1.58 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_227_congruent2D,axiom,
% 2.70/1.58 ! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z12: A,Y22: B,Z23: B] :
% 2.70/1.58 ( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R1 )
% 2.70/1.58 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R22 )
% 2.70/1.58 => ( ( F @ Y1 @ Y22 )
% 2.70/1.58 = ( F @ Z12 @ Z23 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_228_proj__iff,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y4: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 )
% 2.70/1.58 => ( ( ( equiv_proj @ A @ A @ R2 @ X3 )
% 2.70/1.58 = ( equiv_proj @ A @ A @ R2 @ Y4 ) )
% 2.70/1.58 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_229_refines__equiv__class__eq2,axiom,
% 2.70/1.58 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 2.70/1.58 => ( ( equiv_equiv @ A @ A3 @ R )
% 2.70/1.58 => ( ( equiv_equiv @ A @ A3 @ S )
% 2.70/1.58 => ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 = ( image @ A @ A @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_230_refines__equiv__class__eq,axiom,
% 2.70/1.58 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A2: A] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 2.70/1.58 => ( ( equiv_equiv @ A @ A3 @ R )
% 2.70/1.58 => ( ( equiv_equiv @ A @ A3 @ S )
% 2.70/1.58 => ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 = ( image @ A @ A @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_231_congruent2I,axiom,
% 2.70/1.58 ! [C: $tType,B: $tType,A: $tType,A12: set @ A,R1: set @ ( product_prod @ A @ A ),A23: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A12 @ R1 )
% 2.70/1.58 => ( ( equiv_equiv @ B @ A23 @ R22 )
% 2.70/1.58 => ( ! [Y3: A,Z3: A,W: B] :
% 2.70/1.58 ( ( member @ B @ W @ A23 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R1 )
% 2.70/1.58 => ( ( F @ Y3 @ W )
% 2.70/1.58 = ( F @ Z3 @ W ) ) ) )
% 2.70/1.58 => ( ! [Y3: B,Z3: B,W: A] :
% 2.70/1.58 ( ( member @ A @ W @ A12 )
% 2.70/1.58 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z3 ) @ R22 )
% 2.70/1.58 => ( ( F @ W @ Y3 )
% 2.70/1.58 = ( F @ W @ Z3 ) ) ) )
% 2.70/1.58 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_232_congruent2__commuteI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F: A > A > B] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ! [Y3: A,Z3: A] :
% 2.70/1.58 ( ( member @ A @ Y3 @ A3 )
% 2.70/1.58 => ( ( member @ A @ Z3 @ A3 )
% 2.70/1.58 => ( ( F @ Y3 @ Z3 )
% 2.70/1.58 = ( F @ Z3 @ Y3 ) ) ) )
% 2.70/1.58 => ( ! [Y3: A,Z3: A,W: A] :
% 2.70/1.58 ( ( member @ A @ W @ A3 )
% 2.70/1.58 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 2.70/1.58 => ( ( F @ W @ Y3 )
% 2.70/1.58 = ( F @ W @ Z3 ) ) ) )
% 2.70/1.58 => ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_233_subset__equiv__class,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.70/1.58 => ( ( member @ A @ B2 @ A3 )
% 2.70/1.58 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_234_insert__subsetI,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,A3: set @ A,X4: set @ A] :
% 2.70/1.58 ( ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ A3 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X4 ) @ A3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_235_subset__emptyI,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ A] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ~ ( member @ A @ X5 @ A3 )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_236_univ__commute,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F: A > B,X3: A] :
% 2.70/1.58 ( ( equiv_equiv @ A @ A3 @ R2 )
% 2.70/1.58 => ( ( equiv_congruent @ A @ B @ R2 @ F )
% 2.70/1.58 => ( ( member @ A @ X3 @ A3 )
% 2.70/1.58 => ( ( bNF_Greatest_univ @ A @ B @ F @ ( equiv_proj @ A @ A @ R2 @ X3 ) )
% 2.70/1.58 = ( F @ X3 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_237_pred__subset__eq,axiom,
% 2.70/1.58 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.70/1.58 ( ( ord_less_eq @ ( A > $o )
% 2.70/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ R )
% 2.70/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ S ) )
% 2.70/1.58 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_238_less__eq__set__def,axiom,
% 2.70/1.58 ! [A: $tType] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) )
% 2.70/1.58 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.70/1.58 ( ord_less_eq @ ( A > $o )
% 2.70/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ A7 )
% 2.70/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_239_pred__subset__eq2,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.70/1.58 ( ( ord_less_eq @ ( A > B > $o )
% 2.70/1.58 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
% 2.70/1.58 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) )
% 2.70/1.58 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_240_univ__def,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType] :
% 2.70/1.58 ( ( bNF_Greatest_univ @ B @ A )
% 2.70/1.58 = ( ^ [F4: B > A,X8: set @ B] :
% 2.70/1.58 ( F4
% 2.70/1.58 @ ( hilbert_Eps @ B
% 2.70/1.58 @ ^ [X2: B] : ( member @ B @ X2 @ X8 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_241_ssubst__Pair__rhs,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 2.70/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
% 2.70/1.58 => ( ( S3 = S2 )
% 2.70/1.58 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_242_subset__Collect__iff,axiom,
% 2.70/1.58 ! [A: $tType,B3: set @ A,A3: set @ A,P: A > $o] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B3
% 2.70/1.58 @ ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ A3 )
% 2.70/1.58 & ( P @ X2 ) ) ) )
% 2.70/1.58 = ( ! [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ B3 )
% 2.70/1.58 => ( P @ X2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_243_subset__CollectI,axiom,
% 2.70/1.58 ! [A: $tType,B3: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 2.70/1.58 => ( ! [X5: A] :
% 2.70/1.58 ( ( member @ A @ X5 @ B3 )
% 2.70/1.58 => ( ( Q @ X5 )
% 2.70/1.58 => ( P @ X5 ) ) )
% 2.70/1.58 => ( ord_less_eq @ ( set @ A )
% 2.70/1.58 @ ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ B3 )
% 2.70/1.58 & ( Q @ X2 ) ) )
% 2.70/1.58 @ ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ A3 )
% 2.70/1.58 & ( P @ X2 ) ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_244_Collect__restrict,axiom,
% 2.70/1.58 ! [A: $tType,X4: set @ A,P: A > $o] :
% 2.70/1.58 ( ord_less_eq @ ( set @ A )
% 2.70/1.58 @ ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ X4 )
% 2.70/1.58 & ( P @ X2 ) ) )
% 2.70/1.58 @ X4 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_245_prop__restrict,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,Z4: set @ A,X4: set @ A,P: A > $o] :
% 2.70/1.58 ( ( member @ A @ X3 @ Z4 )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 2.70/1.58 @ ( collect @ A
% 2.70/1.58 @ ^ [X2: A] :
% 2.70/1.58 ( ( member @ A @ X2 @ X4 )
% 2.70/1.58 & ( P @ X2 ) ) ) )
% 2.70/1.58 => ( P @ X3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_246_graph__unionI,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ ( product_prod @ B @ B ) ) ) @ ( labeled_edges @ A @ B @ G_1 ) @ ( labeled_edges @ A @ B @ G_2 ) )
% 2.70/1.58 => ( ( ord_less_eq @ ( set @ B ) @ ( labeled_vertices @ A @ B @ G_1 ) @ ( labeled_vertices @ A @ B @ G_2 ) )
% 2.70/1.58 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.70/1.58 = G_2 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_247_less__by__empty,axiom,
% 2.70/1.58 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
% 2.70/1.58 ( ( A3
% 2.70/1.58 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 2.70/1.58 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_248_predicate2I,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 2.70/1.58 ( ! [X5: A,Y3: B] :
% 2.70/1.58 ( ( P @ X5 @ Y3 )
% 2.70/1.58 => ( Q @ X5 @ Y3 ) )
% 2.70/1.58 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_249_predicate1I,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.58 ( ! [X5: A] :
% 2.70/1.58 ( ( P @ X5 )
% 2.70/1.58 => ( Q @ X5 ) )
% 2.70/1.58 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_250_graph__union__idemp_I1_J,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( graph_union @ A @ B @ A3 @ A3 )
% 2.70/1.58 = A3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_251_graph__union__idemp_I2_J,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( graph_union @ A @ B @ A3 @ ( graph_union @ A @ B @ A3 @ B3 ) )
% 2.70/1.58 = ( graph_union @ A @ B @ A3 @ B3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_252_graph__union__idemp_I3_J,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,A3: labeled_graph @ A @ B,B3: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( graph_union @ A @ B @ A3 @ ( graph_union @ A @ B @ B3 @ A3 ) )
% 2.70/1.58 = ( graph_union @ A @ B @ B3 @ A3 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_253_graph__union__preserves__restrict,axiom,
% 2.70/1.58 ! [B: $tType,A: $tType,G_1: labeled_graph @ A @ B,G_2: labeled_graph @ A @ B] :
% 2.70/1.58 ( ( G_1
% 2.70/1.58 = ( restrict @ A @ B @ G_1 ) )
% 2.70/1.58 => ( ( G_2
% 2.70/1.58 = ( restrict @ A @ B @ G_2 ) )
% 2.70/1.58 => ( ( graph_union @ A @ B @ G_1 @ G_2 )
% 2.70/1.58 = ( restrict @ A @ B @ ( graph_union @ A @ B @ G_1 @ G_2 ) ) ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_254_rev__predicate2D,axiom,
% 2.70/1.58 ! [A: $tType,B: $tType,P: A > B > $o,X3: A,Y4: B,Q: A > B > $o] :
% 2.70/1.58 ( ( P @ X3 @ Y4 )
% 2.70/1.58 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.70/1.58 => ( Q @ X3 @ Y4 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(fact_255_rev__predicate1D,axiom,
% 2.70/1.58 ! [A: $tType,P: A > $o,X3: A,Q: A > $o] :
% 2.70/1.58 ( ( P @ X3 )
% 2.70/1.58 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.70/1.58 => ( Q @ X3 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.70/1.58 ! [A9: $tType,A10: $tType] :
% 2.70/1.58 ( ( order_bot @ A10 )
% 2.70/1.58 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.70/1.58 ! [A9: $tType,A10: $tType] :
% 2.70/1.58 ( ( preorder @ A10 )
% 2.70/1.58 => ( preorder @ ( A9 > A10 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.70/1.58 ! [A9: $tType,A10: $tType] :
% 2.70/1.58 ( ( order @ A10 )
% 2.70/1.58 => ( order @ ( A9 > A10 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_fun___Orderings_Oord,axiom,
% 2.70/1.58 ! [A9: $tType,A10: $tType] :
% 2.70/1.58 ( ( ord @ A10 )
% 2.70/1.58 => ( ord @ ( A9 > A10 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_fun___Orderings_Obot,axiom,
% 2.70/1.58 ! [A9: $tType,A10: $tType] :
% 2.70/1.58 ( ( bot @ A10 )
% 2.70/1.58 => ( bot @ ( A9 > A10 ) ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
% 2.70/1.58 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.70/1.58 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 2.70/1.58 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 2.70/1.58 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
% 2.70/1.58 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
% 2.70/1.58 order_bot @ $o ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.70/1.58 preorder @ $o ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.70/1.58 linorder @ $o ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 2.70/1.58 order @ $o ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
% 2.70/1.58 ord @ $o ).
% 2.70/1.58
% 2.70/1.58 thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
% 2.70/1.58 bot @ $o ).
% 2.70/1.58
% 2.70/1.58 thf(help_If_3_1_T,axiom,
% 2.70/1.58 ! [P: $o] :
% 2.70/1.58 ( ( P = $true )
% 2.70/1.58 | ( P = $false ) ) ).
% 2.70/1.58
% 2.70/1.58 thf(help_If_2_1_T,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,Y4: A] :
% 2.70/1.58 ( ( if @ A @ $false @ X3 @ Y4 )
% 2.70/1.58 = Y4 ) ).
% 2.70/1.58
% 2.70/1.58 thf(help_If_1_1_T,axiom,
% 2.70/1.58 ! [A: $tType,X3: A,Y4: A] :
% 2.70/1.58 ( ( if @ A @ $true @ X3 @ Y4 )
% 2.70/1.58 = X3 ) ).
% 2.70/1.58
% 2.70/1.58 thf(conj_0,conjecture,
% 2.70/1.58 ( ( hilbert_Eps @ b
% 2.70/1.58 @ ^ [Y: b] : ( member @ b @ Y @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ x @ ( bot_bot @ ( set @ b ) ) ) ) ) )
% 2.70/1.58 = ( hilbert_Eps @ b
% 2.70/1.58 @ ^ [Y: b] : ( member @ b @ Y @ ( image @ b @ b @ ( getRel @ ( standard_Constant @ a ) @ b @ ( standard_S_Idt @ a ) @ g ) @ ( insert @ b @ y @ ( bot_bot @ ( set @ b ) ) ) ) ) ) ) ).
% 2.70/1.58 ------------------------
% 2.87/1.60 % (16745)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16637 for (2999ds/183Mi)
% 2.87/1.60 % (16748)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16637 for (2999ds/2Mi)
% 2.87/1.60 % (16747)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_16637 for (2999ds/27Mi)
% 2.87/1.60 % (16751)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16637 for (2999ds/18Mi)
% 2.87/1.60 % (16748)Instruction limit reached!
% 2.87/1.60 % (16748)------------------------------
% 2.87/1.60 % (16748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.60 % (16748)Termination reason: Unknown
% 2.87/1.60 % (16748)Termination phase: shuffling
% 2.87/1.60
% 2.87/1.60 % (16748)Memory used [KB]: 1407
% 2.87/1.60 % (16748)Time elapsed: 0.002 s
% 2.87/1.60 % (16748)Instructions burned: 2 (million)
% 2.87/1.60 % (16748)------------------------------
% 2.87/1.60 % (16748)------------------------------
% 2.87/1.60 % (16750)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_16637 for (2999ds/275Mi)
% 2.87/1.61 % (16751)Instruction limit reached!
% 2.87/1.61 % (16751)------------------------------
% 2.87/1.61 % (16751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.61 % (16751)Termination reason: Unknown
% 2.87/1.61 % (16751)Termination phase: shuffling
% 2.87/1.61
% 2.87/1.61 % (16751)Memory used [KB]: 1663
% 2.87/1.61 % (16751)Time elapsed: 0.007 s
% 2.87/1.61 % (16751)Instructions burned: 18 (million)
% 2.87/1.61 % (16751)------------------------------
% 2.87/1.61 % (16751)------------------------------
% 2.87/1.61 % (16746)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_16637 for (2999ds/4Mi)
% 2.87/1.61 % (16749)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_16637 for (2999ds/2Mi)
% 2.87/1.61 % (16747)Instruction limit reached!
% 2.87/1.61 % (16747)------------------------------
% 2.87/1.61 % (16747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.61 % (16747)Termination reason: Unknown
% 2.87/1.61 % (16747)Termination phase: Property scanning
% 2.87/1.61
% 2.87/1.61 % (16747)Memory used [KB]: 1918
% 2.87/1.61 % (16747)Time elapsed: 0.011 s
% 2.87/1.61 % (16747)Instructions burned: 30 (million)
% 2.87/1.61 % (16747)------------------------------
% 2.87/1.61 % (16747)------------------------------
% 2.87/1.61 % (16749)Instruction limit reached!
% 2.87/1.61 % (16749)------------------------------
% 2.87/1.61 % (16749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.61 % (16749)Termination reason: Unknown
% 2.87/1.61 % (16749)Termination phase: shuffling
% 2.87/1.61
% 2.87/1.61 % (16749)Memory used [KB]: 1407
% 2.87/1.61 % (16749)Time elapsed: 0.003 s
% 2.87/1.61 % (16749)Instructions burned: 2 (million)
% 2.87/1.61 % (16749)------------------------------
% 2.87/1.61 % (16749)------------------------------
% 2.87/1.61 % (16752)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16637 for (2999ds/3Mi)
% 2.87/1.61 % (16746)Instruction limit reached!
% 2.87/1.61 % (16746)------------------------------
% 2.87/1.61 % (16746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.61 % (16746)Termination reason: Unknown
% 2.87/1.61 % (16746)Termination phase: shuffling
% 2.87/1.61
% 2.87/1.61 % (16746)Memory used [KB]: 1407
% 2.87/1.61 % (16746)Time elapsed: 0.005 s
% 2.87/1.61 % (16746)Instructions burned: 5 (million)
% 2.87/1.61 % (16746)------------------------------
% 2.87/1.61 % (16746)------------------------------
% 2.87/1.61 % (16752)Instruction limit reached!
% 2.87/1.61 % (16752)------------------------------
% 2.87/1.61 % (16752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.61 % (16752)Termination reason: Unknown
% 2.87/1.61 % (16752)Termination phase: shuffling
% 2.87/1.61
% 2.87/1.61 % (16752)Memory used [KB]: 1407
% 2.87/1.61 % (16752)Time elapsed: 0.002 s
% 2.87/1.61 % (16752)Instructions burned: 3 (million)
% 2.87/1.61 % (16752)------------------------------
% 2.87/1.61 % (16752)------------------------------
% 2.87/1.62 % (16753)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_16637 for (2999ds/37Mi)
% 2.87/1.62 % (16754)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_16637 for (2999ds/15Mi)
% 2.87/1.62 % (16757)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16637 for (2999ds/7Mi)
% 2.87/1.62 % (16757)Instruction limit reached!
% 2.87/1.62 % (16757)------------------------------
% 2.87/1.62 % (16757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.62 % (16757)Termination reason: Unknown
% 2.87/1.62 % (16757)Termination phase: shuffling
% 2.87/1.62
% 2.87/1.62 % (16757)Memory used [KB]: 1535
% 2.87/1.62 % (16757)Time elapsed: 0.004 s
% 2.87/1.62 % (16757)Instructions burned: 8 (million)
% 2.87/1.62 % (16757)------------------------------
% 2.87/1.62 % (16757)------------------------------
% 2.87/1.63 % (16754)Instruction limit reached!
% 2.87/1.63 % (16754)------------------------------
% 2.87/1.63 % (16754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.63 % (16754)Termination reason: Unknown
% 2.87/1.63 % (16754)Termination phase: shuffling
% 2.87/1.63
% 2.87/1.63 % (16754)Memory used [KB]: 1663
% 2.87/1.63 % (16754)Time elapsed: 0.007 s
% 2.87/1.63 % (16754)Instructions burned: 17 (million)
% 2.87/1.63 % (16754)------------------------------
% 2.87/1.63 % (16754)------------------------------
% 2.87/1.63 % (16753)Instruction limit reached!
% 2.87/1.63 % (16753)------------------------------
% 2.87/1.63 % (16753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.63 % (16753)Termination reason: Unknown
% 2.87/1.63 % (16753)Termination phase: shuffling
% 2.87/1.63
% 2.87/1.63 % (16753)Memory used [KB]: 2174
% 2.87/1.63 % (16753)Time elapsed: 0.012 s
% 2.87/1.63 % (16753)Instructions burned: 39 (million)
% 2.87/1.63 % (16753)------------------------------
% 2.87/1.63 % (16753)------------------------------
% 2.87/1.63 % (16755)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_16637 for (2999ds/3Mi)
% 2.87/1.63 % (16756)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_16637 for (2999ds/1041Mi)
% 2.87/1.63 % (16755)Instruction limit reached!
% 2.87/1.63 % (16755)------------------------------
% 2.87/1.63 % (16755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.63 % (16755)Termination reason: Unknown
% 2.87/1.63 % (16755)Termination phase: shuffling
% 2.87/1.63
% 2.87/1.63 % (16755)Memory used [KB]: 1407
% 2.87/1.63 % (16755)Time elapsed: 0.004 s
% 2.87/1.63 % (16755)Instructions burned: 3 (million)
% 2.87/1.63 % (16755)------------------------------
% 2.87/1.63 % (16755)------------------------------
% 2.87/1.63 % (16750)First to succeed.
% 2.87/1.63 % (16750)Refutation found. Thanks to Tanya!
% 2.87/1.63 % SZS status Theorem for DTF2THF_16637
% 2.87/1.63 % SZS output start Proof for DTF2THF_16637
% 2.87/1.63 thf(type_def_6, type, set: ($tType) > $tType).
% 2.87/1.63 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 2.87/1.63 thf(type_def_8, type, standard_Constant: ($tType) > $tType).
% 2.87/1.63 thf(type_def_9, type, labeled_graph: ($tType * $tType) > $tType).
% 2.87/1.63 thf(type_def_10, type, a: $tType).
% 2.87/1.63 thf(type_def_11, type, b: $tType).
% 2.87/1.63 thf(func_def_0, type, standard_Constant: ($tType) > $tType).
% 2.87/1.63 thf(func_def_1, type, labeled_graph: ($tType * $tType) > $tType).
% 2.87/1.63 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 2.87/1.63 thf(func_def_3, type, set: ($tType) > $tType).
% 2.87/1.63 thf(func_def_4, type, b: $tType).
% 2.87/1.63 thf(func_def_5, type, a: $tType).
% 2.87/1.63 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_10, type, linorder: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_11, type, preorder: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_12, type, order_bot: !>[X0: $tType]:($o)).
% 2.87/1.63 thf(func_def_13, type, bNF_Greatest_univ: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 2.87/1.63 thf(func_def_14, type, equiv_congruent: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.87/1.63 thf(func_def_15, type, equiv_congruent2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1 > X2) > $o)).
% 2.87/1.63 thf(func_def_16, type, equiv_equiv: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.87/1.63 thf(func_def_17, type, equiv_proj: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > set @ X1)).
% 2.87/1.63 thf(func_def_18, type, hilbert_Eps: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_19, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.87/1.63 thf(func_def_20, type, standard_S_Idt: !>[X0: $tType]:(standard_Constant @ X0)).
% 2.87/1.63 thf(func_def_21, type, getRel: !>[X0: $tType, X1: $tType]:(X0 > labeled_graph @ X0 @ X1 > set @ product_prod @ X1 @ X1)).
% 2.87/1.63 thf(func_def_22, type, graph_homomorphism: !>[X0: $tType, X1: $tType, X2: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X2 > set @ product_prod @ X1 @ X2 > $o)).
% 2.87/1.63 thf(func_def_23, type, graph_union: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 2.87/1.63 thf(func_def_24, type, labeled_LG: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ product_prod @ X1 @ X1 > set @ X1 > labeled_graph @ X0 @ X1)).
% 2.87/1.63 thf(func_def_25, 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)).
% 2.87/1.63 thf(func_def_26, type, labeled_edges: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ product_prod @ X0 @ product_prod @ X1 @ X1)).
% 2.87/1.63 thf(func_def_27, type, labeled_vertices: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > set @ X1)).
% 2.87/1.63 thf(func_def_28, type, restrict: !>[X0: $tType, X1: $tType]:(labeled_graph @ X0 @ X1 > labeled_graph @ X0 @ X1)).
% 2.87/1.63 thf(func_def_29, type, order_1409979114der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.87/1.63 thf(func_def_30, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.87/1.63 thf(func_def_31, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.87/1.63 thf(func_def_32, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.87/1.63 thf(func_def_33, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.87/1.63 thf(func_def_34, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 2.87/1.63 thf(func_def_35, type, converse: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X0)).
% 2.87/1.63 thf(func_def_36, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.87/1.63 thf(func_def_37, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.87/1.63 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.87/1.63 thf(func_def_39, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.87/1.63 thf(func_def_40, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.87/1.63 thf(func_def_41, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.87/1.63 thf(func_def_42, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.87/1.63 thf(func_def_43, type, g: labeled_graph @ standard_Constant @ a @ b).
% 2.87/1.63 thf(func_def_44, type, p: b > b > $o).
% 2.87/1.63 thf(func_def_45, type, f: b > b).
% 2.87/1.63 thf(func_def_46, type, x: b).
% 2.87/1.63 thf(func_def_47, type, y: b).
% 2.87/1.63 thf(func_def_63, type, sP0: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > $o)).
% 2.87/1.63 thf(func_def_64, type, sP1: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > $o)).
% 2.87/1.63 thf(func_def_65, type, sK2: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.63 thf(func_def_66, type, sK3: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_67, type, sK4: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.87/1.63 thf(func_def_68, type, sK5: !>[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)).
% 2.87/1.63 thf(func_def_69, type, sK6: !>[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)).
% 2.87/1.63 thf(func_def_70, type, sK7: !>[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)).
% 2.87/1.63 thf(func_def_71, type, sK8: !>[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)).
% 2.87/1.63 thf(func_def_72, type, sK9: !>[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)).
% 2.87/1.63 thf(func_def_73, type, sK10: !>[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)).
% 2.87/1.63 thf(func_def_74, type, sK11: !>[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)).
% 2.87/1.63 thf(func_def_75, type, sK12: !>[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)).
% 2.87/1.63 thf(func_def_76, type, sK13: !>[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)).
% 2.87/1.63 thf(func_def_77, type, sK14: !>[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)).
% 2.87/1.63 thf(func_def_78, type, sK15: !>[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)).
% 2.87/1.63 thf(func_def_79, type, sK16: !>[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)).
% 2.87/1.63 thf(func_def_80, type, sK17: !>[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)).
% 2.87/1.63 thf(func_def_81, type, sK18: !>[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)).
% 2.87/1.63 thf(func_def_82, type, sK19: !>[X0: $tType]:(set @ X0 > X0)).
% 2.87/1.63 thf(func_def_83, type, sK20: !>[X0: $tType]:(set @ X0 > X0)).
% 2.87/1.63 thf(func_def_84, type, sK21: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.63 thf(func_def_85, type, sK22: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_86, type, sK23: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_87, type, sK24: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_88, type, sK25: !>[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)).
% 2.87/1.63 thf(func_def_89, type, sK26: !>[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)).
% 2.87/1.63 thf(func_def_90, type, sK27: !>[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)).
% 2.87/1.63 thf(func_def_91, type, sK28: !>[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)).
% 2.87/1.63 thf(func_def_92, type, sK29: !>[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)).
% 2.87/1.63 thf(func_def_93, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.87/1.63 thf(func_def_94, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.87/1.63 thf(func_def_95, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.87/1.63 thf(func_def_96, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.87/1.63 thf(func_def_97, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.87/1.63 thf(func_def_98, type, sK35: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.87/1.63 thf(func_def_99, type, sK36: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.87/1.63 thf(func_def_100, type, sK37: !>[X0: $tType, X1: $tType]:(set @ X1 > set @ product_prod @ X1 @ X0 > X0 > X1)).
% 2.87/1.63 thf(func_def_101, type, sK38: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 2.87/1.63 thf(func_def_102, type, sK39: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_103, type, sK40: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.87/1.63 thf(func_def_104, type, sK41: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.87/1.63 thf(func_def_105, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.87/1.63 thf(func_def_106, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.63 thf(func_def_107, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.63 thf(func_def_108, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_109, type, sK46: !>[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)).
% 2.87/1.63 thf(func_def_110, type, sK47: !>[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)).
% 2.87/1.63 thf(func_def_111, type, sK48: !>[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)).
% 2.87/1.63 thf(func_def_112, type, sK49: !>[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)).
% 2.87/1.63 thf(func_def_113, type, sK50: !>[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)).
% 2.87/1.63 thf(func_def_114, type, sK51: !>[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)).
% 2.87/1.63 thf(func_def_115, type, sK52: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_116, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.87/1.63 thf(func_def_117, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.87/1.63 thf(func_def_118, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.87/1.63 thf(func_def_119, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.87/1.63 thf(func_def_120, type, sK57: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_121, type, sK58: !>[X0: $tType]:(set @ X0 > X0)).
% 2.87/1.63 thf(func_def_122, type, sK59: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.87/1.63 thf(func_def_123, type, sK60: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.87/1.63 thf(func_def_124, type, sK61: !>[X0: $tType]:(set @ X0 > X0)).
% 2.87/1.63 thf(func_def_125, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.63 thf(func_def_126, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.63 thf(func_def_127, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_128, type, sK65: !>[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)).
% 2.87/1.63 thf(func_def_129, type, sK66: !>[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)).
% 2.87/1.63 thf(func_def_130, type, sK67: !>[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)).
% 2.87/1.63 thf(func_def_131, type, sK68: !>[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)).
% 2.87/1.63 thf(func_def_132, type, sK69: !>[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)).
% 2.87/1.63 thf(func_def_133, type, sK70: !>[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)).
% 2.87/1.63 thf(func_def_134, type, sK71: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_135, type, sK72: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.87/1.63 thf(func_def_136, type, sK73: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_137, type, sK74: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.87/1.63 thf(func_def_138, type, sK75: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > set @ X0)).
% 2.87/1.63 thf(func_def_139, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.87/1.63 thf(func_def_140, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.87/1.63 thf(func_def_141, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.87/1.63 thf(func_def_142, type, sK79: !>[X0: $tType, X1: $tType]:(set @ X1 > set @ product_prod @ X1 @ X0 > X0 > X1)).
% 2.87/1.63 thf(func_def_143, type, sK80: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.87/1.63 thf(func_def_144, type, sK81: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.87/1.63 thf(func_def_146, type, ph83: !>[X0: $tType]:(X0)).
% 2.87/1.63 thf(f1192,plain,(
% 2.87/1.63 $false),
% 2.87/1.63 inference(trivial_inequality_removal,[status(thm)],[f1191])).
% 2.87/1.63 thf(f1191,plain,(
% 2.87/1.63 ((hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))) != (hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))))),
% 2.87/1.63 inference(forward_demodulation,[status(thm)],[f1006,f1152])).
% 2.87/1.63 thf(f1152,plain,(
% 2.87/1.63 ((image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b)) = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b)))),
% 2.87/1.63 inference(cnf_transformation,[status(thm)],[f1])).
% 2.87/1.63 thf(f1,axiom,(
% 2.87/1.63 ((image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b)) = (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b)))),
% 2.87/1.63 file('/export/starexec/sandbox/tmp/tmp.27O21xWGGm/DTF2THF_16637.p',fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062)).
% 2.87/1.63 thf(f1006,plain,(
% 2.87/1.63 ((hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b))))) != (hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))))),
% 2.87/1.63 inference(cnf_transformation,[status(thm)],[f733])).
% 2.87/1.63 thf(f733,plain,(
% 2.87/1.63 ((hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b))))) != (hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))))),
% 2.87/1.63 inference(flattening,[status(thm)],[f598])).
% 2.87/1.63 thf(f598,plain,(
% 2.87/1.63 ~((hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b))))) = (hilbert_Eps @ b @ (^[Y0 : b]: (member @ b @ Y0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))))),
% 2.87/1.63 inference(fool_elimination,[status(thm)],[f597])).
% 2.87/1.63 thf(f597,plain,(
% 2.87/1.63 ~((hilbert_Eps @ b @ (^[X0 : b] : (member @ b @ X0 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))) = (hilbert_Eps @ b @ (^[X1 : b] : (member @ b @ X1 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b))))))),
% 2.87/1.63 inference(rectify,[status(thm)],[f277])).
% 2.87/1.63 thf(f277,negated_conjecture,(
% 2.87/1.63 ~((hilbert_Eps @ b @ (^[X8 : b] : (member @ b @ X8 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))) = (hilbert_Eps @ b @ (^[X8 : b] : (member @ b @ X8 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b))))))),
% 2.87/1.63 inference(negated_conjecture,[status(cth)],[f276])).
% 2.87/1.63 thf(f276,conjecture,(
% 2.87/1.63 ((hilbert_Eps @ b @ (^[X8 : b] : (member @ b @ X8 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ x @ bot_bot @ set @ b))))) = (hilbert_Eps @ b @ (^[X8 : b] : (member @ b @ X8 @ (image @ b @ b @ (getRel @ standard_Constant @ a @ b @ standard_S_Idt @ a @ g) @ (insert @ b @ y @ bot_bot @ set @ b))))))),
% 2.87/1.63 file('/export/starexec/sandbox/tmp/tmp.27O21xWGGm/DTF2THF_16637.p',conj_0)).
% 2.87/1.63 % SZS output end Proof for DTF2THF_16637
% 2.87/1.63 % (16750)------------------------------
% 2.87/1.63 % (16750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.63 % (16750)Termination reason: Refutation
% 2.87/1.63
% 2.87/1.63 % (16750)Memory used [KB]: 6908
% 2.87/1.63 % (16750)Time elapsed: 0.032 s
% 2.87/1.63 % (16750)Instructions burned: 88 (million)
% 2.87/1.63 % (16750)------------------------------
% 2.87/1.63 % (16750)------------------------------
% 2.87/1.63 % (16744)Success in time 0.038 s
%------------------------------------------------------------------------------