TPTP Problem File: ITP175^1.p
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%------------------------------------------------------------------------------
% File : ITP175^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_201__5389132_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : StandardRules/prob_201__5389132_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 639 ( 162 unt; 283 typ; 0 def)
% Number of atoms : 976 ( 325 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3193 ( 85 ~; 8 |; 70 &;2641 @)
% ( 0 <=>; 389 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 77 ( 76 usr)
% Number of type conns : 370 ( 370 >; 0 *; 0 +; 0 <<)
% Number of symbols : 208 ( 207 usr; 19 con; 0-3 aty)
% Number of variables : 1046 ( 33 ^; 996 !; 17 ?;1046 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:20.179
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member670041670od_b_b: produc130774173od_b_b > set_Pr2028709629od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member178477328od_b_b: produc300546791od_b_b > set_Pr1391123143od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
member1769525082_a_nat: produc1230257585_a_nat > set_Pr2054759953_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
member1432657168od_b_b: produc1554726631od_b_b > set_Pr1229594311od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1829919504od_b_b: produc1921243879od_b_b > set_Pr454079687od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member442676358od_b_b: produc401211613od_b_b > set_Pr1549208893od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1023210704od_b_b: produc1114535079od_b_b > set_Pr1271557767od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
member817785114_a_nat: produc776320369_a_nat > set_Pr1626978257_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
member1379268944_a_nat: produc2106791463_a_nat > set_Pr2072721159_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
member962139856od_b_b: produc1053464231od_b_b > set_Pr406163591od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1694913744od_b_b: produc197046951od_b_b > set_Pr963356295od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J,type,
member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
member1072844688_a_nat: produc1869574247_a_nat > set_Pr1045675335_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1009730320od_b_b: produc51693287od_b_b > set_Pr773647559od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_Itf__b_J_J,type,
member1199885584_set_b: produc150548199_set_b > set_Pr1463111367_set_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J,type,
member763519277_a_nat: produc126142486_a_nat > set_Pr2010644300_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J,type,
member692218050_a_nat: produc207618585_a_nat > set_Pr352593785_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J,type,
member101535418_a_nat: set_Pr1912306449_a_nat > set_se480333169_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member1050118384od_b_b: set_Product_prod_b_b > set_se668355239od_b_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_G,type,
g: labeled_graph_a_b ).
thf(sy_v_I,type,
i: a ).
thf(sy_v_l,type,
l: a ).
thf(sy_v_y____,type,
y: b ).
thf(sy_v_z____,type,
z: b ).
% Relevant facts (355)
thf(fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062,axiom,
( ( refl_on_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) )
& ! [X: b,Y: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( getRel_a_b @ i @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ X ) @ ( getRel_a_b @ i @ g ) ) )
& ( trans_b @ ( getRel_a_b @ i @ g ) ) ) ).
% \<open>refl_on (vertices G) (getRel I G) \<and> (\<forall>x y. (x, y) \<in> getRel I G \<longrightarrow> (y, x) \<in> getRel I G) \<and> trans (getRel I G)\<close>
thf(fact_1_aI,axiom,
member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ) ).
% aI
thf(fact_2_g,axiom,
( g
= ( restrict_a_b @ g ) ) ).
% g
thf(fact_3_m,axiom,
maintained_a_nat_b @ ( standa414753934rule_a @ i @ l ) @ g ).
% m
thf(fact_4_eq,axiom,
equiv_equiv_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) ).
% eq
thf(fact_5_prod_Oinject,axiom,
! [X1: labeled_graph_a_nat,X2: labeled_graph_a_nat,Y1: labeled_graph_a_nat,Y2: labeled_graph_a_nat] :
( ( ( produc2000589545_a_nat @ X1 @ X2 )
= ( produc2000589545_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_prod_Oinject,axiom,
! [X1: product_prod_b_b,X2: product_prod_b_b,Y1: product_prod_b_b,Y2: product_prod_b_b] :
( ( ( produc546497367od_b_b @ X1 @ X2 )
= ( produc546497367od_b_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: produc598811825_a_nat,X2: produc598811825_a_nat,Y1: produc598811825_a_nat,Y2: produc598811825_a_nat] :
( ( ( produc410388695_a_nat @ X1 @ X2 )
= ( produc410388695_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_prod_Oinject,axiom,
! [X1: a,X2: product_prod_b_b,Y1: a,Y2: product_prod_b_b] :
( ( ( produc2135284224od_b_b @ X1 @ X2 )
= ( produc2135284224od_b_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_prod_Oinject,axiom,
! [X1: b,X2: b,Y1: b,Y2: b] :
( ( ( product_Pair_b_b @ X1 @ X2 )
= ( product_Pair_b_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: labeled_graph_a_nat,B: labeled_graph_a_nat,A2: labeled_graph_a_nat,B2: labeled_graph_a_nat] :
( ( ( produc2000589545_a_nat @ A @ B )
= ( produc2000589545_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A2: product_prod_b_b,B2: product_prod_b_b] :
( ( ( produc546497367od_b_b @ A @ B )
= ( produc546497367od_b_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: produc598811825_a_nat,B: produc598811825_a_nat,A2: produc598811825_a_nat,B2: produc598811825_a_nat] :
( ( ( produc410388695_a_nat @ A @ B )
= ( produc410388695_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
! [A: a,B: product_prod_b_b,A2: a,B2: product_prod_b_b] :
( ( ( produc2135284224od_b_b @ A @ B )
= ( produc2135284224od_b_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
! [A: b,B: b,A2: b,B2: b] :
( ( ( product_Pair_b_b @ A @ B )
= ( product_Pair_b_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_15_surj__pair,axiom,
! [P: produc598811825_a_nat] :
? [X3: labeled_graph_a_nat,Y3: labeled_graph_a_nat] :
( P
= ( produc2000589545_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_16_surj__pair,axiom,
! [P: produc1168037607od_b_b] :
? [X3: product_prod_b_b,Y3: product_prod_b_b] :
( P
= ( produc546497367od_b_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_17_surj__pair,axiom,
! [P: produc1099860071_a_nat] :
? [X3: produc598811825_a_nat,Y3: produc598811825_a_nat] :
( P
= ( produc410388695_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_18_surj__pair,axiom,
! [P: produc1133639312od_b_b] :
? [X3: a,Y3: product_prod_b_b] :
( P
= ( produc2135284224od_b_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_19_surj__pair,axiom,
! [P: product_prod_b_b] :
? [X3: b,Y3: b] :
( P
= ( product_Pair_b_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_20_prod__cases,axiom,
! [P2: produc598811825_a_nat > $o,P: produc598811825_a_nat] :
( ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] : ( P2 @ ( produc2000589545_a_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_21_prod__cases,axiom,
! [P2: produc1168037607od_b_b > $o,P: produc1168037607od_b_b] :
( ! [A3: product_prod_b_b,B3: product_prod_b_b] : ( P2 @ ( produc546497367od_b_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_22_prod__cases,axiom,
! [P2: produc1099860071_a_nat > $o,P: produc1099860071_a_nat] :
( ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_23_prod__cases,axiom,
! [P2: produc1133639312od_b_b > $o,P: produc1133639312od_b_b] :
( ! [A3: a,B3: product_prod_b_b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_24_prod__cases,axiom,
! [P2: product_prod_b_b > $o,P: product_prod_b_b] :
( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_25_Pair__inject,axiom,
! [A: labeled_graph_a_nat,B: labeled_graph_a_nat,A2: labeled_graph_a_nat,B2: labeled_graph_a_nat] :
( ( ( produc2000589545_a_nat @ A @ B )
= ( produc2000589545_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_26_Pair__inject,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A2: product_prod_b_b,B2: product_prod_b_b] :
( ( ( produc546497367od_b_b @ A @ B )
= ( produc546497367od_b_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_27_Pair__inject,axiom,
! [A: produc598811825_a_nat,B: produc598811825_a_nat,A2: produc598811825_a_nat,B2: produc598811825_a_nat] :
( ( ( produc410388695_a_nat @ A @ B )
= ( produc410388695_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_28_Pair__inject,axiom,
! [A: a,B: product_prod_b_b,A2: a,B2: product_prod_b_b] :
( ( ( produc2135284224od_b_b @ A @ B )
= ( produc2135284224od_b_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_29_Pair__inject,axiom,
! [A: b,B: b,A2: b,B2: b] :
( ( ( product_Pair_b_b @ A @ B )
= ( product_Pair_b_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_30_prod__cases3,axiom,
! [Y4: produc1168037607od_b_b] :
~ ! [A3: product_prod_b_b,B3: b,C: b] :
( Y4
!= ( produc546497367od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_31_prod__cases3,axiom,
! [Y4: produc1099860071_a_nat] :
~ ! [A3: produc598811825_a_nat,B3: labeled_graph_a_nat,C: labeled_graph_a_nat] :
( Y4
!= ( produc410388695_a_nat @ A3 @ ( produc2000589545_a_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_32_prod__cases3,axiom,
! [Y4: produc1133639312od_b_b] :
~ ! [A3: a,B3: b,C: b] :
( Y4
!= ( produc2135284224od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_33_old_Oprod_Oinducts,axiom,
! [P2: produc598811825_a_nat > $o,Prod: produc598811825_a_nat] :
( ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] : ( P2 @ ( produc2000589545_a_nat @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_34_old_Oprod_Oinducts,axiom,
! [P2: produc1168037607od_b_b > $o,Prod: produc1168037607od_b_b] :
( ! [A3: product_prod_b_b,B3: product_prod_b_b] : ( P2 @ ( produc546497367od_b_b @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_35_old_Oprod_Oinducts,axiom,
! [P2: produc1099860071_a_nat > $o,Prod: produc1099860071_a_nat] :
( ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_36_old_Oprod_Oinducts,axiom,
! [P2: produc1133639312od_b_b > $o,Prod: produc1133639312od_b_b] :
( ! [A3: a,B3: product_prod_b_b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_37_old_Oprod_Oinducts,axiom,
! [P2: product_prod_b_b > $o,Prod: product_prod_b_b] :
( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_38_old_Oprod_Oexhaust,axiom,
! [Y4: produc598811825_a_nat] :
~ ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] :
( Y4
!= ( produc2000589545_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_39_old_Oprod_Oexhaust,axiom,
! [Y4: produc1168037607od_b_b] :
~ ! [A3: product_prod_b_b,B3: product_prod_b_b] :
( Y4
!= ( produc546497367od_b_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_40_old_Oprod_Oexhaust,axiom,
! [Y4: produc1099860071_a_nat] :
~ ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] :
( Y4
!= ( produc410388695_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_41_old_Oprod_Oexhaust,axiom,
! [Y4: produc1133639312od_b_b] :
~ ! [A3: a,B3: product_prod_b_b] :
( Y4
!= ( produc2135284224od_b_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_42_old_Oprod_Oexhaust,axiom,
! [Y4: product_prod_b_b] :
~ ! [A3: b,B3: b] :
( Y4
!= ( product_Pair_b_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_43_prod__induct3,axiom,
! [P2: produc1168037607od_b_b > $o,X4: produc1168037607od_b_b] :
( ! [A3: product_prod_b_b,B3: b,C: b] : ( P2 @ ( produc546497367od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct3
thf(fact_44_prod__induct3,axiom,
! [P2: produc1099860071_a_nat > $o,X4: produc1099860071_a_nat] :
( ! [A3: produc598811825_a_nat,B3: labeled_graph_a_nat,C: labeled_graph_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ ( produc2000589545_a_nat @ B3 @ C ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct3
thf(fact_45_prod__induct3,axiom,
! [P2: produc1133639312od_b_b > $o,X4: produc1133639312od_b_b] :
( ! [A3: a,B3: b,C: b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct3
thf(fact_46_vertices__restrict,axiom,
! [G: labeled_graph_a_b] :
( ( labeled_vertices_a_b @ ( restrict_a_b @ G ) )
= ( labeled_vertices_a_b @ G ) ) ).
% vertices_restrict
thf(fact_47_getRel__dom_I1_J,axiom,
! [G: labeled_graph_a_b,A: b,B: b,L: a] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel_a_b @ L @ G ) )
=> ( member_b @ A @ ( labeled_vertices_a_b @ G ) ) ) ) ).
% getRel_dom(1)
thf(fact_48_getRel__dom_I2_J,axiom,
! [G: labeled_graph_a_b,A: b,B: b,L: a] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel_a_b @ L @ G ) )
=> ( member_b @ B @ ( labeled_vertices_a_b @ G ) ) ) ) ).
% getRel_dom(2)
thf(fact_49_transitive__rule,axiom,
! [G: labele553847424_a_nat,L: a] :
( ( G
= ( restri1808752350_a_nat @ G ) )
=> ( ( mainta535344367_a_nat @ ( standa390703484rule_a @ L ) @ G )
=> ( trans_1002055497_a_nat @ ( getRel1657740669_a_nat @ L @ G ) ) ) ) ).
% transitive_rule
thf(fact_50_transitive__rule,axiom,
! [G: labeled_graph_a_b,L: a] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ( maintained_a_nat_b @ ( standa390703484rule_a @ L ) @ G )
=> ( trans_b @ ( getRel_a_b @ L @ G ) ) ) ) ).
% transitive_rule
thf(fact_51_maintained__refl,axiom,
! [R: labeled_graph_a_nat,G: labeled_graph_a_b] : ( maintained_a_nat_b @ ( produc2000589545_a_nat @ R @ R ) @ G ) ).
% maintained_refl
thf(fact_52_reflexivity__rule,axiom,
! [G: labele488176869od_b_b,L: a] :
( ( G
= ( restri431340145od_b_b @ G ) )
=> ( ( mainta552012832od_b_b @ ( standa1329480012rule_a @ L ) @ G )
=> ( refl_o2134473190od_b_b @ ( labele240831545od_b_b @ G ) @ ( getRel1183868946od_b_b @ L @ G ) ) ) ) ).
% reflexivity_rule
thf(fact_53_reflexivity__rule,axiom,
! [G: labele1586099759_a_nat,L: a] :
( ( G
= ( restri147273531_a_nat @ G ) )
=> ( ( mainta1627462506_a_nat @ ( standa1329480012rule_a @ L ) @ G )
=> ( refl_o206400048_a_nat @ ( labele198153475_a_nat @ G ) @ ( getRel1248627292_a_nat @ L @ G ) ) ) ) ).
% reflexivity_rule
thf(fact_54_reflexivity__rule,axiom,
! [G: labele553847424_a_nat,L: a] :
( ( G
= ( restri1808752350_a_nat @ G ) )
=> ( ( mainta535344367_a_nat @ ( standa1329480012rule_a @ L ) @ G )
=> ( refl_o1018941609_a_nat @ ( labele953802518_a_nat @ G ) @ ( getRel1657740669_a_nat @ L @ G ) ) ) ) ).
% reflexivity_rule
thf(fact_55_reflexivity__rule,axiom,
! [G: labeled_graph_a_b,L: a] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ( maintained_a_nat_b @ ( standa1329480012rule_a @ L ) @ G )
=> ( refl_on_b @ ( labeled_vertices_a_b @ G ) @ ( getRel_a_b @ L @ G ) ) ) ) ).
% reflexivity_rule
thf(fact_56_restrict__idemp,axiom,
! [X4: labeled_graph_a_b] :
( ( restrict_a_b @ ( restrict_a_b @ X4 ) )
= ( restrict_a_b @ X4 ) ) ).
% restrict_idemp
thf(fact_57_refl__on__domain,axiom,
! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,A: produc1168037607od_b_b,B: produc1168037607od_b_b] :
( ( refl_o1941611494od_b_b @ A4 @ R2 )
=> ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ A @ B ) @ R2 )
=> ( ( member1652792080od_b_b @ A @ A4 )
& ( member1652792080od_b_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_58_refl__on__domain,axiom,
! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,A: set_Pr1912306449_a_nat,B: set_Pr1912306449_a_nat] :
( ( refl_o730481808_a_nat @ A4 @ R2 )
=> ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ A @ B ) @ R2 )
=> ( ( member101535418_a_nat @ A @ A4 )
& ( member101535418_a_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_59_refl__on__domain,axiom,
! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
( ( refl_o1899808198od_b_b @ A4 @ R2 )
=> ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ A @ B ) @ R2 )
=> ( ( member1050118384od_b_b @ A @ A4 )
& ( member1050118384od_b_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_60_refl__on__domain,axiom,
! [A4: set_set_b,R2: set_Pr1463111367_set_b,A: set_b,B: set_b] :
( ( refl_on_set_b @ A4 @ R2 )
=> ( ( member1199885584_set_b @ ( produc1403713879_set_b @ A @ B ) @ R2 )
=> ( ( member_set_b @ A @ A4 )
& ( member_set_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_61_refl__on__domain,axiom,
! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,A: labeled_graph_a_nat,B: labeled_graph_a_nat] :
( ( refl_o1018941609_a_nat @ A4 @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ A @ B ) @ R2 )
=> ( ( member154916863_a_nat @ A @ A4 )
& ( member154916863_a_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_62_refl__on__domain,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,A: produc598811825_a_nat,B: produc598811825_a_nat] :
( ( refl_o206400048_a_nat @ A4 @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ A @ B ) @ R2 )
=> ( ( member327031898_a_nat @ A @ A4 )
& ( member327031898_a_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_63_refl__on__domain,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A4 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
=> ( ( member1285940496od_b_b @ A @ A4 )
& ( member1285940496od_b_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_64_refl__on__domain,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
( ( refl_on_b @ A4 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A4 )
& ( member_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_65_refl__onD2,axiom,
! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,X4: produc1168037607od_b_b,Y4: produc1168037607od_b_b] :
( ( refl_o1941611494od_b_b @ A4 @ R2 )
=> ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1652792080od_b_b @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_66_refl__onD2,axiom,
! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,X4: set_Pr1912306449_a_nat,Y4: set_Pr1912306449_a_nat] :
( ( refl_o730481808_a_nat @ A4 @ R2 )
=> ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ X4 @ Y4 ) @ R2 )
=> ( member101535418_a_nat @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_67_refl__onD2,axiom,
! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,X4: set_Product_prod_b_b,Y4: set_Product_prod_b_b] :
( ( refl_o1899808198od_b_b @ A4 @ R2 )
=> ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1050118384od_b_b @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_68_refl__onD2,axiom,
! [A4: set_set_b,R2: set_Pr1463111367_set_b,X4: set_b,Y4: set_b] :
( ( refl_on_set_b @ A4 @ R2 )
=> ( ( member1199885584_set_b @ ( produc1403713879_set_b @ X4 @ Y4 ) @ R2 )
=> ( member_set_b @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_69_refl__onD2,axiom,
! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat] :
( ( refl_o1018941609_a_nat @ A4 @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
=> ( member154916863_a_nat @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_70_refl__onD2,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
( ( refl_o206400048_a_nat @ A4 @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
=> ( member327031898_a_nat @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_71_refl__onD2,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A4 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1285940496od_b_b @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_72_refl__onD2,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
( ( refl_on_b @ A4 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
=> ( member_b @ Y4 @ A4 ) ) ) ).
% refl_onD2
thf(fact_73_transD,axiom,
! [R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat,Z: labeled_graph_a_nat] :
( ( trans_1002055497_a_nat @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y4 @ Z ) @ R2 )
=> ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_74_transD,axiom,
! [R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b,Z: product_prod_b_b] :
( ( trans_1808857926od_b_b @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y4 @ Z ) @ R2 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_75_transD,axiom,
! [R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat,Z: produc598811825_a_nat] :
( ( trans_444496784_a_nat @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y4 @ Z ) @ R2 )
=> ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_76_transD,axiom,
! [R2: set_Product_prod_b_b,X4: b,Y4: b,Z: b] :
( ( trans_b @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_77_transE,axiom,
! [R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat,Z: labeled_graph_a_nat] :
( ( trans_1002055497_a_nat @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y4 @ Z ) @ R2 )
=> ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_78_transE,axiom,
! [R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b,Z: product_prod_b_b] :
( ( trans_1808857926od_b_b @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y4 @ Z ) @ R2 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_79_transE,axiom,
! [R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat,Z: produc598811825_a_nat] :
( ( trans_444496784_a_nat @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y4 @ Z ) @ R2 )
=> ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_80_transE,axiom,
! [R2: set_Product_prod_b_b,X4: b,Y4: b,Z: b] :
( ( trans_b @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_81_transI,axiom,
! [R2: set_Pr1912306449_a_nat] :
( ! [X3: labeled_graph_a_nat,Y3: labeled_graph_a_nat,Z2: labeled_graph_a_nat] :
( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X3 @ Y3 ) @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y3 @ Z2 ) @ R2 )
=> ( member327031898_a_nat @ ( produc2000589545_a_nat @ X3 @ Z2 ) @ R2 ) ) )
=> ( trans_1002055497_a_nat @ R2 ) ) ).
% transI
thf(fact_82_transI,axiom,
! [R2: set_Pr2007082183od_b_b] :
( ! [X3: product_prod_b_b,Y3: product_prod_b_b,Z2: product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Y3 ) @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ Z2 ) @ R2 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Z2 ) @ R2 ) ) )
=> ( trans_1808857926od_b_b @ R2 ) ) ).
% transI
thf(fact_83_transI,axiom,
! [R2: set_Pr2074180935_a_nat] :
( ! [X3: produc598811825_a_nat,Y3: produc598811825_a_nat,Z2: produc598811825_a_nat] :
( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X3 @ Y3 ) @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y3 @ Z2 ) @ R2 )
=> ( member1542696848_a_nat @ ( produc410388695_a_nat @ X3 @ Z2 ) @ R2 ) ) )
=> ( trans_444496784_a_nat @ R2 ) ) ).
% transI
thf(fact_84_transI,axiom,
! [R2: set_Product_prod_b_b] :
( ! [X3: b,Y3: b,Z2: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z2 ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Z2 ) @ R2 ) ) )
=> ( trans_b @ R2 ) ) ).
% transI
thf(fact_85_Relation_Otrans__def,axiom,
( trans_1002055497_a_nat
= ( ^ [R3: set_Pr1912306449_a_nat] :
! [X5: labeled_graph_a_nat,Y5: labeled_graph_a_nat,Z3: labeled_graph_a_nat] :
( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X5 @ Y5 ) @ R3 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y5 @ Z3 ) @ R3 )
=> ( member327031898_a_nat @ ( produc2000589545_a_nat @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% Relation.trans_def
thf(fact_86_Relation_Otrans__def,axiom,
( trans_1808857926od_b_b
= ( ^ [R3: set_Pr2007082183od_b_b] :
! [X5: product_prod_b_b,Y5: product_prod_b_b,Z3: product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X5 @ Y5 ) @ R3 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y5 @ Z3 ) @ R3 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% Relation.trans_def
thf(fact_87_Relation_Otrans__def,axiom,
( trans_444496784_a_nat
= ( ^ [R3: set_Pr2074180935_a_nat] :
! [X5: produc598811825_a_nat,Y5: produc598811825_a_nat,Z3: produc598811825_a_nat] :
( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X5 @ Y5 ) @ R3 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y5 @ Z3 ) @ R3 )
=> ( member1542696848_a_nat @ ( produc410388695_a_nat @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% Relation.trans_def
thf(fact_88_Relation_Otrans__def,axiom,
( trans_b
= ( ^ [R3: set_Product_prod_b_b] :
! [X5: b,Y5: b,Z3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ Z3 ) @ R3 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% Relation.trans_def
thf(fact_89_refl__onD,axiom,
! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,A: produc1168037607od_b_b] :
( ( refl_o1941611494od_b_b @ A4 @ R2 )
=> ( ( member1652792080od_b_b @ A @ A4 )
=> ( member1829919504od_b_b @ ( produc1403229015od_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_90_refl__onD,axiom,
! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,A: set_Pr1912306449_a_nat] :
( ( refl_o730481808_a_nat @ A4 @ R2 )
=> ( ( member101535418_a_nat @ A @ A4 )
=> ( member1072844688_a_nat @ ( produc1767291095_a_nat @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_91_refl__onD,axiom,
! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,A: set_Product_prod_b_b] :
( ( refl_o1899808198od_b_b @ A4 @ R2 )
=> ( ( member1050118384od_b_b @ A @ A4 )
=> ( member1009730320od_b_b @ ( produc1027169111od_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_92_refl__onD,axiom,
! [A4: set_set_b,R2: set_Pr1463111367_set_b,A: set_b] :
( ( refl_on_set_b @ A4 @ R2 )
=> ( ( member_set_b @ A @ A4 )
=> ( member1199885584_set_b @ ( produc1403713879_set_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_93_refl__onD,axiom,
! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,A: labeled_graph_a_nat] :
( ( refl_o1018941609_a_nat @ A4 @ R2 )
=> ( ( member154916863_a_nat @ A @ A4 )
=> ( member327031898_a_nat @ ( produc2000589545_a_nat @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_94_refl__onD,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,A: produc598811825_a_nat] :
( ( refl_o206400048_a_nat @ A4 @ R2 )
=> ( ( member327031898_a_nat @ A @ A4 )
=> ( member1542696848_a_nat @ ( produc410388695_a_nat @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_95_refl__onD,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A4 @ R2 )
=> ( ( member1285940496od_b_b @ A @ A4 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_96_refl__onD,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,A: b] :
( ( refl_on_b @ A4 @ R2 )
=> ( ( member_b @ A @ A4 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_97_refl__onD1,axiom,
! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,X4: produc1168037607od_b_b,Y4: produc1168037607od_b_b] :
( ( refl_o1941611494od_b_b @ A4 @ R2 )
=> ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1652792080od_b_b @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_98_refl__onD1,axiom,
! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,X4: set_Pr1912306449_a_nat,Y4: set_Pr1912306449_a_nat] :
( ( refl_o730481808_a_nat @ A4 @ R2 )
=> ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ X4 @ Y4 ) @ R2 )
=> ( member101535418_a_nat @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_99_refl__onD1,axiom,
! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,X4: set_Product_prod_b_b,Y4: set_Product_prod_b_b] :
( ( refl_o1899808198od_b_b @ A4 @ R2 )
=> ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1050118384od_b_b @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_100_refl__onD1,axiom,
! [A4: set_set_b,R2: set_Pr1463111367_set_b,X4: set_b,Y4: set_b] :
( ( refl_on_set_b @ A4 @ R2 )
=> ( ( member1199885584_set_b @ ( produc1403713879_set_b @ X4 @ Y4 ) @ R2 )
=> ( member_set_b @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_101_refl__onD1,axiom,
! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat] :
( ( refl_o1018941609_a_nat @ A4 @ R2 )
=> ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
=> ( member154916863_a_nat @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_102_refl__onD1,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
( ( refl_o206400048_a_nat @ A4 @ R2 )
=> ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
=> ( member327031898_a_nat @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_103_refl__onD1,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A4 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_104_refl__onD1,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
( ( refl_on_b @ A4 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
=> ( member_b @ X4 @ A4 ) ) ) ).
% refl_onD1
thf(fact_105_mA,axiom,
! [X: produc598811825_a_nat] :
( ( member327031898_a_nat @ X @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ i ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ i ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ i ) @ bot_bo2066531197_a_nat ) ) ) )
=> ( maintained_a_nat_b @ X @ g ) ) ).
% mA
thf(fact_106_preorder__on__def,axiom,
( order_1195372542od_b_b
= ( ^ [A5: set_Product_prod_b_b,R3: set_Pr2007082183od_b_b] :
( ( refl_o2134473190od_b_b @ A5 @ R3 )
& ( trans_1808857926od_b_b @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_107_preorder__on__def,axiom,
( order_145089608_a_nat
= ( ^ [A5: set_Pr1912306449_a_nat,R3: set_Pr2074180935_a_nat] :
( ( refl_o206400048_a_nat @ A5 @ R3 )
& ( trans_444496784_a_nat @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_108_preorder__on__def,axiom,
( order_989282449_a_nat
= ( ^ [A5: set_la1108361502_a_nat,R3: set_Pr1912306449_a_nat] :
( ( refl_o1018941609_a_nat @ A5 @ R3 )
& ( trans_1002055497_a_nat @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_109_preorder__on__def,axiom,
( order_preorder_on_b
= ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
( ( refl_on_b @ A5 @ R3 )
& ( trans_b @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_110_on__triple,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
= ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_111_on__triple,axiom,
! [L1: a,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: a,V3: b,V4: b,R: set_Pr471017666_nat_b] :
( ( member442676358od_b_b @ ( produc179745357od_b_b @ ( produc1110849866_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1993240081at_b_a @ R ) )
= ( ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V1 @ V3 ) @ R )
& ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_112_on__triple,axiom,
! [L1: a,V1: product_prod_b_b,V2: product_prod_b_b,L2: a,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
( ( member1023210704od_b_b @ ( produc1116925335od_b_b @ ( produc690925568od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1764442556_b_b_a @ R ) )
= ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V1 @ V3 ) @ R )
& ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_113_on__triple,axiom,
! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: b,V4: b,R: set_Product_prod_b_b] :
( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri111564221od_b_b @ R ) )
= ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_114_on__triple,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
= ( ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R )
& ( member763519277_a_nat @ ( produc1311571856_a_nat @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_115_on__triple,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
= ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R )
& ( member641857272od_b_b @ ( produc1257047359od_b_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_116_on__triple,axiom,
! [L1: product_prod_b_b,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr471017666_nat_b] :
( ( member670041670od_b_b @ ( produc926922637od_b_b @ ( produc90796321_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1706508026od_b_b @ R ) )
= ( ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V1 @ V3 ) @ R )
& ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_117_on__triple,axiom,
! [L1: product_prod_b_b,V1: product_prod_b_b,V2: product_prod_b_b,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
( ( member178477328od_b_b @ ( produc885982551od_b_b @ ( produc1850071895od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri97791909od_b_b @ R ) )
= ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V1 @ V3 ) @ R )
& ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_118_on__triple,axiom,
! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
( ( member1769525082_a_nat @ ( produc1783652257_a_nat @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc90796321_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1837931688od_b_b @ R ) )
= ( ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R )
& ( member763519277_a_nat @ ( produc1311571856_a_nat @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_119_on__triple,axiom,
! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
( ( member1432657168od_b_b @ ( produc962089303od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1850071895od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri927008213od_b_b @ R ) )
= ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R )
& ( member641857272od_b_b @ ( produc1257047359od_b_b @ V2 @ V4 ) @ R )
& ( L1 = L2 ) ) ) ).
% on_triple
thf(fact_120_maintained__then__fin__maintained,axiom,
! [R: produc598811825_a_nat,G: labeled_graph_a_b] :
( ( maintained_a_nat_b @ R @ G )
=> ( fin_ma603645735_nat_b @ R @ G ) ) ).
% maintained_then_fin_maintained
thf(fact_121_refl__trans__impl__idempotent,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
( ( refl_o2134473190od_b_b @ A4 @ R2 )
=> ( ( trans_1808857926od_b_b @ R2 )
=> ( idempo371692375od_b_b @ R2 ) ) ) ).
% refl_trans_impl_idempotent
thf(fact_122_refl__trans__impl__idempotent,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
( ( refl_o206400048_a_nat @ A4 @ R2 )
=> ( ( trans_444496784_a_nat @ R2 )
=> ( idempo365306145_a_nat @ R2 ) ) ) ).
% refl_trans_impl_idempotent
thf(fact_123_refl__trans__impl__idempotent,axiom,
! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat] :
( ( refl_o1018941609_a_nat @ A4 @ R2 )
=> ( ( trans_1002055497_a_nat @ R2 )
=> ( idempo848975736_a_nat @ R2 ) ) ) ).
% refl_trans_impl_idempotent
thf(fact_124_refl__trans__impl__idempotent,axiom,
! [A4: set_b,R2: set_Product_prod_b_b] :
( ( refl_on_b @ A4 @ R2 )
=> ( ( trans_b @ R2 )
=> ( idempotent_b @ R2 ) ) ) ).
% refl_trans_impl_idempotent
thf(fact_125_equivalence,axiom,
! [G: labele1586099759_a_nat,I: a] :
( ( G
= ( restri147273531_a_nat @ G ) )
=> ( ! [X3: produc598811825_a_nat] :
( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
=> ( mainta1627462506_a_nat @ X3 @ G ) )
=> ( equiv_281634663_a_nat @ ( labele198153475_a_nat @ G ) @ ( getRel1248627292_a_nat @ I @ G ) ) ) ) ).
% equivalence
thf(fact_126_equivalence,axiom,
! [G: labele488176869od_b_b,I: a] :
( ( G
= ( restri431340145od_b_b @ G ) )
=> ( ! [X3: produc598811825_a_nat] :
( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
=> ( mainta552012832od_b_b @ X3 @ G ) )
=> ( equiv_1125628061od_b_b @ ( labele240831545od_b_b @ G ) @ ( getRel1183868946od_b_b @ I @ G ) ) ) ) ).
% equivalence
thf(fact_127_equivalence,axiom,
! [G: labeled_graph_a_b,I: a] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ! [X3: produc598811825_a_nat] :
( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
=> ( maintained_a_nat_b @ X3 @ G ) )
=> ( equiv_equiv_b @ ( labeled_vertices_a_b @ G ) @ ( getRel_a_b @ I @ G ) ) ) ) ).
% equivalence
thf(fact_128_mem__Collect__eq,axiom,
! [A: produc1168037607od_b_b,P2: produc1168037607od_b_b > $o] :
( ( member1652792080od_b_b @ A @ ( collec1231052114od_b_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
! [A: set_Pr1912306449_a_nat,P2: set_Pr1912306449_a_nat > $o] :
( ( member101535418_a_nat @ A @ ( collec971438076_a_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
! [A: set_Product_prod_b_b,P2: set_Product_prod_b_b > $o] :
( ( member1050118384od_b_b @ A @ ( collec1543202610od_b_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
! [A: set_b,P2: set_b > $o] :
( ( member_set_b @ A @ ( collect_set_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
! [A: b,P2: b > $o] :
( ( member_b @ A @ ( collect_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A: produc598811825_a_nat,P2: produc598811825_a_nat > $o] :
( ( member327031898_a_nat @ A @ ( collec978177436_a_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A: product_prod_b_b,P2: product_prod_b_b > $o] :
( ( member1285940496od_b_b @ A @ ( collec1481886546od_b_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A4: set_Pr2007082183od_b_b] :
( ( collec1231052114od_b_b
@ ^ [X5: produc1168037607od_b_b] : ( member1652792080od_b_b @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A4: set_se480333169_a_nat] :
( ( collec971438076_a_nat
@ ^ [X5: set_Pr1912306449_a_nat] : ( member101535418_a_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A4: set_se668355239od_b_b] :
( ( collec1543202610od_b_b
@ ^ [X5: set_Product_prod_b_b] : ( member1050118384od_b_b @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A4: set_set_b] :
( ( collect_set_b
@ ^ [X5: set_b] : ( member_set_b @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A4: set_b] :
( ( collect_b
@ ^ [X5: b] : ( member_b @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ( collec978177436_a_nat
@ ^ [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A4: set_Product_prod_b_b] :
( ( collec1481886546od_b_b
@ ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_142_Collect__cong,axiom,
! [P2: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
( ! [X3: product_prod_b_b] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1481886546od_b_b @ P2 )
= ( collec1481886546od_b_b @ Q ) ) ) ).
% Collect_cong
thf(fact_143_Collect__cong,axiom,
! [P2: produc598811825_a_nat > $o,Q: produc598811825_a_nat > $o] :
( ! [X3: produc598811825_a_nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec978177436_a_nat @ P2 )
= ( collec978177436_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_144_semantics__in__vertices_I1_J,axiom,
! [A4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
( ( A4
= ( restrict_a_b @ A4 ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
=> ( member_b @ A @ ( labeled_vertices_a_b @ A4 ) ) ) ) ).
% semantics_in_vertices(1)
thf(fact_145_semantics__in__vertices_I2_J,axiom,
! [A4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
( ( A4
= ( restrict_a_b @ A4 ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
=> ( member_b @ B @ ( labeled_vertices_a_b @ A4 ) ) ) ) ).
% semantics_in_vertices(2)
thf(fact_146_refl__on__singleton,axiom,
! [X4: produc1099860071_a_nat] : ( refl_o2093460070_a_nat @ ( insert354391607_a_nat @ X4 @ bot_bo1803599795_a_nat ) @ ( insert1772835511_a_nat @ ( produc404908887_a_nat @ X4 @ X4 ) @ bot_bo896299827_a_nat ) ) ).
% refl_on_singleton
thf(fact_147_refl__on__singleton,axiom,
! [X4: product_prod_b_b] : ( refl_o2134473190od_b_b @ ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X4 @ X4 ) @ bot_bo1174586163od_b_b ) ) ).
% refl_on_singleton
thf(fact_148_refl__on__singleton,axiom,
! [X4: produc1133639312od_b_b] : ( refl_o686103183od_b_b @ ( insert1655137376od_b_b @ X4 @ bot_bo519661404od_b_b ) @ ( insert428421239od_b_b @ ( produc1037687191od_b_b @ X4 @ X4 ) @ bot_bo1856908531od_b_b ) ) ).
% refl_on_singleton
thf(fact_149_refl__on__singleton,axiom,
! [X4: set_Pr1912306449_a_nat] : ( refl_o730481808_a_nat @ ( insert1220735201_a_nat @ X4 @ bot_bo1746593245_a_nat ) @ ( insert885128759_a_nat @ ( produc1767291095_a_nat @ X4 @ X4 ) @ bot_bo2142478259_a_nat ) ) ).
% refl_on_singleton
thf(fact_150_refl__on__singleton,axiom,
! [X4: b] : ( refl_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% refl_on_singleton
thf(fact_151_refl__on__singleton,axiom,
! [X4: labeled_graph_a_nat] : ( refl_o1018941609_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% refl_on_singleton
thf(fact_152_refl__on__singleton,axiom,
! [X4: produc598811825_a_nat] : ( refl_o206400048_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% refl_on_singleton
thf(fact_153_preorder__on__empty,axiom,
order_265664679od_b_b @ bot_bo519661404od_b_b @ bot_bo1856908531od_b_b ).
% preorder_on_empty
thf(fact_154_preorder__on__empty,axiom,
order_640573096_a_nat @ bot_bo1746593245_a_nat @ bot_bo2142478259_a_nat ).
% preorder_on_empty
thf(fact_155_preorder__on__empty,axiom,
order_preorder_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% preorder_on_empty
thf(fact_156_preorder__on__empty,axiom,
order_989282449_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% preorder_on_empty
thf(fact_157_preorder__on__empty,axiom,
order_145089608_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% preorder_on_empty
thf(fact_158_refl__on__empty,axiom,
refl_o2134473190od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% refl_on_empty
thf(fact_159_refl__on__empty,axiom,
refl_o686103183od_b_b @ bot_bo519661404od_b_b @ bot_bo1856908531od_b_b ).
% refl_on_empty
thf(fact_160_refl__on__empty,axiom,
refl_o730481808_a_nat @ bot_bo1746593245_a_nat @ bot_bo2142478259_a_nat ).
% refl_on_empty
thf(fact_161_refl__on__empty,axiom,
refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% refl_on_empty
thf(fact_162_refl__on__empty,axiom,
refl_o1018941609_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% refl_on_empty
thf(fact_163_refl__on__empty,axiom,
refl_o206400048_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% refl_on_empty
thf(fact_164_trans__singleton,axiom,
! [A: product_prod_b_b] : ( trans_1808857926od_b_b @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ A @ A ) @ bot_bo1174586163od_b_b ) ) ).
% trans_singleton
thf(fact_165_trans__singleton,axiom,
! [A: produc598811825_a_nat] : ( trans_444496784_a_nat @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ A @ A ) @ bot_bo1803599795_a_nat ) ) ).
% trans_singleton
thf(fact_166_trans__singleton,axiom,
! [A: b] : ( trans_b @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ A @ A ) @ bot_bo1343651123od_b_b ) ) ).
% trans_singleton
thf(fact_167_trans__singleton,axiom,
! [A: labeled_graph_a_nat] : ( trans_1002055497_a_nat @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ A @ A ) @ bot_bo2066531197_a_nat ) ) ).
% trans_singleton
thf(fact_168_idempotent__impl__trans,axiom,
! [R2: set_Pr1912306449_a_nat] :
( ( idempo848975736_a_nat @ R2 )
=> ( trans_1002055497_a_nat @ R2 ) ) ).
% idempotent_impl_trans
thf(fact_169_idempotent__impl__trans,axiom,
! [R2: set_Product_prod_b_b] :
( ( idempotent_b @ R2 )
=> ( trans_b @ R2 ) ) ).
% idempotent_impl_trans
thf(fact_170_trans__empty,axiom,
trans_b @ bot_bo1343651123od_b_b ).
% trans_empty
thf(fact_171_trans__empty,axiom,
trans_1002055497_a_nat @ bot_bo2066531197_a_nat ).
% trans_empty
thf(fact_172_on__tripleD_I1_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_173_on__tripleD_I1_J,axiom,
! [L1: a,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: a,V3: b,V4: b,R: set_Pr471017666_nat_b] :
( ( member442676358od_b_b @ ( produc179745357od_b_b @ ( produc1110849866_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1993240081at_b_a @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_174_on__tripleD_I1_J,axiom,
! [L1: a,V1: product_prod_b_b,V2: product_prod_b_b,L2: a,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
( ( member1023210704od_b_b @ ( produc1116925335od_b_b @ ( produc690925568od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1764442556_b_b_a @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_175_on__tripleD_I1_J,axiom,
! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: b,V4: b,R: set_Product_prod_b_b] :
( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri111564221od_b_b @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_176_on__tripleD_I1_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_177_on__tripleD_I1_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_178_on__tripleD_I1_J,axiom,
! [L1: product_prod_b_b,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr471017666_nat_b] :
( ( member670041670od_b_b @ ( produc926922637od_b_b @ ( produc90796321_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1706508026od_b_b @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_179_on__tripleD_I1_J,axiom,
! [L1: product_prod_b_b,V1: product_prod_b_b,V2: product_prod_b_b,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
( ( member178477328od_b_b @ ( produc885982551od_b_b @ ( produc1850071895od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri97791909od_b_b @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_180_on__tripleD_I1_J,axiom,
! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
( ( member1769525082_a_nat @ ( produc1783652257_a_nat @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc90796321_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1837931688od_b_b @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_181_on__tripleD_I1_J,axiom,
! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
( ( member1432657168od_b_b @ ( produc962089303od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1850071895od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri927008213od_b_b @ R ) )
=> ( L2 = L1 ) ) ).
% on_tripleD(1)
thf(fact_182_on__tripleD_I2_J,axiom,
! [L1: produc598811825_a_nat,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: produc598811825_a_nat,V3: produc598811825_a_nat,V4: produc598811825_a_nat,R: set_Pr1870121716_a_nat] :
( ( member428378182_a_nat @ ( produc1645191821_a_nat @ ( produc410388695_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc639906189_a_nat @ L2 @ ( produc410388695_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1766439058_a_nat @ R ) )
=> ( member2103088853_a_nat @ ( produc2092742640_a_nat @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_183_on__tripleD_I2_J,axiom,
! [L1: produc598811825_a_nat,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: produc598811825_a_nat,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr1912306449_a_nat] :
( ( member121756432_a_nat @ ( produc404908887_a_nat @ ( produc410388695_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc410388695_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri232494965_a_nat @ R ) )
=> ( member327031898_a_nat @ ( produc2000589545_a_nat @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_184_on__tripleD_I2_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
=> ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_185_on__tripleD_I2_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
=> ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_186_on__tripleD_I2_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: produc598811825_a_nat,V4: produc598811825_a_nat,R: set_Pr352593785_a_nat] :
( ( member1379268944_a_nat @ ( produc797892631_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1904940160_a_nat @ L2 @ ( produc410388695_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1794757026_nat_a @ R ) )
=> ( member692218050_a_nat @ ( produc255168265_a_nat @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_187_on__tripleD_I2_J,axiom,
! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R ) ) ).
% on_tripleD(2)
thf(fact_188_singletonI,axiom,
! [A: product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% singletonI
thf(fact_189_singletonI,axiom,
! [A: produc598811825_a_nat] : ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) ) ).
% singletonI
thf(fact_190_nonempty__rule,axiom,
! [G: labeled_graph_a_b] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ( maintained_a_nat_b @ standa879863265rule_a @ G )
= ( ( labeled_vertices_a_b @ G )
!= bot_bot_set_b ) ) ) ).
% nonempty_rule
thf(fact_191_insertCI,axiom,
! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
( ( ~ ( member327031898_a_nat @ A @ B4 )
=> ( A = B ) )
=> ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_192_insertCI,axiom,
! [A: product_prod_b_b,B4: set_Product_prod_b_b,B: product_prod_b_b] :
( ( ~ ( member1285940496od_b_b @ A @ B4 )
=> ( A = B ) )
=> ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B4 ) ) ) ).
% insertCI
thf(fact_193_insert__iff,axiom,
! [A: produc598811825_a_nat,B: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ A4 ) )
= ( ( A = B )
| ( member327031898_a_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_194_insert__iff,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A4 ) )
= ( ( A = B )
| ( member1285940496od_b_b @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_195_insert__absorb2,axiom,
! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ X4 @ A4 ) )
= ( insert2058869377_a_nat @ X4 @ A4 ) ) ).
% insert_absorb2
thf(fact_196_empty__iff,axiom,
! [C2: product_prod_b_b] :
~ ( member1285940496od_b_b @ C2 @ bot_bo1343651123od_b_b ) ).
% empty_iff
thf(fact_197_empty__iff,axiom,
! [C2: produc598811825_a_nat] :
~ ( member327031898_a_nat @ C2 @ bot_bo2066531197_a_nat ) ).
% empty_iff
thf(fact_198_all__not__in__conv,axiom,
! [A4: set_Product_prod_b_b] :
( ( ! [X5: product_prod_b_b] :
~ ( member1285940496od_b_b @ X5 @ A4 ) )
= ( A4 = bot_bo1343651123od_b_b ) ) ).
% all_not_in_conv
thf(fact_199_all__not__in__conv,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ( ! [X5: produc598811825_a_nat] :
~ ( member327031898_a_nat @ X5 @ A4 ) )
= ( A4 = bot_bo2066531197_a_nat ) ) ).
% all_not_in_conv
thf(fact_200_Collect__empty__eq,axiom,
! [P2: produc598811825_a_nat > $o] :
( ( ( collec978177436_a_nat @ P2 )
= bot_bo2066531197_a_nat )
= ( ! [X5: produc598811825_a_nat] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_201_empty__Collect__eq,axiom,
! [P2: produc598811825_a_nat > $o] :
( ( bot_bo2066531197_a_nat
= ( collec978177436_a_nat @ P2 ) )
= ( ! [X5: produc598811825_a_nat] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_202_ex__in__conv,axiom,
! [A4: set_Product_prod_b_b] :
( ( ? [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A4 ) )
= ( A4 != bot_bo1343651123od_b_b ) ) ).
% ex_in_conv
thf(fact_203_ex__in__conv,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ( ? [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ A4 ) )
= ( A4 != bot_bo2066531197_a_nat ) ) ).
% ex_in_conv
thf(fact_204_equals0I,axiom,
! [A4: set_Product_prod_b_b] :
( ! [Y3: product_prod_b_b] :
~ ( member1285940496od_b_b @ Y3 @ A4 )
=> ( A4 = bot_bo1343651123od_b_b ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ! [Y3: produc598811825_a_nat] :
~ ( member327031898_a_nat @ Y3 @ A4 )
=> ( A4 = bot_bo2066531197_a_nat ) ) ).
% equals0I
thf(fact_206_equals0D,axiom,
! [A4: set_Product_prod_b_b,A: product_prod_b_b] :
( ( A4 = bot_bo1343651123od_b_b )
=> ~ ( member1285940496od_b_b @ A @ A4 ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A4: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
( ( A4 = bot_bo2066531197_a_nat )
=> ~ ( member327031898_a_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_208_emptyE,axiom,
! [A: product_prod_b_b] :
~ ( member1285940496od_b_b @ A @ bot_bo1343651123od_b_b ) ).
% emptyE
thf(fact_209_emptyE,axiom,
! [A: produc598811825_a_nat] :
~ ( member327031898_a_nat @ A @ bot_bo2066531197_a_nat ) ).
% emptyE
thf(fact_210_mk__disjoint__insert,axiom,
! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( member327031898_a_nat @ A @ A4 )
=> ? [B5: set_Pr1912306449_a_nat] :
( ( A4
= ( insert2058869377_a_nat @ A @ B5 ) )
& ~ ( member327031898_a_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_211_mk__disjoint__insert,axiom,
! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ A4 )
=> ? [B5: set_Product_prod_b_b] :
( ( A4
= ( insert1952693431od_b_b @ A @ B5 ) )
& ~ ( member1285940496od_b_b @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_212_insert__commute,axiom,
! [X4: produc598811825_a_nat,Y4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ A4 ) )
= ( insert2058869377_a_nat @ Y4 @ ( insert2058869377_a_nat @ X4 @ A4 ) ) ) ).
% insert_commute
thf(fact_213_insert__eq__iff,axiom,
! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B: produc598811825_a_nat,B4: set_Pr1912306449_a_nat] :
( ~ ( member327031898_a_nat @ A @ A4 )
=> ( ~ ( member327031898_a_nat @ B @ B4 )
=> ( ( ( insert2058869377_a_nat @ A @ A4 )
= ( insert2058869377_a_nat @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C3: set_Pr1912306449_a_nat] :
( ( A4
= ( insert2058869377_a_nat @ B @ C3 ) )
& ~ ( member327031898_a_nat @ B @ C3 )
& ( B4
= ( insert2058869377_a_nat @ A @ C3 ) )
& ~ ( member327031898_a_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_214_insert__eq__iff,axiom,
! [A: product_prod_b_b,A4: set_Product_prod_b_b,B: product_prod_b_b,B4: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ A @ A4 )
=> ( ~ ( member1285940496od_b_b @ B @ B4 )
=> ( ( ( insert1952693431od_b_b @ A @ A4 )
= ( insert1952693431od_b_b @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_b_b] :
( ( A4
= ( insert1952693431od_b_b @ B @ C3 ) )
& ~ ( member1285940496od_b_b @ B @ C3 )
& ( B4
= ( insert1952693431od_b_b @ A @ C3 ) )
& ~ ( member1285940496od_b_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_215_insert__absorb,axiom,
! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( member327031898_a_nat @ A @ A4 )
=> ( ( insert2058869377_a_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_216_insert__absorb,axiom,
! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ A4 )
=> ( ( insert1952693431od_b_b @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_217_insert__ident,axiom,
! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
( ~ ( member327031898_a_nat @ X4 @ A4 )
=> ( ~ ( member327031898_a_nat @ X4 @ B4 )
=> ( ( ( insert2058869377_a_nat @ X4 @ A4 )
= ( insert2058869377_a_nat @ X4 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_218_insert__ident,axiom,
! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ X4 @ A4 )
=> ( ~ ( member1285940496od_b_b @ X4 @ B4 )
=> ( ( ( insert1952693431od_b_b @ X4 @ A4 )
= ( insert1952693431od_b_b @ X4 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_219_Set_Oset__insert,axiom,
! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( member327031898_a_nat @ X4 @ A4 )
=> ~ ! [B5: set_Pr1912306449_a_nat] :
( ( A4
= ( insert2058869377_a_nat @ X4 @ B5 ) )
=> ( member327031898_a_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_220_Set_Oset__insert,axiom,
! [X4: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A4 )
=> ~ ! [B5: set_Product_prod_b_b] :
( ( A4
= ( insert1952693431od_b_b @ X4 @ B5 ) )
=> ( member1285940496od_b_b @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_221_insertI2,axiom,
! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
( ( member327031898_a_nat @ A @ B4 )
=> ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_222_insertI2,axiom,
! [A: product_prod_b_b,B4: set_Product_prod_b_b,B: product_prod_b_b] :
( ( member1285940496od_b_b @ A @ B4 )
=> ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B4 ) ) ) ).
% insertI2
thf(fact_223_insertI1,axiom,
! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat] : ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ A @ B4 ) ) ).
% insertI1
thf(fact_224_insertI1,axiom,
! [A: product_prod_b_b,B4: set_Product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ B4 ) ) ).
% insertI1
thf(fact_225_insertE,axiom,
! [A: produc598811825_a_nat,B: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member327031898_a_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_226_insertE,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A4 ) )
=> ( ( A != B )
=> ( member1285940496od_b_b @ A @ A4 ) ) ) ).
% insertE
thf(fact_227_singleton__inject,axiom,
! [A: produc598811825_a_nat,B: produc598811825_a_nat] :
( ( ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat )
= ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_228_insert__not__empty,axiom,
! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( insert2058869377_a_nat @ A @ A4 )
!= bot_bo2066531197_a_nat ) ).
% insert_not_empty
thf(fact_229_doubleton__eq__iff,axiom,
! [A: produc598811825_a_nat,B: produc598811825_a_nat,C2: produc598811825_a_nat,D: produc598811825_a_nat] :
( ( ( insert2058869377_a_nat @ A @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
= ( insert2058869377_a_nat @ C2 @ ( insert2058869377_a_nat @ D @ bot_bo2066531197_a_nat ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_230_singleton__iff,axiom,
! [B: product_prod_b_b,A: product_prod_b_b] :
( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_231_singleton__iff,axiom,
! [B: produc598811825_a_nat,A: produc598811825_a_nat] :
( ( member327031898_a_nat @ B @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_232_singletonD,axiom,
! [B: product_prod_b_b,A: product_prod_b_b] :
( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_233_singletonD,axiom,
! [B: produc598811825_a_nat,A: produc598811825_a_nat] :
( ( member327031898_a_nat @ B @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_234_the__elem__eq,axiom,
! [X4: produc598811825_a_nat] :
( ( the_el537460006_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_235_graph__single,axiom,
! [A: a,B: b,C2: b] :
( ( labeled_LG_a_b @ ( insert1655137376od_b_b @ ( produc2135284224od_b_b @ A @ ( product_Pair_b_b @ B @ C2 ) ) @ bot_bo519661404od_b_b ) @ ( insert_b @ B @ ( insert_b @ C2 @ bot_bot_set_b ) ) )
= ( restrict_a_b @ ( labeled_LG_a_b @ ( insert1655137376od_b_b @ ( produc2135284224od_b_b @ A @ ( product_Pair_b_b @ B @ C2 ) ) @ bot_bo519661404od_b_b ) @ ( insert_b @ B @ ( insert_b @ C2 @ bot_bot_set_b ) ) ) ) ) ).
% graph_single
thf(fact_236_is__singletonI,axiom,
! [X4: produc598811825_a_nat] : ( is_sin1787449765_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) ) ).
% is_singletonI
thf(fact_237_linear__order__on__singleton,axiom,
! [X4: b] : ( order_653178932r_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% linear_order_on_singleton
thf(fact_238_linear__order__on__singleton,axiom,
! [X4: labeled_graph_a_nat] : ( order_1770871027_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% linear_order_on_singleton
thf(fact_239_linear__order__on__singleton,axiom,
! [X4: produc598811825_a_nat] : ( order_1458491494_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% linear_order_on_singleton
thf(fact_240_total__on__singleton,axiom,
! [X4: b] : ( total_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% total_on_singleton
thf(fact_241_total__on__singleton,axiom,
! [X4: labeled_graph_a_nat] : ( total_6094180_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% total_on_singleton
thf(fact_242_total__on__singleton,axiom,
! [X4: produc598811825_a_nat] : ( total_2145009205_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% total_on_singleton
thf(fact_243_symmetry__rule,axiom,
! [G: labeled_graph_a_b,L: a] :
( ( G
= ( restrict_a_b @ G ) )
=> ( ( maintained_a_nat_b @ ( standa709637829rule_a @ L ) @ G )
=> ( sym_b @ ( getRel_a_b @ L @ G ) ) ) ) ).
% symmetry_rule
thf(fact_244_graph__empty__e,axiom,
! [V: set_b] :
( ( labeled_LG_a_b @ bot_bo519661404od_b_b @ V )
= ( restrict_a_b @ ( labeled_LG_a_b @ bot_bo519661404od_b_b @ V ) ) ) ).
% graph_empty_e
thf(fact_245_bot__empty__eq,axiom,
( bot_bo1302443370_b_b_o
= ( ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ bot_bo1343651123od_b_b ) ) ) ).
% bot_empty_eq
thf(fact_246_bot__empty__eq,axiom,
( bot_bo186023328_nat_o
= ( ^ [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ bot_bo2066531197_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_247_bot__set__def,axiom,
( bot_bo2066531197_a_nat
= ( collec978177436_a_nat @ bot_bo186023328_nat_o ) ) ).
% bot_set_def
thf(fact_248_sym__def,axiom,
( sym_b
= ( ^ [R3: set_Product_prod_b_b] :
! [X5: b,Y5: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ X5 ) @ R3 ) ) ) ) ).
% sym_def
thf(fact_249_symI,axiom,
! [R2: set_Product_prod_b_b] :
( ! [A3: b,B3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ B3 @ A3 ) @ R2 ) )
=> ( sym_b @ R2 ) ) ).
% symI
thf(fact_250_symE,axiom,
! [R2: set_Product_prod_b_b,B: b,A: b] :
( ( sym_b @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ).
% symE
thf(fact_251_symD,axiom,
! [R2: set_Product_prod_b_b,B: b,A: b] :
( ( sym_b @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ).
% symD
thf(fact_252_total__on__def,axiom,
( total_on_b
= ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
! [X5: b] :
( ( member_b @ X5 @ A5 )
=> ! [Y5: b] :
( ( member_b @ Y5 @ A5 )
=> ( ( X5 != Y5 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
| ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ X5 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_253_total__onI,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
( ! [X3: product_prod_b_b,Y3: product_prod_b_b] :
( ( member1285940496od_b_b @ X3 @ A4 )
=> ( ( member1285940496od_b_b @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Y3 ) @ R2 )
| ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_178717291od_b_b @ A4 @ R2 ) ) ).
% total_onI
thf(fact_254_total__onI,axiom,
! [A4: set_b,R2: set_Product_prod_b_b] :
( ! [X3: b,Y3: b] :
( ( member_b @ X3 @ A4 )
=> ( ( member_b @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
| ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_b @ A4 @ R2 ) ) ).
% total_onI
thf(fact_255_total__on__empty,axiom,
! [R2: set_Pr2074180935_a_nat] : ( total_2145009205_a_nat @ bot_bo2066531197_a_nat @ R2 ) ).
% total_on_empty
thf(fact_256_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr1900477680od_b_b,X2: set_b] :
( ( labeled_vertices_a_b @ ( labeled_LG_a_b @ X1 @ X2 ) )
= X2 ) ).
% labeled_graph.sel(2)
thf(fact_257_is__singleton__the__elem,axiom,
( is_sin1787449765_a_nat
= ( ^ [A5: set_Pr1912306449_a_nat] :
( A5
= ( insert2058869377_a_nat @ ( the_el537460006_a_nat @ A5 ) @ bot_bo2066531197_a_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_258_is__singletonI_H,axiom,
! [A4: set_Product_prod_b_b] :
( ( A4 != bot_bo1343651123od_b_b )
=> ( ! [X3: product_prod_b_b,Y3: product_prod_b_b] :
( ( member1285940496od_b_b @ X3 @ A4 )
=> ( ( member1285940496od_b_b @ Y3 @ A4 )
=> ( X3 = Y3 ) ) )
=> ( is_sin1713057243od_b_b @ A4 ) ) ) ).
% is_singletonI'
thf(fact_259_is__singletonI_H,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ( A4 != bot_bo2066531197_a_nat )
=> ( ! [X3: produc598811825_a_nat,Y3: produc598811825_a_nat] :
( ( member327031898_a_nat @ X3 @ A4 )
=> ( ( member327031898_a_nat @ Y3 @ A4 )
=> ( X3 = Y3 ) ) )
=> ( is_sin1787449765_a_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_260_lnear__order__on__empty,axiom,
order_1770871027_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% lnear_order_on_empty
thf(fact_261_lnear__order__on__empty,axiom,
order_1458491494_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% lnear_order_on_empty
thf(fact_262_is__singletonE,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ( is_sin1787449765_a_nat @ A4 )
=> ~ ! [X3: produc598811825_a_nat] :
( A4
!= ( insert2058869377_a_nat @ X3 @ bot_bo2066531197_a_nat ) ) ) ).
% is_singletonE
thf(fact_263_is__singleton__def,axiom,
( is_sin1787449765_a_nat
= ( ^ [A5: set_Pr1912306449_a_nat] :
? [X5: produc598811825_a_nat] :
( A5
= ( insert2058869377_a_nat @ X5 @ bot_bo2066531197_a_nat ) ) ) ) ).
% is_singleton_def
thf(fact_264_equivE,axiom,
! [A4: set_b,R2: set_Product_prod_b_b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ~ ( ( refl_on_b @ A4 @ R2 )
=> ( ( sym_b @ R2 )
=> ~ ( trans_b @ R2 ) ) ) ) ).
% equivE
thf(fact_265_equivI,axiom,
! [A4: set_b,R2: set_Product_prod_b_b] :
( ( refl_on_b @ A4 @ R2 )
=> ( ( sym_b @ R2 )
=> ( ( trans_b @ R2 )
=> ( equiv_equiv_b @ A4 @ R2 ) ) ) ) ).
% equivI
thf(fact_266_equiv__def,axiom,
( equiv_equiv_b
= ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
( ( refl_on_b @ A5 @ R3 )
& ( sym_b @ R3 )
& ( trans_b @ R3 ) ) ) ) ).
% equiv_def
thf(fact_267_graph__homomorphism__empty,axiom,
! [G: labeled_graph_a_b,F: set_Pr1914572181_nat_b] :
( ( graph_23275493_nat_b @ ( labele1171788421_a_nat @ bot_bo1463793628_a_nat @ bot_bo2066531197_a_nat ) @ G @ F )
= ( ( F = bot_bo1936722433_nat_b )
& ( G
= ( restrict_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_268_Collect__empty__eq__bot,axiom,
! [P2: produc598811825_a_nat > $o] :
( ( ( collec978177436_a_nat @ P2 )
= bot_bo2066531197_a_nat )
= ( P2 = bot_bo186023328_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_269_maintainedI,axiom,
! [A4: labeled_graph_a_nat,G: labeled_graph_a_b,B4: labeled_graph_a_nat] :
( ! [F2: set_Pr2106913242_nat_b] :
( ( graph_914848022_nat_b @ A4 @ G @ F2 )
=> ( extensible_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G @ F2 ) )
=> ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G ) ) ).
% maintainedI
thf(fact_270_maintainedD,axiom,
! [A4: labeled_graph_a_nat,B4: labeled_graph_a_nat,G: labeled_graph_a_b,F: set_Pr2106913242_nat_b] :
( ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G )
=> ( ( graph_914848022_nat_b @ A4 @ G @ F )
=> ( extensible_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G @ F ) ) ) ).
% maintainedD
thf(fact_271_Set_Ois__empty__def,axiom,
( is_emp1506632323_a_nat
= ( ^ [A5: set_Pr1912306449_a_nat] : A5 = bot_bo2066531197_a_nat ) ) ).
% Set.is_empty_def
thf(fact_272_eq__equiv__class__iff2,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A4 @ R2 )
=> ( ( member1285940496od_b_b @ X4 @ A4 )
=> ( ( member1285940496od_b_b @ Y4 @ A4 )
=> ( ( ( equiv_1530858158od_b_b @ ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) @ R2 )
= ( equiv_1530858158od_b_b @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) @ R2 ) )
= ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_273_eq__equiv__class__iff2,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_b @ X4 @ A4 )
=> ( ( member_b @ Y4 @ A4 )
=> ( ( ( equiv_quotient_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ R2 )
= ( equiv_quotient_b @ ( insert_b @ Y4 @ bot_bot_set_b ) @ R2 ) )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_274_eq__equiv__class__iff2,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
( ( equiv_281634663_a_nat @ A4 @ R2 )
=> ( ( member327031898_a_nat @ X4 @ A4 )
=> ( ( member327031898_a_nat @ Y4 @ A4 )
=> ( ( ( equiv_1338033912_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ R2 )
= ( equiv_1338033912_a_nat @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) @ R2 ) )
= ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_275_quotient__is__empty2,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
( ( bot_bo1746593245_a_nat
= ( equiv_1338033912_a_nat @ A4 @ R2 ) )
= ( A4 = bot_bo2066531197_a_nat ) ) ).
% quotient_is_empty2
thf(fact_276_quotient__is__empty,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
( ( ( equiv_1338033912_a_nat @ A4 @ R2 )
= bot_bo1746593245_a_nat )
= ( A4 = bot_bo2066531197_a_nat ) ) ).
% quotient_is_empty
thf(fact_277_quotient__empty,axiom,
! [R2: set_Pr2074180935_a_nat] :
( ( equiv_1338033912_a_nat @ bot_bo2066531197_a_nat @ R2 )
= bot_bo1746593245_a_nat ) ).
% quotient_empty
thf(fact_278_in__quotient__imp__closed,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A4 @ R2 )
=> ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
=> ( ( member1285940496od_b_b @ X4 @ X6 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
=> ( member1285940496od_b_b @ Y4 @ X6 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_279_in__quotient__imp__closed,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,X4: b,Y4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ( member_b @ X4 @ X6 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
=> ( member_b @ Y4 @ X6 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_280_quotient__eq__iff,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,Y6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A4 @ R2 )
=> ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
=> ( ( member1050118384od_b_b @ Y6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
=> ( ( member1285940496od_b_b @ X4 @ X6 )
=> ( ( member1285940496od_b_b @ Y4 @ Y6 )
=> ( ( X6 = Y6 )
= ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_281_quotient__eq__iff,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,Y6: set_b,X4: b,Y4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ( member_set_b @ Y6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ( member_b @ X4 @ X6 )
=> ( ( member_b @ Y4 @ Y6 )
=> ( ( X6 = Y6 )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_282_quotient__eqI,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,Y6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A4 @ R2 )
=> ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
=> ( ( member1050118384od_b_b @ Y6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
=> ( ( member1285940496od_b_b @ X4 @ X6 )
=> ( ( member1285940496od_b_b @ Y4 @ Y6 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
=> ( X6 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_283_quotient__eqI,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,Y6: set_b,X4: b,Y4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ( member_set_b @ Y6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ( member_b @ X4 @ X6 )
=> ( ( member_b @ Y4 @ Y6 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
=> ( X6 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_284_in__quotient__imp__non__empty,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( X6 != bot_bot_set_b ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_285_in__quotient__imp__non__empty,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X6: set_Pr1912306449_a_nat] :
( ( equiv_281634663_a_nat @ A4 @ R2 )
=> ( ( member101535418_a_nat @ X6 @ ( equiv_1338033912_a_nat @ A4 @ R2 ) )
=> ( X6 != bot_bo2066531197_a_nat ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_286_in__quotient__imp__in__rel,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,X4: b,Y4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ( ord_less_eq_set_b @ ( insert_b @ X4 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) @ X6 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_287_in__quotient__imp__in__rel,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X6: set_Pr1912306449_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
( ( equiv_281634663_a_nat @ A4 @ R2 )
=> ( ( member101535418_a_nat @ X6 @ ( equiv_1338033912_a_nat @ A4 @ R2 ) )
=> ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) ) @ X6 )
=> ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_288_same__fstI,axiom,
! [P2: b > $o,X4: b,Y7: b,Y4: b,R: b > set_Product_prod_b_b] :
( ( P2 @ X4 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y7 @ Y4 ) @ ( R @ X4 ) )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ ( product_Pair_b_b @ X4 @ Y7 ) @ ( product_Pair_b_b @ X4 @ Y4 ) ) @ ( same_fst_b_b @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_289_getRel__subgraph,axiom,
! [Y4: b,Z: b,L: a,G: labeled_graph_a_b,G2: labeled_graph_a_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ ( getRel_a_b @ L @ G ) )
=> ( ( graph_1322245399_a_b_b @ G @ G2 @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ ( getRel_a_b @ L @ G2 ) ) ) ) ).
% getRel_subgraph
thf(fact_290_subsetI,axiom,
! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
( ! [X3: product_prod_b_b] :
( ( member1285940496od_b_b @ X3 @ A4 )
=> ( member1285940496od_b_b @ X3 @ B4 ) )
=> ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ).
% subsetI
thf(fact_291_empty__subsetI,axiom,
! [A4: set_Pr1912306449_a_nat] : ( ord_le1374266033_a_nat @ bot_bo2066531197_a_nat @ A4 ) ).
% empty_subsetI
thf(fact_292_subset__empty,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ( ord_le1374266033_a_nat @ A4 @ bot_bo2066531197_a_nat )
= ( A4 = bot_bo2066531197_a_nat ) ) ).
% subset_empty
thf(fact_293_insert__subset,axiom,
! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ A4 ) @ B4 )
= ( ( member327031898_a_nat @ X4 @ B4 )
& ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_294_insert__subset,axiom,
! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X4 @ A4 ) @ B4 )
= ( ( member1285940496od_b_b @ X4 @ B4 )
& ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_295_Id__onI,axiom,
! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ A4 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ ( id_on_2019020932od_b_b @ A4 ) ) ) ).
% Id_onI
thf(fact_296_Id__onI,axiom,
! [A: b,A4: set_b] :
( ( member_b @ A @ A4 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ ( id_on_b @ A4 ) ) ) ).
% Id_onI
thf(fact_297_singleton__insert__inj__eq,axiom,
! [B: produc598811825_a_nat,A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
( ( ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat )
= ( insert2058869377_a_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_298_singleton__insert__inj__eq_H,axiom,
! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
( ( ( insert2058869377_a_nat @ A @ A4 )
= ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
= ( ( A = B )
& ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_299_Id__on__empty,axiom,
( ( id_on_1401723275_a_nat @ bot_bo1119737266_a_nat )
= bot_bo2066531197_a_nat ) ).
% Id_on_empty
thf(fact_300_Id__on__empty,axiom,
( ( id_on_1649591758_a_nat @ bot_bo2066531197_a_nat )
= bot_bo1803599795_a_nat ) ).
% Id_on_empty
thf(fact_301_subgraph__refl,axiom,
! [G: labeled_graph_a_b] :
( ( graph_1322245399_a_b_b @ G @ G @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
= ( G
= ( restrict_a_b @ G ) ) ) ).
% subgraph_refl
thf(fact_302_subgraph__restrict,axiom,
! [G: labeled_graph_a_b] :
( ( graph_1322245399_a_b_b @ G @ ( restrict_a_b @ G ) @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
= ( G
= ( restrict_a_b @ G ) ) ) ).
% subgraph_restrict
thf(fact_303_graph__homomorphism__Id,axiom,
! [A: labeled_graph_a_b] : ( graph_1322245399_a_b_b @ ( restrict_a_b @ A ) @ ( restrict_a_b @ A ) @ ( id_on_b @ ( labeled_vertices_a_b @ A ) ) ) ).
% graph_homomorphism_Id
thf(fact_304_subgraph__subset_I1_J,axiom,
! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
=> ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) ) ) ).
% subgraph_subset(1)
thf(fact_305_Id__onE,axiom,
! [C2: produc1168037607od_b_b,A4: set_Product_prod_b_b] :
( ( member1652792080od_b_b @ C2 @ ( id_on_2019020932od_b_b @ A4 ) )
=> ~ ! [X3: product_prod_b_b] :
( ( member1285940496od_b_b @ X3 @ A4 )
=> ( C2
!= ( produc546497367od_b_b @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_306_Id__onE,axiom,
! [C2: product_prod_b_b,A4: set_b] :
( ( member1285940496od_b_b @ C2 @ ( id_on_b @ A4 ) )
=> ~ ! [X3: b] :
( ( member_b @ X3 @ A4 )
=> ( C2
!= ( product_Pair_b_b @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_307_Id__on__eqI,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( A = B )
=> ( ( member1285940496od_b_b @ A @ A4 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ ( id_on_2019020932od_b_b @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_308_Id__on__eqI,axiom,
! [A: b,B: b,A4: set_b] :
( ( A = B )
=> ( ( member_b @ A @ A4 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( id_on_b @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_309_Id__on__iff,axiom,
! [X4: product_prod_b_b,Y4: product_prod_b_b,A4: set_Product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ ( id_on_2019020932od_b_b @ A4 ) )
= ( ( X4 = Y4 )
& ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_310_Id__on__iff,axiom,
! [X4: b,Y4: b,A4: set_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ ( id_on_b @ A4 ) )
= ( ( X4 = Y4 )
& ( member_b @ X4 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_311_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr1912306449_a_nat] :
( ( ord_le1374266033_a_nat @ A @ bot_bo2066531197_a_nat )
=> ( A = bot_bo2066531197_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_312_bot_Oextremum__unique,axiom,
! [A: set_Pr1912306449_a_nat] :
( ( ord_le1374266033_a_nat @ A @ bot_bo2066531197_a_nat )
= ( A = bot_bo2066531197_a_nat ) ) ).
% bot.extremum_unique
thf(fact_313_bot_Oextremum,axiom,
! [A: set_Pr1912306449_a_nat] : ( ord_le1374266033_a_nat @ bot_bo2066531197_a_nat @ A ) ).
% bot.extremum
thf(fact_314_subrelI,axiom,
! [R2: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ! [X3: b,Y3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le1036320359od_b_b @ R2 @ S ) ) ).
% subrelI
thf(fact_315_insert__mono,axiom,
! [C4: set_Pr1912306449_a_nat,D2: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
( ( ord_le1374266033_a_nat @ C4 @ D2 )
=> ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ A @ C4 ) @ ( insert2058869377_a_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_316_subset__insert,axiom,
! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
( ~ ( member327031898_a_nat @ X4 @ A4 )
=> ( ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ X4 @ B4 ) )
= ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_317_subset__insert,axiom,
! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ X4 @ A4 )
=> ( ( ord_le1036320359od_b_b @ A4 @ ( insert1952693431od_b_b @ X4 @ B4 ) )
= ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_318_subset__insertI,axiom,
! [B4: set_Pr1912306449_a_nat,A: produc598811825_a_nat] : ( ord_le1374266033_a_nat @ B4 @ ( insert2058869377_a_nat @ A @ B4 ) ) ).
% subset_insertI
thf(fact_319_subset__insertI2,axiom,
! [A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
( ( ord_le1374266033_a_nat @ A4 @ B4 )
=> ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_320_trans__Id__on,axiom,
! [A4: set_b] : ( trans_b @ ( id_on_b @ A4 ) ) ).
% trans_Id_on
thf(fact_321_refl__on__Id__on,axiom,
! [A4: set_b] : ( refl_on_b @ A4 @ ( id_on_b @ A4 ) ) ).
% refl_on_Id_on
thf(fact_322_subset__iff,axiom,
( ord_le1036320359od_b_b
= ( ^ [A5: set_Product_prod_b_b,B6: set_Product_prod_b_b] :
! [T: product_prod_b_b] :
( ( member1285940496od_b_b @ T @ A5 )
=> ( member1285940496od_b_b @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_323_subset__eq,axiom,
( ord_le1036320359od_b_b
= ( ^ [A5: set_Product_prod_b_b,B6: set_Product_prod_b_b] :
! [X5: product_prod_b_b] :
( ( member1285940496od_b_b @ X5 @ A5 )
=> ( member1285940496od_b_b @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_324_subsetD,axiom,
! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b,C2: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A4 @ B4 )
=> ( ( member1285940496od_b_b @ C2 @ A4 )
=> ( member1285940496od_b_b @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_325_in__mono,axiom,
! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b,X4: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A4 @ B4 )
=> ( ( member1285940496od_b_b @ X4 @ A4 )
=> ( member1285940496od_b_b @ X4 @ B4 ) ) ) ).
% in_mono
thf(fact_326_subset__singletonD,axiom,
! [A4: set_Pr1912306449_a_nat,X4: produc598811825_a_nat] :
( ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) )
=> ( ( A4 = bot_bo2066531197_a_nat )
| ( A4
= ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_327_subset__singleton__iff,axiom,
! [X6: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
( ( ord_le1374266033_a_nat @ X6 @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
= ( ( X6 = bot_bo2066531197_a_nat )
| ( X6
= ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_328_subgraph__trans,axiom,
! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b,G_3: labeled_graph_a_b] :
( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
=> ( ( graph_1322245399_a_b_b @ G_2 @ G_3 @ ( id_on_b @ ( labeled_vertices_a_b @ G_2 ) ) )
=> ( graph_1322245399_a_b_b @ G_1 @ G_3 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) ) ) ) ).
% subgraph_trans
thf(fact_329_in__quotient__imp__subset,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( ord_less_eq_set_b @ X6 @ A4 ) ) ) ).
% in_quotient_imp_subset
thf(fact_330_maintainedD2,axiom,
! [A4: labeled_graph_a_nat,B4: labeled_graph_a_nat,G: labeled_graph_a_b,F: set_Pr2106913242_nat_b] :
( ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G )
=> ( ( graph_914848022_nat_b @ A4 @ G @ F )
=> ~ ! [G3: set_Pr2106913242_nat_b] :
( ( graph_914848022_nat_b @ B4 @ G @ G3 )
=> ~ ( ord_le1484132922_nat_b @ F @ G3 ) ) ) ) ).
% maintainedD2
thf(fact_331_subgraph__semantics,axiom,
! [A4: labeled_graph_a_b,B4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
( ( graph_1322245399_a_b_b @ A4 @ B4 @ ( id_on_b @ ( labeled_vertices_a_b @ A4 ) ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ B4 @ E ) ) ) ) ).
% subgraph_semantics
thf(fact_332_proj__iff,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( ord_less_eq_set_b @ ( insert_b @ X4 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) @ A4 )
=> ( ( ( equiv_proj_b_b @ R2 @ X4 )
= ( equiv_proj_b_b @ R2 @ Y4 ) )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_333_proj__iff,axiom,
! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
( ( equiv_281634663_a_nat @ A4 @ R2 )
=> ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) ) @ A4 )
=> ( ( ( equiv_1873824970_a_nat @ R2 @ X4 )
= ( equiv_1873824970_a_nat @ R2 @ Y4 ) )
= ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_334_consequence__graphD_I1_J,axiom,
! [Rs: set_Pr1912306449_a_nat,G: labeled_graph_a_b,R: produc598811825_a_nat] :
( ( conseq1058228089_nat_b @ Rs @ G )
=> ( ( member327031898_a_nat @ R @ Rs )
=> ( maintained_a_nat_b @ R @ G ) ) ) ).
% consequence_graphD(1)
thf(fact_335_proj__in__iff,axiom,
! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A4 @ R2 )
=> ( ( member1050118384od_b_b @ ( equiv_270653002od_b_b @ R2 @ X4 ) @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
= ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% proj_in_iff
thf(fact_336_proj__in__iff,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,X4: b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( member_set_b @ ( equiv_proj_b_b @ R2 @ X4 ) @ ( equiv_quotient_b @ A4 @ R2 ) )
= ( member_b @ X4 @ A4 ) ) ) ).
% proj_in_iff
thf(fact_337_less__by__empty,axiom,
! [A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
( ( A4 = bot_bo2066531197_a_nat )
=> ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ).
% less_by_empty
thf(fact_338_insert__subsetI,axiom,
! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,X6: set_Pr1912306449_a_nat] :
( ( member327031898_a_nat @ X4 @ A4 )
=> ( ( ord_le1374266033_a_nat @ X6 @ A4 )
=> ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_339_insert__subsetI,axiom,
! [X4: product_prod_b_b,A4: set_Product_prod_b_b,X6: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A4 )
=> ( ( ord_le1036320359od_b_b @ X6 @ A4 )
=> ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X4 @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_340_ssubst__Pair__rhs,axiom,
! [R2: b,S: b,R: set_Product_prod_b_b,S2: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ R2 @ S ) @ R )
=> ( ( S2 = S )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ R2 @ S2 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_341_subset__emptyI,axiom,
! [A4: set_Product_prod_b_b] :
( ! [X3: product_prod_b_b] :
~ ( member1285940496od_b_b @ X3 @ A4 )
=> ( ord_le1036320359od_b_b @ A4 @ bot_bo1343651123od_b_b ) ) ).
% subset_emptyI
thf(fact_342_subset__emptyI,axiom,
! [A4: set_Pr1912306449_a_nat] :
( ! [X3: produc598811825_a_nat] :
~ ( member327031898_a_nat @ X3 @ A4 )
=> ( ord_le1374266033_a_nat @ A4 @ bot_bo2066531197_a_nat ) ) ).
% subset_emptyI
thf(fact_343_subgraph__def,axiom,
! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
= ( ( G_1
= ( restrict_a_b @ G_1 ) )
& ( G_2
= ( restrict_a_b @ G_2 ) )
& ( ( graph_union_a_b @ G_1 @ G_2 )
= G_2 ) ) ) ).
% subgraph_def
thf(fact_344_graph__union__preserves__restrict,axiom,
! [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 ) ) ) ) ) ).
% graph_union_preserves_restrict
thf(fact_345_univ__preserves,axiom,
! [A4: set_b,R2: set_Product_prod_b_b,F: b > product_prod_b_b,B4: set_Product_prod_b_b] :
( ( equiv_equiv_b @ A4 @ R2 )
=> ( ( equiv_1778431934od_b_b @ R2 @ F )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A4 )
=> ( member1285940496od_b_b @ ( F @ X3 ) @ B4 ) )
=> ! [X: set_b] :
( ( member_set_b @ X @ ( equiv_quotient_b @ A4 @ R2 ) )
=> ( member1285940496od_b_b @ ( bNF_Gr1693995344od_b_b @ F @ X ) @ B4 ) ) ) ) ) ).
% univ_preserves
thf(fact_346_relImage__proj,axiom,
! [A4: set_b,R: set_Product_prod_b_b] :
( ( equiv_equiv_b @ A4 @ R )
=> ( ord_le2142668391_set_b @ ( bNF_Gr345950468_set_b @ R @ ( equiv_proj_b_b @ R ) ) @ ( id_on_set_b @ ( equiv_quotient_b @ A4 @ R ) ) ) ) ).
% relImage_proj
thf(fact_347_subgraph__def2,axiom,
! [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_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
= ( ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) )
& ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) ) ) ) ) ) ).
% subgraph_def2
thf(fact_348_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labeled_graph_a_b] :
( ( labeled_LG_a_b @ ( labeled_edges_a_b @ Labeled_graph ) @ ( labeled_vertices_a_b @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_349_graph__unionI,axiom,
! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
( ( ord_le887919760od_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 ) ) ) ).
% graph_unionI
thf(fact_350_restrict__subsD,axiom,
! [G: labeled_graph_a_b] :
( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G ) @ ( labeled_edges_a_b @ ( restrict_a_b @ G ) ) )
=> ( G
= ( restrict_a_b @ G ) ) ) ).
% restrict_subsD
thf(fact_351_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labeled_graph_a_b] :
( Labeled_graph
= ( labeled_LG_a_b @ ( labeled_edges_a_b @ Labeled_graph ) @ ( labeled_vertices_a_b @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_352_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labeled_graph_a_b,Labeled_graph2: labeled_graph_a_b] :
( ( ( ( labeled_edges_a_b @ Labeled_graph )
= ( labeled_edges_a_b @ Labeled_graph2 ) )
& ( ( labeled_vertices_a_b @ Labeled_graph )
= ( labeled_vertices_a_b @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_353_graph__union__iff,axiom,
! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
( ( ( graph_union_a_b @ G_1 @ G_2 )
= G_2 )
= ( ( ord_le887919760od_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_iff
thf(fact_354_subgraph__subset_I2_J,axiom,
! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
=> ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ ( restrict_a_b @ G_1 ) ) @ ( labeled_edges_a_b @ G_2 ) ) ) ).
% subgraph_subset(2)
% Conjectures (1)
thf(conj_0,conjecture,
member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) ).
%------------------------------------------------------------------------------