TPTP Problem File: ITP180^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP180^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_651__5393624_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_651__5393624_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 537 ( 195 unt; 181 typ; 0 def)
% Number of atoms : 924 ( 468 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2959 ( 141 ~; 28 |; 78 &;2417 @)
% ( 0 <=>; 295 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 44 ( 43 usr)
% Number of type conns : 300 ( 300 >; 0 *; 0 +; 0 <<)
% Number of symbols : 139 ( 138 usr; 24 con; 0-3 aty)
% Number of variables : 888 ( 24 ^; 839 !; 25 ?; 888 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:39:26.524
%------------------------------------------------------------------------------
% Could-be-implicit typings (43)
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% Explicit typings (138)
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thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J,type,
mainta1342478555od_V_V: produc778275879_V_nat > labele201426742od_V_V > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_J,type,
mainta922614952_V_nat: produc778275879_V_nat > labele506073123_V_nat > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J,type,
mainta1984786828od_V_V: produc778275879_V_nat > labele236903175od_V_V > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_001t__Nat__Onat_001tf__V,type,
mainta1026355316_nat_V: produc778275879_V_nat > labele788688367nt_V_V > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_001t__Nat__Onat_001tf__V__H,type,
mainta197426964_nat_V: produc778275879_V_nat > labele2115946735nt_V_V > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_001tf__V__H_001tf__V__H,type,
mainta1699210777_V_V_V: produc651414759nt_V_V > labele2115946735nt_V_V > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J,type,
collec1840263537od_V_V: ( produc1699571228od_V_V > $o ) > set_Pr744279122od_V_V ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_J,type,
collec1411521682_V_nat: ( produc778275879_V_nat > $o ) > set_Pr1058435079_V_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J,type,
collec2040633974od_V_V: ( product_prod_V_V > $o ) > set_Product_prod_V_V ).
thf(sy_c_Set_OCollect_001tf__V,type,
collect_V: ( v2 > $o ) > set_V2 ).
thf(sy_c_Set_OCollect_001tf__V__H,type,
collect_V2: ( v > $o ) > set_V ).
thf(sy_c_Set_Oinsert_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J,type,
insert1818834324tant_V: standard_Constant_V > set_St1111633946tant_V > set_St1111633946tant_V ).
thf(sy_c_Set_Oinsert_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J,type,
insert213439034_V_nat: labele2128733290_V_nat > set_la14611914_V_nat > set_la14611914_V_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J,type,
insert135842892od_V_V: produc1699571228od_V_V > set_Pr744279122od_V_V > set_Pr744279122od_V_V ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_J,type,
insert481364471_V_nat: produc778275879_V_nat > set_Pr1058435079_V_nat > set_Pr1058435079_V_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J,type,
insert2024285275od_V_V: product_prod_V_V > set_Product_prod_V_V > set_Product_prod_V_V ).
thf(sy_c_Set_Oinsert_001tf__V,type,
insert_V: v2 > set_V2 > set_V2 ).
thf(sy_c_Set_Oinsert_001tf__V__H,type,
insert_V2: v > set_V > set_V ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oconstant__rules_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_J,type,
standa434642387_V_nat: set_Pr1058435079_V_nat > set_Pr361461255at_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oconstant__rules_001t__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J,type,
standa1571708471od_V_V: set_Product_prod_V_V > set_Pr597600647_V_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oconstant__rules_001tf__V,type,
standa1897115807ules_V: set_V2 > set_Pr1058435079_V_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oconstant__rules_001tf__V__H,type,
standa294833961ules_V: set_V > set_Pr473545225_V_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oidentity__rules_001tf__V,type,
standa1568205529ules_V: set_St1111633946tant_V > set_Pr1058435079_V_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ononempty__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J,type,
standa1319953089tant_V: produc778275879_V_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ostandard__rules_001tf__V,type,
standa157301464ules_V: set_V2 > set_St1111633946tant_V > set_Pr1058435079_V_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Otop__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J,type,
standa214871990tant_V: standard_Constant_V > produc778275879_V_nat ).
thf(sy_c_member_001t__LabeledGraphSemantics__OStandard____Constant_Itf__V_J,type,
member1542015739tant_V: standard_Constant_V > set_St1111633946tant_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J_J_J,type,
member600445121od_V_V: produc1269720746od_V_V > set_Pr872354016od_V_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J_J,type,
member361435663od_V_V: produc1443401592od_V_V > set_Pr951255342od_V_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V_Mtf__V_J_J,type,
member1416643087od_V_V: produc1113554040od_V_V > set_Pr1498723630od_V_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J,type,
member1010391347od_V_V: produc1699571228od_V_V > set_Pr744279122od_V_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Nat__Onat_J_J,type,
member1886821968_V_nat: produc778275879_V_nat > set_Pr1058435079_V_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__V_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J_J,type,
member785416642od_V_V: produc1082089497od_V_V > set_Pr2015814777od_V_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_Mt__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J_J,type,
member134497296od_V_V: produc522115815od_V_V > set_Pr2022779335od_V_V > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__V_Mtf__V_J,type,
member1990987792od_V_V: product_prod_V_V2 > set_Product_prod_V_V3 > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__V__H_Mtf__V__H_J,type,
member2015049524od_V_V: product_prod_V_V > set_Product_prod_V_V > $o ).
thf(sy_c_member_001tf__V,type,
member_V: v2 > set_V2 > $o ).
thf(sy_c_member_001tf__V__H,type,
member_V2: v > set_V > $o ).
thf(sy_v_C,type,
c: set_V2 ).
thf(sy_v_G_H,type,
g: labele2115946735nt_V_V ).
thf(sy_v_L,type,
l: set_St1111633946tant_V ).
thf(sy_v_h____,type,
h: v > v ).
thf(sy_v_m____,type,
m: v2 > v ).
thf(sy_v_x____,type,
x: v ).
thf(sy_v_xa____,type,
xa: v2 ).
thf(sy_v_y____,type,
y: v2 ).
% Relevant facts (355)
thf(fact_0_x,axiom,
member_V2 @ x @ ( labele1134902411nt_V_V @ g ) ).
% x
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: v] :
~ ( member_V2 @ X @ ( labele1134902411nt_V_V @ g ) ) ).
% \<open>\<And>thesis. (\<And>x. x \<in> vertices G' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2__C1_C_I3_J,axiom,
( ( m @ xa )
= ( m @ y ) ) ).
% "1"(3)
thf(fact_3__C1_C_I1_J,axiom,
member_V @ xa @ c ).
% "1"(1)
thf(fact_4__C1_C_I2_J,axiom,
member_V @ y @ c ).
% "1"(2)
thf(fact_5__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
member2015049524od_V_V @ ( product_Pair_V_V2 @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ xa ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ).
% \<open>(m x, m x) \<in> getRel (S_Const x) (map_graph_fn G' h)\<close>
thf(fact_6__092_060open_062_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ay_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
member2015049524od_V_V @ ( product_Pair_V_V2 @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ y ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ).
% \<open>(m x, m x) \<in> getRel (S_Const y) (map_graph_fn G' h)\<close>
thf(fact_7_gr_I1_J,axiom,
( g
= ( restri1305980611nt_V_V @ g ) ) ).
% gr(1)
thf(fact_8_h_I3_J,axiom,
graph_1808119_V_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) @ g @ ( id_on_V2 @ ( labele1134902411nt_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ).
% h(3)
thf(fact_9_Standard__Constant_Oinject,axiom,
! [X4: v2,Y4: v2] :
( ( ( standard_S_Const_V @ X4 )
= ( standard_S_Const_V @ Y4 ) )
= ( X4 = Y4 ) ) ).
% Standard_Constant.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: v,X2: v,Y1: v,Y2: v] :
( ( ( product_Pair_V_V2 @ X1 @ X2 )
= ( product_Pair_V_V2 @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: standard_Constant_V,X2: product_prod_V_V,Y1: standard_Constant_V,Y2: product_prod_V_V] :
( ( ( produc107991630od_V_V @ X1 @ X2 )
= ( produc107991630od_V_V @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: v,B: v,A2: v,B2: v] :
( ( ( product_Pair_V_V2 @ A @ B )
= ( product_Pair_V_V2 @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
! [A: standard_Constant_V,B: product_prod_V_V,A2: standard_Constant_V,B2: product_prod_V_V] :
( ( ( produc107991630od_V_V @ A @ B )
= ( produc107991630od_V_V @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_m,axiom,
! [X3: v2] :
( ( member_V @ X3 @ c )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( m @ X3 ) @ ( m @ X3 ) ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ X3 ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ) ).
% m
thf(fact_15_top__nonempty,axiom,
member2015049524od_V_V @ ( product_Pair_V_V2 @ x @ x ) @ ( getRel1432786916nt_V_V @ standard_S_Top_V @ g ) ).
% top_nonempty
thf(fact_16_cf,axiom,
( ( getRel1432786916nt_V_V @ standard_S_Bot_V @ g )
= bot_bo1389414743od_V_V ) ).
% cf
thf(fact_17__092_060open_062_092_060And_062c_O_Ac_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_092_060exists_062v_O_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
! [C: v2] :
( ( member_V @ C @ c )
=> ? [V: v] : ( member2015049524od_V_V @ ( product_Pair_V_V2 @ V @ V ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ C ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<And>c. c \<in> C \<Longrightarrow> \<exists>v. (v, v) \<in> getRel (S_Const c) (map_graph_fn G' h)\<close>
thf(fact_18__092_060open_062_092_060forall_062c_O_A_092_060exists_062v_O_Ac_A_092_060in_062_AC_A_092_060longrightarrow_062_A_Iv_M_Av_J_A_092_060in_062_AgetRel_A_IS__Const_Ac_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
! [C2: v2] :
? [V: v] :
( ( member_V @ C2 @ c )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ V @ V ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ C2 ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<forall>c. \<exists>v. c \<in> C \<longrightarrow> (v, v) \<in> getRel (S_Const c) (map_graph_fn G' h)\<close>
thf(fact_19__092_060open_062_092_060exists_062f_O_A_092_060forall_062x_O_Ax_A_092_060in_062_AC_A_092_060longrightarrow_062_A_If_Ax_M_Af_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_092_060close_062,axiom,
? [F: v2 > v] :
! [X5: v2] :
( ( member_V @ X5 @ c )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( F @ X5 ) @ ( F @ X5 ) ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ X5 ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<exists>f. \<forall>x. x \<in> C \<longrightarrow> (f x, f x) \<in> getRel (S_Const x) (map_graph_fn G' h)\<close>
thf(fact_20__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AC_A_092_060Longrightarrow_062_A_Im_Ax_M_Am_Ax_J_A_092_060in_062_AgetRel_A_IS__Const_Ax_J_A_Imap__graph__fn_AG_H_Ah_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: v2 > v] :
~ ! [X5: v2] :
( ( member_V @ X5 @ c )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( M @ X5 ) @ ( M @ X5 ) ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ X5 ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ) ).
% \<open>\<And>thesis. (\<And>m. (\<And>x. x \<in> C \<Longrightarrow> (m x, m x) \<in> getRel (S_Const x) (map_graph_fn G' h)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_21_getRel__homR,axiom,
! [Y: v,Z: v,L: standard_Constant_V,G: labele2115946735nt_V_V,U: v,F2: set_Product_prod_V_V,V2: v] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ Y @ Z ) @ ( getRel1432786916nt_V_V @ L @ G ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ Y @ U ) @ F2 )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ Z @ V2 ) @ F2 )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ U @ V2 ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr907434255tant_V @ F2 @ G ) ) ) ) ) ) ).
% getRel_homR
thf(fact_22_surj__pair,axiom,
! [P: product_prod_V_V] :
? [X: v,Y3: v] :
( P
= ( product_Pair_V_V2 @ X @ Y3 ) ) ).
% surj_pair
thf(fact_23_surj__pair,axiom,
! [P: produc1699571228od_V_V] :
? [X: standard_Constant_V,Y3: product_prod_V_V] :
( P
= ( produc107991630od_V_V @ X @ Y3 ) ) ).
% surj_pair
thf(fact_24_h_I1_J,axiom,
( ( comp_V_V_V2 @ h @ h )
= h ) ).
% h(1)
thf(fact_25_getRel__map__fn,axiom,
! [A22: product_prod_V_V,G: labele236903175od_V_V,B22: product_prod_V_V,L: standard_Constant_V,F2: product_prod_V_V > v,A: v,B: v] :
( ( member2015049524od_V_V @ A22 @ ( labele981584981od_V_V @ G ) )
=> ( ( member2015049524od_V_V @ B22 @ ( labele981584981od_V_V @ G ) )
=> ( ( member134497296od_V_V @ ( produc1905750615od_V_V @ A22 @ B22 ) @ ( getRel1902920892od_V_V @ L @ G ) )
=> ( ( ( F2 @ A22 )
= A )
=> ( ( ( F2 @ B22 )
= B )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr875022471tant_V @ ( bNF_Gr1069958334_V_V_V @ ( labele981584981od_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_26_getRel__map__fn,axiom,
! [A22: v2,G: labele788688367nt_V_V,B22: v2,L: standard_Constant_V,F2: v2 > v,A: v,B: v] :
( ( member_V @ A22 @ ( labele1152678333nt_V_V @ G ) )
=> ( ( member_V @ B22 @ ( labele1152678333nt_V_V @ G ) )
=> ( ( member1990987792od_V_V @ ( product_Pair_V_V @ A22 @ B22 ) @ ( getRel632961956nt_V_V @ L @ G ) )
=> ( ( ( F2 @ A22 )
= A )
=> ( ( ( F2 @ B22 )
= B )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr530397039tant_V @ ( bNF_Gr_V_V @ ( labele1152678333nt_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_27_getRel__map__fn,axiom,
! [A22: produc1699571228od_V_V,G: labele201426742od_V_V,B22: produc1699571228od_V_V,L: standard_Constant_V,F2: produc1699571228od_V_V > v,A: v,B: v] :
( ( member1010391347od_V_V @ A22 @ ( labele1328382418od_V_V @ G ) )
=> ( ( member1010391347od_V_V @ B22 @ ( labele1328382418od_V_V @ G ) )
=> ( ( member785416642od_V_V @ ( produc1736057425od_V_V @ A22 @ B22 ) @ ( getRel2057312683od_V_V @ L @ G ) )
=> ( ( ( F2 @ A22 )
= A )
=> ( ( ( F2 @ B22 )
= B )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr157381846tant_V @ ( bNF_Gr480427157_V_V_V @ ( labele1328382418od_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_28_getRel__map__fn,axiom,
! [A22: v,G: labele2115946735nt_V_V,B22: v,L: standard_Constant_V,F2: v > v,A: v,B: v] :
( ( member_V2 @ A22 @ ( labele1134902411nt_V_V @ G ) )
=> ( ( member_V2 @ B22 @ ( labele1134902411nt_V_V @ G ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A22 @ B22 ) @ ( getRel1432786916nt_V_V @ L @ G ) )
=> ( ( ( F2 @ A22 )
= A )
=> ( ( ( F2 @ B22 )
= B )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_29_getRel__hom,axiom,
! [Y: product_prod_V_V,Z: product_prod_V_V,L: standard_Constant_V,G: labele236903175od_V_V,F2: product_prod_V_V > v] :
( ( member134497296od_V_V @ ( produc1905750615od_V_V @ Y @ Z ) @ ( getRel1902920892od_V_V @ L @ G ) )
=> ( ( member2015049524od_V_V @ Y @ ( labele981584981od_V_V @ G ) )
=> ( ( member2015049524od_V_V @ Z @ ( labele981584981od_V_V @ G ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr875022471tant_V @ ( bNF_Gr1069958334_V_V_V @ ( labele981584981od_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_30_getRel__hom,axiom,
! [Y: v2,Z: v2,L: standard_Constant_V,G: labele788688367nt_V_V,F2: v2 > v] :
( ( member1990987792od_V_V @ ( product_Pair_V_V @ Y @ Z ) @ ( getRel632961956nt_V_V @ L @ G ) )
=> ( ( member_V @ Y @ ( labele1152678333nt_V_V @ G ) )
=> ( ( member_V @ Z @ ( labele1152678333nt_V_V @ G ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr530397039tant_V @ ( bNF_Gr_V_V @ ( labele1152678333nt_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_31_getRel__hom,axiom,
! [Y: produc1699571228od_V_V,Z: produc1699571228od_V_V,L: standard_Constant_V,G: labele201426742od_V_V,F2: produc1699571228od_V_V > v] :
( ( member785416642od_V_V @ ( produc1736057425od_V_V @ Y @ Z ) @ ( getRel2057312683od_V_V @ L @ G ) )
=> ( ( member1010391347od_V_V @ Y @ ( labele1328382418od_V_V @ G ) )
=> ( ( member1010391347od_V_V @ Z @ ( labele1328382418od_V_V @ G ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr157381846tant_V @ ( bNF_Gr480427157_V_V_V @ ( labele1328382418od_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_32_getRel__hom,axiom,
! [Y: v,Z: v,L: standard_Constant_V,G: labele2115946735nt_V_V,F2: v > v] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ Y @ Z ) @ ( getRel1432786916nt_V_V @ L @ G ) )
=> ( ( member_V2 @ Y @ ( labele1134902411nt_V_V @ G ) )
=> ( ( member_V2 @ Z @ ( labele1134902411nt_V_V @ G ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( F2 @ Y ) @ ( F2 @ Z ) ) @ ( getRel1432786916nt_V_V @ L @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_33_Standard__Constant_Odistinct_I1_J,axiom,
standard_S_Top_V != standard_S_Bot_V ).
% Standard_Constant.distinct(1)
thf(fact_34_Standard__Constant_Odistinct_I5_J,axiom,
! [X4: v2] :
( standard_S_Top_V
!= ( standard_S_Const_V @ X4 ) ) ).
% Standard_Constant.distinct(5)
thf(fact_35_Standard__Constant_Odistinct_I9_J,axiom,
! [X4: v2] :
( standard_S_Bot_V
!= ( standard_S_Const_V @ X4 ) ) ).
% Standard_Constant.distinct(9)
thf(fact_36_getRel__dom_I1_J,axiom,
! [G: labele2115946735nt_V_V,A: v,B: v,L: standard_Constant_V] :
( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( getRel1432786916nt_V_V @ L @ G ) )
=> ( member_V2 @ A @ ( labele1134902411nt_V_V @ G ) ) ) ) ).
% getRel_dom(1)
thf(fact_37_getRel__dom_I2_J,axiom,
! [G: labele2115946735nt_V_V,A: v,B: v,L: standard_Constant_V] :
( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( getRel1432786916nt_V_V @ L @ G ) )
=> ( member_V2 @ B @ ( labele1134902411nt_V_V @ G ) ) ) ) ).
% getRel_dom(2)
thf(fact_38_getRel__subgraph,axiom,
! [Y: v,Z: v,L: standard_Constant_V,G: labele2115946735nt_V_V,G2: labele2115946735nt_V_V] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ Y @ Z ) @ ( getRel1432786916nt_V_V @ L @ G ) )
=> ( ( graph_1808119_V_V_V @ G @ G2 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G ) ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ Y @ Z ) @ ( getRel1432786916nt_V_V @ L @ G2 ) ) ) ) ).
% getRel_subgraph
thf(fact_39_old_Oprod_Oinducts,axiom,
! [P2: product_prod_V_V > $o,Prod: product_prod_V_V] :
( ! [A3: v,B3: v] : ( P2 @ ( product_Pair_V_V2 @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_40_old_Oprod_Oinducts,axiom,
! [P2: produc1699571228od_V_V > $o,Prod: produc1699571228od_V_V] :
( ! [A3: standard_Constant_V,B3: product_prod_V_V] : ( P2 @ ( produc107991630od_V_V @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_41_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_V_V] :
~ ! [A3: v,B3: v] :
( Y
!= ( product_Pair_V_V2 @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_42_old_Oprod_Oexhaust,axiom,
! [Y: produc1699571228od_V_V] :
~ ! [A3: standard_Constant_V,B3: product_prod_V_V] :
( Y
!= ( produc107991630od_V_V @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_43_prod__induct3,axiom,
! [P2: produc1699571228od_V_V > $o,X3: produc1699571228od_V_V] :
( ! [A3: standard_Constant_V,B3: v,C3: v] : ( P2 @ ( produc107991630od_V_V @ A3 @ ( product_Pair_V_V2 @ B3 @ C3 ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_44_prod__cases3,axiom,
! [Y: produc1699571228od_V_V] :
~ ! [A3: standard_Constant_V,B3: v,C3: v] :
( Y
!= ( produc107991630od_V_V @ A3 @ ( product_Pair_V_V2 @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_45_Pair__inject,axiom,
! [A: v,B: v,A2: v,B2: v] :
( ( ( product_Pair_V_V2 @ A @ B )
= ( product_Pair_V_V2 @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_46_Pair__inject,axiom,
! [A: standard_Constant_V,B: product_prod_V_V,A2: standard_Constant_V,B2: product_prod_V_V] :
( ( ( produc107991630od_V_V @ A @ B )
= ( produc107991630od_V_V @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_47_prod__cases,axiom,
! [P2: product_prod_V_V > $o,P: product_prod_V_V] :
( ! [A3: v,B3: v] : ( P2 @ ( product_Pair_V_V2 @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_48_prod__cases,axiom,
! [P2: produc1699571228od_V_V > $o,P: produc1699571228od_V_V] :
( ! [A3: standard_Constant_V,B3: product_prod_V_V] : ( P2 @ ( produc107991630od_V_V @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_49_mg,axiom,
! [R: produc778275879_V_nat] :
( ( mainta197426964_nat_V @ R @ g )
=> ( mainta197426964_nat_V @ R @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ g ) @ h ) @ g ) ) ) ).
% mg
thf(fact_50_mem__Collect__eq,axiom,
! [A: product_prod_V_V,P2: product_prod_V_V > $o] :
( ( member2015049524od_V_V @ A @ ( collec2040633974od_V_V @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: v,P2: v > $o] :
( ( member_V2 @ A @ ( collect_V2 @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: v2,P2: v2 > $o] :
( ( member_V @ A @ ( collect_V @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: produc1699571228od_V_V,P2: produc1699571228od_V_V > $o] :
( ( member1010391347od_V_V @ A @ ( collec1840263537od_V_V @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A4: set_Product_prod_V_V] :
( ( collec2040633974od_V_V
@ ^ [X6: product_prod_V_V] : ( member2015049524od_V_V @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A4: set_V] :
( ( collect_V2
@ ^ [X6: v] : ( member_V2 @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A4: set_V2] :
( ( collect_V
@ ^ [X6: v2] : ( member_V @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A4: set_Pr744279122od_V_V] :
( ( collec1840263537od_V_V
@ ^ [X6: produc1699571228od_V_V] : ( member1010391347od_V_V @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_58_graph__homo,axiom,
! [G: labele2115946735nt_V_V,F2: v > v] :
( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( graph_1808119_V_V_V @ G @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) ) ) ).
% graph_homo
thf(fact_59_map__graph__fn__id_I2_J,axiom,
! [X7: labele2115946735nt_V_V] :
( ( map_gr907434255tant_V @ ( id_on_V2 @ ( labele1134902411nt_V_V @ X7 ) ) @ X7 )
= ( restri1305980611nt_V_V @ X7 ) ) ).
% map_graph_fn_id(2)
thf(fact_60_map__graph__fn__graphI,axiom,
! [G: labele2115946735nt_V_V,F2: v > v] :
( ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G )
= ( restri1305980611nt_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) ) ) ).
% map_graph_fn_graphI
thf(fact_61_map__graph__preserves__subgraph,axiom,
! [A4: labele2115946735nt_V_V,B4: labele2115946735nt_V_V,F2: set_Product_prod_V_V] :
( ( graph_1808119_V_V_V @ A4 @ B4 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ A4 ) ) )
=> ( graph_1808119_V_V_V @ ( map_gr907434255tant_V @ F2 @ A4 ) @ ( map_gr907434255tant_V @ F2 @ B4 ) @ ( id_on_V2 @ ( labele1134902411nt_V_V @ ( map_gr907434255tant_V @ F2 @ A4 ) ) ) ) ) ).
% map_graph_preserves_subgraph
thf(fact_62_subgraph__refl,axiom,
! [G: labele2115946735nt_V_V] :
( ( graph_1808119_V_V_V @ G @ G @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G ) ) )
= ( G
= ( restri1305980611nt_V_V @ G ) ) ) ).
% subgraph_refl
thf(fact_63_subgraph__restrict,axiom,
! [G: labele2115946735nt_V_V] :
( ( graph_1808119_V_V_V @ G @ ( restri1305980611nt_V_V @ G ) @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G ) ) )
= ( G
= ( restri1305980611nt_V_V @ G ) ) ) ).
% subgraph_restrict
thf(fact_64_graph__homomorphism__Id,axiom,
! [A: labele2115946735nt_V_V] : ( graph_1808119_V_V_V @ ( restri1305980611nt_V_V @ A ) @ ( restri1305980611nt_V_V @ A ) @ ( id_on_V2 @ ( labele1134902411nt_V_V @ A ) ) ) ).
% graph_homomorphism_Id
thf(fact_65_Id__on__empty,axiom,
( ( id_on_1882925319_V_nat @ bot_bo1053950006_V_nat )
= bot_bo907687539_V_nat ) ).
% Id_on_empty
thf(fact_66_Id__on__empty,axiom,
( ( id_on_2125509800od_V_V @ bot_bo1389414743od_V_V )
= bot_bo12878899od_V_V ) ).
% Id_on_empty
thf(fact_67_Id__on__empty,axiom,
( ( id_on_V2 @ bot_bot_set_V2 )
= bot_bo1389414743od_V_V ) ).
% Id_on_empty
thf(fact_68_Id__on__empty,axiom,
( ( id_on_558037444_V_nat @ bot_bo907687539_V_nat )
= bot_bo2057106227_V_nat ) ).
% Id_on_empty
thf(fact_69_Gr__empty,axiom,
! [F2: labele2128733290_V_nat > labele2128733290_V_nat] :
( ( bNF_Gr17960482_V_nat @ bot_bo1053950006_V_nat @ F2 )
= bot_bo907687539_V_nat ) ).
% Gr_empty
thf(fact_70_Gr__empty,axiom,
! [F2: v > v] :
( ( bNF_Gr_V_V2 @ bot_bot_set_V2 @ F2 )
= bot_bo1389414743od_V_V ) ).
% Gr_empty
thf(fact_71_restrict__idemp,axiom,
! [X3: labele2115946735nt_V_V] :
( ( restri1305980611nt_V_V @ ( restri1305980611nt_V_V @ X3 ) )
= ( restri1305980611nt_V_V @ X3 ) ) ).
% restrict_idemp
thf(fact_72_in__Gr,axiom,
! [X3: v,Y: v,A4: set_V,F2: v > v] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ X3 @ Y ) @ ( bNF_Gr_V_V2 @ A4 @ F2 ) )
= ( ( member_V2 @ X3 @ A4 )
& ( ( F2 @ X3 )
= Y ) ) ) ).
% in_Gr
thf(fact_73_in__Gr,axiom,
! [X3: standard_Constant_V,Y: product_prod_V_V,A4: set_St1111633946tant_V,F2: standard_Constant_V > product_prod_V_V] :
( ( member1010391347od_V_V @ ( produc107991630od_V_V @ X3 @ Y ) @ ( bNF_Gr320641027od_V_V @ A4 @ F2 ) )
= ( ( member1542015739tant_V @ X3 @ A4 )
& ( ( F2 @ X3 )
= Y ) ) ) ).
% in_Gr
thf(fact_74_Gr__not__in,axiom,
! [X3: v,F3: set_V,F2: v > v,Y: v] :
( ( ~ ( member_V2 @ X3 @ F3 )
| ( ( F2 @ X3 )
!= Y ) )
=> ~ ( member2015049524od_V_V @ ( product_Pair_V_V2 @ X3 @ Y ) @ ( bNF_Gr_V_V2 @ F3 @ F2 ) ) ) ).
% Gr_not_in
thf(fact_75_Gr__not__in,axiom,
! [X3: standard_Constant_V,F3: set_St1111633946tant_V,F2: standard_Constant_V > product_prod_V_V,Y: product_prod_V_V] :
( ( ~ ( member1542015739tant_V @ X3 @ F3 )
| ( ( F2 @ X3 )
!= Y ) )
=> ~ ( member1010391347od_V_V @ ( produc107991630od_V_V @ X3 @ Y ) @ ( bNF_Gr320641027od_V_V @ F3 @ F2 ) ) ) ).
% Gr_not_in
thf(fact_76_Id__onI,axiom,
! [A: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ A @ A4 )
=> ( member134497296od_V_V @ ( produc1905750615od_V_V @ A @ A ) @ ( id_on_2125509800od_V_V @ A4 ) ) ) ).
% Id_onI
thf(fact_77_Id__onI,axiom,
! [A: v2,A4: set_V2] :
( ( member_V @ A @ A4 )
=> ( member1990987792od_V_V @ ( product_Pair_V_V @ A @ A ) @ ( id_on_V @ A4 ) ) ) ).
% Id_onI
thf(fact_78_Id__onI,axiom,
! [A: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ A @ A4 )
=> ( member785416642od_V_V @ ( produc1736057425od_V_V @ A @ A ) @ ( id_on_1491482303od_V_V @ A4 ) ) ) ).
% Id_onI
thf(fact_79_Id__onI,axiom,
! [A: v,A4: set_V] :
( ( member_V2 @ A @ A4 )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ A ) @ ( id_on_V2 @ A4 ) ) ) ).
% Id_onI
thf(fact_80_vertices__restrict,axiom,
! [G: labele2115946735nt_V_V] :
( ( labele1134902411nt_V_V @ ( restri1305980611nt_V_V @ G ) )
= ( labele1134902411nt_V_V @ G ) ) ).
% vertices_restrict
thf(fact_81_map__graph__preserves__restricted,axiom,
! [G: labele2115946735nt_V_V,F2: set_Product_prod_V_V] :
( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( ( map_gr907434255tant_V @ F2 @ G )
= ( restri1305980611nt_V_V @ ( map_gr907434255tant_V @ F2 @ G ) ) ) ) ).
% map_graph_preserves_restricted
thf(fact_82_map__graph__fn__comp,axiom,
! [G: labele2115946735nt_V_V,F2: v > v,G3: v > v] :
( ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ ( comp_V_V_V2 @ F2 @ G3 ) ) @ G )
= ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ G3 ) @ G ) ) @ F2 ) @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ G3 ) @ G ) ) ) ).
% map_graph_fn_comp
thf(fact_83_assms_I1_J,axiom,
! [X5: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X5 @ ( standa157301464ules_V @ c @ l ) )
=> ( mainta197426964_nat_V @ X5 @ g ) ) ).
% assms(1)
thf(fact_84_ne,axiom,
mainta197426964_nat_V @ standa1319953089tant_V @ g ).
% ne
thf(fact_85__092_060open_062maintained_Anonempty__rule_AG_H_A_061_A_Ivertices_AG_H_A_092_060noteq_062_A_123_125_J_092_060close_062,axiom,
( ( mainta197426964_nat_V @ standa1319953089tant_V @ g )
= ( ( labele1134902411nt_V_V @ g )
!= bot_bot_set_V2 ) ) ).
% \<open>maintained nonempty_rule G' = (vertices G' \<noteq> {})\<close>
thf(fact_86_maintained__preserved__by__isomorphism,axiom,
! [G: labele236903175od_V_V,F2: v > product_prod_V_V,G3: product_prod_V_V > v,R: produc778275879_V_nat] :
( ! [X: product_prod_V_V] :
( ( member2015049524od_V_V @ X @ ( labele981584981od_V_V @ G ) )
=> ( ( comp_V772749048od_V_V @ F2 @ G3 @ X )
= X ) )
=> ( ( G
= ( restri1502575645od_V_V @ G ) )
=> ( ( mainta197426964_nat_V @ R @ ( map_gr875022471tant_V @ ( bNF_Gr1069958334_V_V_V @ ( labele981584981od_V_V @ G ) @ G3 ) @ G ) )
=> ( mainta1984786828od_V_V @ R @ G ) ) ) ) ).
% maintained_preserved_by_isomorphism
thf(fact_87_maintained__preserved__by__isomorphism,axiom,
! [G: labele788688367nt_V_V,F2: v > v2,G3: v2 > v,R: produc778275879_V_nat] :
( ! [X: v2] :
( ( member_V @ X @ ( labele1152678333nt_V_V @ G ) )
=> ( ( comp_V_V_V @ F2 @ G3 @ X )
= X ) )
=> ( ( G
= ( restri175070597nt_V_V @ G ) )
=> ( ( mainta197426964_nat_V @ R @ ( map_gr530397039tant_V @ ( bNF_Gr_V_V @ ( labele1152678333nt_V_V @ G ) @ G3 ) @ G ) )
=> ( mainta1026355316_nat_V @ R @ G ) ) ) ) ).
% maintained_preserved_by_isomorphism
thf(fact_88_maintained__preserved__by__isomorphism,axiom,
! [G: labele201426742od_V_V,F2: v > produc1699571228od_V_V,G3: produc1699571228od_V_V > v,R: produc778275879_V_nat] :
( ! [X: produc1699571228od_V_V] :
( ( member1010391347od_V_V @ X @ ( labele1328382418od_V_V @ G ) )
=> ( ( comp_V1480932210od_V_V @ F2 @ G3 @ X )
= X ) )
=> ( ( G
= ( restri1699516426od_V_V @ G ) )
=> ( ( mainta197426964_nat_V @ R @ ( map_gr157381846tant_V @ ( bNF_Gr480427157_V_V_V @ ( labele1328382418od_V_V @ G ) @ G3 ) @ G ) )
=> ( mainta1342478555od_V_V @ R @ G ) ) ) ) ).
% maintained_preserved_by_isomorphism
thf(fact_89_maintained__preserved__by__isomorphism,axiom,
! [G: labele2115946735nt_V_V,F2: v > v,G3: v > v,R: produc778275879_V_nat] :
( ! [X: v] :
( ( member_V2 @ X @ ( labele1134902411nt_V_V @ G ) )
=> ( ( comp_V_V_V2 @ F2 @ G3 @ X )
= X ) )
=> ( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( ( mainta197426964_nat_V @ R @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ G3 ) @ G ) )
=> ( mainta197426964_nat_V @ R @ G ) ) ) ) ).
% maintained_preserved_by_isomorphism
thf(fact_90_idemp__embedding__maintained__preserved,axiom,
! [G: labele2115946735nt_V_V,F2: v > v,R: produc778275879_V_nat] :
( ( graph_1808119_V_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) @ G @ ( id_on_V2 @ ( labele1134902411nt_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) ) ) )
=> ( ! [X: v] :
( ( member_V2 @ X @ ( labele1134902411nt_V_V @ G ) )
=> ( ( comp_V_V_V2 @ F2 @ F2 @ X )
= ( F2 @ X ) ) )
=> ( ( mainta197426964_nat_V @ R @ G )
=> ( mainta197426964_nat_V @ R @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) ) ) ) ) ).
% idemp_embedding_maintained_preserved
thf(fact_91_Id__onE,axiom,
! [C: produc522115815od_V_V,A4: set_Product_prod_V_V] :
( ( member134497296od_V_V @ C @ ( id_on_2125509800od_V_V @ A4 ) )
=> ~ ! [X: product_prod_V_V] :
( ( member2015049524od_V_V @ X @ A4 )
=> ( C
!= ( produc1905750615od_V_V @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_92_Id__onE,axiom,
! [C: product_prod_V_V2,A4: set_V2] :
( ( member1990987792od_V_V @ C @ ( id_on_V @ A4 ) )
=> ~ ! [X: v2] :
( ( member_V @ X @ A4 )
=> ( C
!= ( product_Pair_V_V @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_93_Id__onE,axiom,
! [C: produc1082089497od_V_V,A4: set_Pr744279122od_V_V] :
( ( member785416642od_V_V @ C @ ( id_on_1491482303od_V_V @ A4 ) )
=> ~ ! [X: produc1699571228od_V_V] :
( ( member1010391347od_V_V @ X @ A4 )
=> ( C
!= ( produc1736057425od_V_V @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_94_Id__onE,axiom,
! [C: product_prod_V_V,A4: set_V] :
( ( member2015049524od_V_V @ C @ ( id_on_V2 @ A4 ) )
=> ~ ! [X: v] :
( ( member_V2 @ X @ A4 )
=> ( C
!= ( product_Pair_V_V2 @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_95_Id__on__eqI,axiom,
! [A: product_prod_V_V,B: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( A = B )
=> ( ( member2015049524od_V_V @ A @ A4 )
=> ( member134497296od_V_V @ ( produc1905750615od_V_V @ A @ B ) @ ( id_on_2125509800od_V_V @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_96_Id__on__eqI,axiom,
! [A: v2,B: v2,A4: set_V2] :
( ( A = B )
=> ( ( member_V @ A @ A4 )
=> ( member1990987792od_V_V @ ( product_Pair_V_V @ A @ B ) @ ( id_on_V @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_97_Id__on__eqI,axiom,
! [A: produc1699571228od_V_V,B: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( A = B )
=> ( ( member1010391347od_V_V @ A @ A4 )
=> ( member785416642od_V_V @ ( produc1736057425od_V_V @ A @ B ) @ ( id_on_1491482303od_V_V @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_98_Id__on__eqI,axiom,
! [A: v,B: v,A4: set_V] :
( ( A = B )
=> ( ( member_V2 @ A @ A4 )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( id_on_V2 @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_99_Id__on__iff,axiom,
! [X3: product_prod_V_V,Y: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member134497296od_V_V @ ( produc1905750615od_V_V @ X3 @ Y ) @ ( id_on_2125509800od_V_V @ A4 ) )
= ( ( X3 = Y )
& ( member2015049524od_V_V @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_100_Id__on__iff,axiom,
! [X3: v2,Y: v2,A4: set_V2] :
( ( member1990987792od_V_V @ ( product_Pair_V_V @ X3 @ Y ) @ ( id_on_V @ A4 ) )
= ( ( X3 = Y )
& ( member_V @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_101_Id__on__iff,axiom,
! [X3: produc1699571228od_V_V,Y: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member785416642od_V_V @ ( produc1736057425od_V_V @ X3 @ Y ) @ ( id_on_1491482303od_V_V @ A4 ) )
= ( ( X3 = Y )
& ( member1010391347od_V_V @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_102_Id__on__iff,axiom,
! [X3: v,Y: v,A4: set_V] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ X3 @ Y ) @ ( id_on_V2 @ A4 ) )
= ( ( X3 = Y )
& ( member_V2 @ X3 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_103_subgraph__preserves__hom,axiom,
! [A4: labele2115946735nt_V_V,B4: labele2115946735nt_V_V,X7: labele2115946735nt_V_V,H: set_Product_prod_V_V] :
( ( graph_1808119_V_V_V @ A4 @ B4 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ A4 ) ) )
=> ( ( graph_1808119_V_V_V @ X7 @ A4 @ H )
=> ( graph_1808119_V_V_V @ X7 @ B4 @ H ) ) ) ).
% subgraph_preserves_hom
thf(fact_104_subgraph__trans,axiom,
! [G_1: labele2115946735nt_V_V,G_2: labele2115946735nt_V_V,G_3: labele2115946735nt_V_V] :
( ( graph_1808119_V_V_V @ G_1 @ G_2 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G_1 ) ) )
=> ( ( graph_1808119_V_V_V @ G_2 @ G_3 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G_2 ) ) )
=> ( graph_1808119_V_V_V @ G_1 @ G_3 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G_1 ) ) ) ) ) ).
% subgraph_trans
thf(fact_105_map__graph__fn__eqI,axiom,
! [G: labele2115946735nt_V_V,F2: v > v,G3: v > v] :
( ! [X: v] :
( ( member_V2 @ X @ ( labele1134902411nt_V_V @ G ) )
=> ( ( F2 @ X )
= ( G3 @ X ) ) )
=> ( ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G )
= ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ G3 ) @ G ) ) ) ).
% map_graph_fn_eqI
thf(fact_106_tr,axiom,
mainta197426964_nat_V @ ( standa214871990tant_V @ standard_S_Top_V ) @ g ).
% tr
thf(fact_107_h_I4_J,axiom,
! [L: standard_Constant_V,X3: v,Y: v] :
( ( member1010391347od_V_V @ ( produc107991630od_V_V @ L @ ( product_Pair_V_V2 @ X3 @ Y ) ) @ ( labele515735386nt_V_V @ g ) )
= ( member1010391347od_V_V @ ( produc107991630od_V_V @ L @ ( product_Pair_V_V2 @ ( h @ X3 ) @ ( h @ Y ) ) ) @ ( labele515735386nt_V_V @ g ) ) ) ).
% h(4)
thf(fact_108_maintained__refl,axiom,
! [R2: labele2128733290_V_nat,G: labele2115946735nt_V_V] : ( mainta197426964_nat_V @ ( produc698429463_V_nat @ R2 @ R2 ) @ G ) ).
% maintained_refl
thf(fact_109_comp__apply,axiom,
( comp_V_V_V2
= ( ^ [F4: v > v,G4: v > v,X6: v] : ( F4 @ ( G4 @ X6 ) ) ) ) ).
% comp_apply
thf(fact_110_empty__Collect__eq,axiom,
! [P2: product_prod_V_V > $o] :
( ( bot_bo1389414743od_V_V
= ( collec2040633974od_V_V @ P2 ) )
= ( ! [X6: product_prod_V_V] :
~ ( P2 @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_111_empty__Collect__eq,axiom,
! [P2: v > $o] :
( ( bot_bot_set_V2
= ( collect_V2 @ P2 ) )
= ( ! [X6: v] :
~ ( P2 @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_112_empty__Collect__eq,axiom,
! [P2: produc778275879_V_nat > $o] :
( ( bot_bo907687539_V_nat
= ( collec1411521682_V_nat @ P2 ) )
= ( ! [X6: produc778275879_V_nat] :
~ ( P2 @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_113_Collect__empty__eq,axiom,
! [P2: product_prod_V_V > $o] :
( ( ( collec2040633974od_V_V @ P2 )
= bot_bo1389414743od_V_V )
= ( ! [X6: product_prod_V_V] :
~ ( P2 @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_114_Collect__empty__eq,axiom,
! [P2: v > $o] :
( ( ( collect_V2 @ P2 )
= bot_bot_set_V2 )
= ( ! [X6: v] :
~ ( P2 @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_115_Collect__empty__eq,axiom,
! [P2: produc778275879_V_nat > $o] :
( ( ( collec1411521682_V_nat @ P2 )
= bot_bo907687539_V_nat )
= ( ! [X6: produc778275879_V_nat] :
~ ( P2 @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_116_empty__iff,axiom,
! [C: v2] :
~ ( member_V @ C @ bot_bot_set_V ) ).
% empty_iff
thf(fact_117_empty__iff,axiom,
! [C: produc1699571228od_V_V] :
~ ( member1010391347od_V_V @ C @ bot_bo667804902od_V_V ) ).
% empty_iff
thf(fact_118_empty__iff,axiom,
! [C: product_prod_V_V] :
~ ( member2015049524od_V_V @ C @ bot_bo1389414743od_V_V ) ).
% empty_iff
thf(fact_119_empty__iff,axiom,
! [C: v] :
~ ( member_V2 @ C @ bot_bot_set_V2 ) ).
% empty_iff
thf(fact_120_empty__iff,axiom,
! [C: produc778275879_V_nat] :
~ ( member1886821968_V_nat @ C @ bot_bo907687539_V_nat ) ).
% empty_iff
thf(fact_121_all__not__in__conv,axiom,
! [A4: set_V2] :
( ( ! [X6: v2] :
~ ( member_V @ X6 @ A4 ) )
= ( A4 = bot_bot_set_V ) ) ).
% all_not_in_conv
thf(fact_122_all__not__in__conv,axiom,
! [A4: set_Pr744279122od_V_V] :
( ( ! [X6: produc1699571228od_V_V] :
~ ( member1010391347od_V_V @ X6 @ A4 ) )
= ( A4 = bot_bo667804902od_V_V ) ) ).
% all_not_in_conv
thf(fact_123_all__not__in__conv,axiom,
! [A4: set_Product_prod_V_V] :
( ( ! [X6: product_prod_V_V] :
~ ( member2015049524od_V_V @ X6 @ A4 ) )
= ( A4 = bot_bo1389414743od_V_V ) ) ).
% all_not_in_conv
thf(fact_124_all__not__in__conv,axiom,
! [A4: set_V] :
( ( ! [X6: v] :
~ ( member_V2 @ X6 @ A4 ) )
= ( A4 = bot_bot_set_V2 ) ) ).
% all_not_in_conv
thf(fact_125_all__not__in__conv,axiom,
! [A4: set_Pr1058435079_V_nat] :
( ( ! [X6: produc778275879_V_nat] :
~ ( member1886821968_V_nat @ X6 @ A4 ) )
= ( A4 = bot_bo907687539_V_nat ) ) ).
% all_not_in_conv
thf(fact_126_nonempty__rule,axiom,
! [G: labele236903175od_V_V] :
( ( G
= ( restri1502575645od_V_V @ G ) )
=> ( ( mainta1984786828od_V_V @ standa1319953089tant_V @ G )
= ( ( labele981584981od_V_V @ G )
!= bot_bo1389414743od_V_V ) ) ) ).
% nonempty_rule
thf(fact_127_nonempty__rule,axiom,
! [G: labele506073123_V_nat] :
( ( G
= ( restri1932147897_V_nat @ G ) )
=> ( ( mainta922614952_V_nat @ standa1319953089tant_V @ G )
= ( ( labele1822884081_V_nat @ G )
!= bot_bo907687539_V_nat ) ) ) ).
% nonempty_rule
thf(fact_128_nonempty__rule,axiom,
! [G: labele2115946735nt_V_V] :
( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( ( mainta197426964_nat_V @ standa1319953089tant_V @ G )
= ( ( labele1134902411nt_V_V @ G )
!= bot_bot_set_V2 ) ) ) ).
% nonempty_rule
thf(fact_129__092_060open_062maintainedA_A_Iidentity__rules_AL_J_AG_H_092_060close_062,axiom,
! [X5: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X5 @ ( standa1568205529ules_V @ l ) )
=> ( mainta197426964_nat_V @ X5 @ g ) ) ).
% \<open>maintainedA (identity_rules L) G'\<close>
thf(fact_130_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labele2115946735nt_V_V,Labeled_graph2: labele2115946735nt_V_V] :
( ( ( ( labele515735386nt_V_V @ Labeled_graph )
= ( labele515735386nt_V_V @ Labeled_graph2 ) )
& ( ( labele1134902411nt_V_V @ Labeled_graph )
= ( labele1134902411nt_V_V @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_131_emptyE,axiom,
! [A: v2] :
~ ( member_V @ A @ bot_bot_set_V ) ).
% emptyE
thf(fact_132_emptyE,axiom,
! [A: produc1699571228od_V_V] :
~ ( member1010391347od_V_V @ A @ bot_bo667804902od_V_V ) ).
% emptyE
thf(fact_133_emptyE,axiom,
! [A: product_prod_V_V] :
~ ( member2015049524od_V_V @ A @ bot_bo1389414743od_V_V ) ).
% emptyE
thf(fact_134_emptyE,axiom,
! [A: v] :
~ ( member_V2 @ A @ bot_bot_set_V2 ) ).
% emptyE
thf(fact_135_emptyE,axiom,
! [A: produc778275879_V_nat] :
~ ( member1886821968_V_nat @ A @ bot_bo907687539_V_nat ) ).
% emptyE
thf(fact_136_equals0D,axiom,
! [A4: set_V2,A: v2] :
( ( A4 = bot_bot_set_V )
=> ~ ( member_V @ A @ A4 ) ) ).
% equals0D
thf(fact_137_equals0D,axiom,
! [A4: set_Pr744279122od_V_V,A: produc1699571228od_V_V] :
( ( A4 = bot_bo667804902od_V_V )
=> ~ ( member1010391347od_V_V @ A @ A4 ) ) ).
% equals0D
thf(fact_138_equals0D,axiom,
! [A4: set_Product_prod_V_V,A: product_prod_V_V] :
( ( A4 = bot_bo1389414743od_V_V )
=> ~ ( member2015049524od_V_V @ A @ A4 ) ) ).
% equals0D
thf(fact_139_equals0D,axiom,
! [A4: set_V,A: v] :
( ( A4 = bot_bot_set_V2 )
=> ~ ( member_V2 @ A @ A4 ) ) ).
% equals0D
thf(fact_140_equals0D,axiom,
! [A4: set_Pr1058435079_V_nat,A: produc778275879_V_nat] :
( ( A4 = bot_bo907687539_V_nat )
=> ~ ( member1886821968_V_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_141_equals0I,axiom,
! [A4: set_V2] :
( ! [Y3: v2] :
~ ( member_V @ Y3 @ A4 )
=> ( A4 = bot_bot_set_V ) ) ).
% equals0I
thf(fact_142_equals0I,axiom,
! [A4: set_Pr744279122od_V_V] :
( ! [Y3: produc1699571228od_V_V] :
~ ( member1010391347od_V_V @ Y3 @ A4 )
=> ( A4 = bot_bo667804902od_V_V ) ) ).
% equals0I
thf(fact_143_equals0I,axiom,
! [A4: set_Product_prod_V_V] :
( ! [Y3: product_prod_V_V] :
~ ( member2015049524od_V_V @ Y3 @ A4 )
=> ( A4 = bot_bo1389414743od_V_V ) ) ).
% equals0I
thf(fact_144_equals0I,axiom,
! [A4: set_V] :
( ! [Y3: v] :
~ ( member_V2 @ Y3 @ A4 )
=> ( A4 = bot_bot_set_V2 ) ) ).
% equals0I
thf(fact_145_equals0I,axiom,
! [A4: set_Pr1058435079_V_nat] :
( ! [Y3: produc778275879_V_nat] :
~ ( member1886821968_V_nat @ Y3 @ A4 )
=> ( A4 = bot_bo907687539_V_nat ) ) ).
% equals0I
thf(fact_146_ex__in__conv,axiom,
! [A4: set_V2] :
( ( ? [X6: v2] : ( member_V @ X6 @ A4 ) )
= ( A4 != bot_bot_set_V ) ) ).
% ex_in_conv
thf(fact_147_ex__in__conv,axiom,
! [A4: set_Pr744279122od_V_V] :
( ( ? [X6: produc1699571228od_V_V] : ( member1010391347od_V_V @ X6 @ A4 ) )
= ( A4 != bot_bo667804902od_V_V ) ) ).
% ex_in_conv
thf(fact_148_ex__in__conv,axiom,
! [A4: set_Product_prod_V_V] :
( ( ? [X6: product_prod_V_V] : ( member2015049524od_V_V @ X6 @ A4 ) )
= ( A4 != bot_bo1389414743od_V_V ) ) ).
% ex_in_conv
thf(fact_149_ex__in__conv,axiom,
! [A4: set_V] :
( ( ? [X6: v] : ( member_V2 @ X6 @ A4 ) )
= ( A4 != bot_bot_set_V2 ) ) ).
% ex_in_conv
thf(fact_150_ex__in__conv,axiom,
! [A4: set_Pr1058435079_V_nat] :
( ( ? [X6: produc778275879_V_nat] : ( member1886821968_V_nat @ X6 @ A4 ) )
= ( A4 != bot_bo907687539_V_nat ) ) ).
% ex_in_conv
thf(fact_151_comp__def,axiom,
( comp_V_V_V2
= ( ^ [F4: v > v,G4: v > v,X6: v] : ( F4 @ ( G4 @ X6 ) ) ) ) ).
% comp_def
thf(fact_152_comp__assoc,axiom,
! [F2: v > v,G3: v > v,H: v > v] :
( ( comp_V_V_V2 @ ( comp_V_V_V2 @ F2 @ G3 ) @ H )
= ( comp_V_V_V2 @ F2 @ ( comp_V_V_V2 @ G3 @ H ) ) ) ).
% comp_assoc
thf(fact_153_comp__eq__dest,axiom,
! [A: v > v,B: v > v,C: v > v,D: v > v,V2: v] :
( ( ( comp_V_V_V2 @ A @ B )
= ( comp_V_V_V2 @ C @ D ) )
=> ( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_dest
thf(fact_154_comp__eq__elim,axiom,
! [A: v > v,B: v > v,C: v > v,D: v > v] :
( ( ( comp_V_V_V2 @ A @ B )
= ( comp_V_V_V2 @ C @ D ) )
=> ! [V3: v] :
( ( A @ ( B @ V3 ) )
= ( C @ ( D @ V3 ) ) ) ) ).
% comp_eq_elim
thf(fact_155_comp__eq__dest__lhs,axiom,
! [A: v > v,B: v > v,C: v > v,V2: v] :
( ( ( comp_V_V_V2 @ A @ B )
= C )
=> ( ( A @ ( B @ V2 ) )
= ( C @ V2 ) ) ) ).
% comp_eq_dest_lhs
thf(fact_156_edge__preserving__on__graphI,axiom,
! [X7: labele236903175od_V_V,F2: product_prod_V_V > v,Y5: set_Pr744279122od_V_V] :
( ! [L2: standard_Constant_V,X: product_prod_V_V,Y3: product_prod_V_V] :
( ( member361435663od_V_V @ ( produc1482441770od_V_V @ L2 @ ( produc1905750615od_V_V @ X @ Y3 ) ) @ ( labele668941254od_V_V @ X7 ) )
=> ( ( member2015049524od_V_V @ X @ ( labele981584981od_V_V @ X7 ) )
=> ( ( member2015049524od_V_V @ Y3 @ ( labele981584981od_V_V @ X7 ) )
=> ( member1010391347od_V_V @ ( produc107991630od_V_V @ L2 @ ( product_Pair_V_V2 @ ( F2 @ X ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
=> ( edge_p1776314003tant_V @ ( bNF_Gr1069958334_V_V_V @ ( labele981584981od_V_V @ X7 ) @ F2 ) @ ( labele668941254od_V_V @ X7 ) @ Y5 ) ) ).
% edge_preserving_on_graphI
thf(fact_157_edge__preserving__on__graphI,axiom,
! [X7: labele788688367nt_V_V,F2: v2 > v,Y5: set_Pr744279122od_V_V] :
( ! [L2: standard_Constant_V,X: v2,Y3: v2] :
( ( member1416643087od_V_V @ ( produc1171747370od_V_V @ L2 @ ( product_Pair_V_V @ X @ Y3 ) ) @ ( labele1469545390nt_V_V @ X7 ) )
=> ( ( member_V @ X @ ( labele1152678333nt_V_V @ X7 ) )
=> ( ( member_V @ Y3 @ ( labele1152678333nt_V_V @ X7 ) )
=> ( member1010391347od_V_V @ ( produc107991630od_V_V @ L2 @ ( product_Pair_V_V2 @ ( F2 @ X ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
=> ( edge_p1322161019tant_V @ ( bNF_Gr_V_V @ ( labele1152678333nt_V_V @ X7 ) @ F2 ) @ ( labele1469545390nt_V_V @ X7 ) @ Y5 ) ) ).
% edge_preserving_on_graphI
thf(fact_158_edge__preserving__on__graphI,axiom,
! [X7: labele201426742od_V_V,F2: produc1699571228od_V_V > v,Y5: set_Pr744279122od_V_V] :
( ! [L2: standard_Constant_V,X: produc1699571228od_V_V,Y3: produc1699571228od_V_V] :
( ( member600445121od_V_V @ ( produc2132932316od_V_V @ L2 @ ( produc1736057425od_V_V @ X @ Y3 ) ) @ ( labele1794513953od_V_V @ X7 ) )
=> ( ( member1010391347od_V_V @ X @ ( labele1328382418od_V_V @ X7 ) )
=> ( ( member1010391347od_V_V @ Y3 @ ( labele1328382418od_V_V @ X7 ) )
=> ( member1010391347od_V_V @ ( produc107991630od_V_V @ L2 @ ( product_Pair_V_V2 @ ( F2 @ X ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
=> ( edge_p669293514tant_V @ ( bNF_Gr480427157_V_V_V @ ( labele1328382418od_V_V @ X7 ) @ F2 ) @ ( labele1794513953od_V_V @ X7 ) @ Y5 ) ) ).
% edge_preserving_on_graphI
thf(fact_159_edge__preserving__on__graphI,axiom,
! [X7: labele2115946735nt_V_V,F2: v > v,Y5: set_Pr744279122od_V_V] :
( ! [L2: standard_Constant_V,X: v,Y3: v] :
( ( member1010391347od_V_V @ ( produc107991630od_V_V @ L2 @ ( product_Pair_V_V2 @ X @ Y3 ) ) @ ( labele515735386nt_V_V @ X7 ) )
=> ( ( member_V2 @ X @ ( labele1134902411nt_V_V @ X7 ) )
=> ( ( member_V2 @ Y3 @ ( labele1134902411nt_V_V @ X7 ) )
=> ( member1010391347od_V_V @ ( produc107991630od_V_V @ L2 @ ( product_Pair_V_V2 @ ( F2 @ X ) @ ( F2 @ Y3 ) ) ) @ Y5 ) ) ) )
=> ( edge_p1560004099tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ X7 ) @ F2 ) @ ( labele515735386nt_V_V @ X7 ) @ Y5 ) ) ).
% edge_preserving_on_graphI
thf(fact_160_GrD2,axiom,
! [X3: v,Fx: v,A4: set_V,F2: v > v] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ X3 @ Fx ) @ ( bNF_Gr_V_V2 @ A4 @ F2 ) )
=> ( ( F2 @ X3 )
= Fx ) ) ).
% GrD2
thf(fact_161_GrD2,axiom,
! [X3: standard_Constant_V,Fx: product_prod_V_V,A4: set_St1111633946tant_V,F2: standard_Constant_V > product_prod_V_V] :
( ( member1010391347od_V_V @ ( produc107991630od_V_V @ X3 @ Fx ) @ ( bNF_Gr320641027od_V_V @ A4 @ F2 ) )
=> ( ( F2 @ X3 )
= Fx ) ) ).
% GrD2
thf(fact_162_GrD1,axiom,
! [X3: v,Fx: v,A4: set_V,F2: v > v] :
( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ X3 @ Fx ) @ ( bNF_Gr_V_V2 @ A4 @ F2 ) )
=> ( member_V2 @ X3 @ A4 ) ) ).
% GrD1
thf(fact_163_GrD1,axiom,
! [X3: standard_Constant_V,Fx: product_prod_V_V,A4: set_St1111633946tant_V,F2: standard_Constant_V > product_prod_V_V] :
( ( member1010391347od_V_V @ ( produc107991630od_V_V @ X3 @ Fx ) @ ( bNF_Gr320641027od_V_V @ A4 @ F2 ) )
=> ( member1542015739tant_V @ X3 @ A4 ) ) ).
% GrD1
thf(fact_164_map__graph__edge__preserving,axiom,
! [F2: set_Product_prod_V_V,G: labele2115946735nt_V_V] : ( edge_p1560004099tant_V @ F2 @ ( labele515735386nt_V_V @ G ) @ ( labele515735386nt_V_V @ ( map_gr907434255tant_V @ F2 @ G ) ) ) ).
% map_graph_edge_preserving
thf(fact_165_bot__empty__eq,axiom,
( bot_bot_V_o
= ( ^ [X6: v2] : ( member_V @ X6 @ bot_bot_set_V ) ) ) ).
% bot_empty_eq
thf(fact_166_bot__empty__eq,axiom,
( bot_bo2024487199_V_V_o
= ( ^ [X6: produc1699571228od_V_V] : ( member1010391347od_V_V @ X6 @ bot_bo667804902od_V_V ) ) ) ).
% bot_empty_eq
thf(fact_167_bot__empty__eq,axiom,
( bot_bo1168561222_V_V_o
= ( ^ [X6: product_prod_V_V] : ( member2015049524od_V_V @ X6 @ bot_bo1389414743od_V_V ) ) ) ).
% bot_empty_eq
thf(fact_168_bot__empty__eq,axiom,
( bot_bot_V_o2
= ( ^ [X6: v] : ( member_V2 @ X6 @ bot_bot_set_V2 ) ) ) ).
% bot_empty_eq
thf(fact_169_bot__empty__eq,axiom,
( bot_bo1078500394_nat_o
= ( ^ [X6: produc778275879_V_nat] : ( member1886821968_V_nat @ X6 @ bot_bo907687539_V_nat ) ) ) ).
% bot_empty_eq
thf(fact_170_bot__set__def,axiom,
( bot_bo1389414743od_V_V
= ( collec2040633974od_V_V @ bot_bo1168561222_V_V_o ) ) ).
% bot_set_def
thf(fact_171_bot__set__def,axiom,
( bot_bot_set_V2
= ( collect_V2 @ bot_bot_V_o2 ) ) ).
% bot_set_def
thf(fact_172_bot__set__def,axiom,
( bot_bo907687539_V_nat
= ( collec1411521682_V_nat @ bot_bo1078500394_nat_o ) ) ).
% bot_set_def
thf(fact_173_consequence__graphD_I1_J,axiom,
! [Rs: set_Pr1058435079_V_nat,G: labele2115946735nt_V_V,R2: produc778275879_V_nat] :
( ( conseq1236301781_nat_V @ Rs @ G )
=> ( ( member1886821968_V_nat @ R2 @ Rs )
=> ( mainta197426964_nat_V @ R2 @ G ) ) ) ).
% consequence_graphD(1)
thf(fact_174_edge__preserving__atomic,axiom,
! [H1: set_Product_prod_V_V,E1: set_Pr744279122od_V_V,E2: set_Pr744279122od_V_V,V1: v,V12: v,V22: v,V23: v,K: standard_Constant_V] :
( ( edge_p1560004099tant_V @ H1 @ E1 @ E2 )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ V1 @ V12 ) @ H1 )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ V22 @ V23 ) @ H1 )
=> ( ( member1010391347od_V_V @ ( produc107991630od_V_V @ K @ ( product_Pair_V_V2 @ V1 @ V22 ) ) @ E1 )
=> ( member1010391347od_V_V @ ( produc107991630od_V_V @ K @ ( product_Pair_V_V2 @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_175_mnt,axiom,
! [X5: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X5 @ ( sup_su1043134939_V_nat @ ( sup_su1043134939_V_nat @ ( standa1897115807ules_V @ c ) @ ( standa1568205529ules_V @ l ) ) @ ( insert481364471_V_nat @ ( standa214871990tant_V @ standard_S_Top_V ) @ ( insert481364471_V_nat @ standa1319953089tant_V @ bot_bo907687539_V_nat ) ) ) )
=> ( mainta197426964_nat_V @ X5 @ g ) ) ).
% mnt
thf(fact_176_map__graph__in,axiom,
! [G: labele2115946735nt_V_V,A: v,B: v,E: allego859987871tant_V,F2: v > v] :
( ( G
= ( restri1305980611nt_V_V @ G ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( semant993842370nt_V_V @ G @ E ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ ( F2 @ A ) @ ( F2 @ B ) ) @ ( semant993842370nt_V_V @ ( map_gr907434255tant_V @ ( bNF_Gr_V_V2 @ ( labele1134902411nt_V_V @ G ) @ F2 ) @ G ) @ E ) ) ) ) ).
% map_graph_in
thf(fact_177_maintainedI,axiom,
! [A4: labele2128733290_V_nat,G: labele2115946735nt_V_V,B4: labele2128733290_V_nat] :
( ! [F: set_Pr53436518_nat_V] :
( ( graph_1541575154_nat_V @ A4 @ G @ F )
=> ( extens1072773157_nat_V @ ( produc698429463_V_nat @ A4 @ B4 ) @ G @ F ) )
=> ( mainta197426964_nat_V @ ( produc698429463_V_nat @ A4 @ B4 ) @ G ) ) ).
% maintainedI
thf(fact_178_maintainedI,axiom,
! [A4: labele2115946735nt_V_V,G: labele2115946735nt_V_V,B4: labele2115946735nt_V_V] :
( ! [F: set_Product_prod_V_V] :
( ( graph_1808119_V_V_V @ A4 @ G @ F )
=> ( extens1533825066_V_V_V @ ( produc533118295nt_V_V @ A4 @ B4 ) @ G @ F ) )
=> ( mainta1699210777_V_V_V @ ( produc533118295nt_V_V @ A4 @ B4 ) @ G ) ) ).
% maintainedI
thf(fact_179_graph__homomorphism__empty,axiom,
! [G: labele2115946735nt_V_V,F2: set_Pr1228022213_V_V_V] :
( ( graph_840721237_V_V_V @ ( labele94156371od_V_V @ bot_bo451843010od_V_V @ bot_bo1389414743od_V_V ) @ G @ F2 )
= ( ( F2 = bot_bo688400985_V_V_V )
& ( G
= ( restri1305980611nt_V_V @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_180_graph__homomorphism__empty,axiom,
! [G: labele2115946735nt_V_V,F2: set_Pr1056235617_nat_V] :
( ( graph_227683057_nat_V @ ( labele1193521135_V_nat @ bot_bo634818242_V_nat @ bot_bo907687539_V_nat ) @ G @ F2 )
= ( ( F2 = bot_bo1708783349_nat_V )
& ( G
= ( restri1305980611nt_V_V @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_181_graph__homomorphism__empty,axiom,
! [G: labele2115946735nt_V_V,F2: set_Product_prod_V_V] :
( ( graph_1808119_V_V_V @ ( labele712009229nt_V_V @ bot_bo667804902od_V_V @ bot_bot_set_V2 ) @ G @ F2 )
= ( ( F2 = bot_bo1389414743od_V_V )
& ( G
= ( restri1305980611nt_V_V @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_182_subgraph__def,axiom,
! [G_1: labele2115946735nt_V_V,G_2: labele2115946735nt_V_V] :
( ( graph_1808119_V_V_V @ G_1 @ G_2 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ G_1 ) ) )
= ( ( G_1
= ( restri1305980611nt_V_V @ G_1 ) )
& ( G_2
= ( restri1305980611nt_V_V @ G_2 ) )
& ( ( graph_1141886696nt_V_V @ G_1 @ G_2 )
= G_2 ) ) ) ).
% subgraph_def
thf(fact_183_insert__absorb2,axiom,
! [X3: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( insert481364471_V_nat @ X3 @ ( insert481364471_V_nat @ X3 @ A4 ) )
= ( insert481364471_V_nat @ X3 @ A4 ) ) ).
% insert_absorb2
thf(fact_184_insert__iff,axiom,
! [A: produc778275879_V_nat,B: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ A @ ( insert481364471_V_nat @ B @ A4 ) )
= ( ( A = B )
| ( member1886821968_V_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_185_insert__iff,axiom,
! [A: product_prod_V_V,B: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ A @ ( insert2024285275od_V_V @ B @ A4 ) )
= ( ( A = B )
| ( member2015049524od_V_V @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_186_insert__iff,axiom,
! [A: v,B: v,A4: set_V] :
( ( member_V2 @ A @ ( insert_V2 @ B @ A4 ) )
= ( ( A = B )
| ( member_V2 @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_187_insert__iff,axiom,
! [A: v2,B: v2,A4: set_V2] :
( ( member_V @ A @ ( insert_V @ B @ A4 ) )
= ( ( A = B )
| ( member_V @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_188_insert__iff,axiom,
! [A: produc1699571228od_V_V,B: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ A @ ( insert135842892od_V_V @ B @ A4 ) )
= ( ( A = B )
| ( member1010391347od_V_V @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_189_insertCI,axiom,
! [A: produc778275879_V_nat,B4: set_Pr1058435079_V_nat,B: produc778275879_V_nat] :
( ( ~ ( member1886821968_V_nat @ A @ B4 )
=> ( A = B ) )
=> ( member1886821968_V_nat @ A @ ( insert481364471_V_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_190_insertCI,axiom,
! [A: product_prod_V_V,B4: set_Product_prod_V_V,B: product_prod_V_V] :
( ( ~ ( member2015049524od_V_V @ A @ B4 )
=> ( A = B ) )
=> ( member2015049524od_V_V @ A @ ( insert2024285275od_V_V @ B @ B4 ) ) ) ).
% insertCI
thf(fact_191_insertCI,axiom,
! [A: v,B4: set_V,B: v] :
( ( ~ ( member_V2 @ A @ B4 )
=> ( A = B ) )
=> ( member_V2 @ A @ ( insert_V2 @ B @ B4 ) ) ) ).
% insertCI
thf(fact_192_insertCI,axiom,
! [A: v2,B4: set_V2,B: v2] :
( ( ~ ( member_V @ A @ B4 )
=> ( A = B ) )
=> ( member_V @ A @ ( insert_V @ B @ B4 ) ) ) ).
% insertCI
thf(fact_193_insertCI,axiom,
! [A: produc1699571228od_V_V,B4: set_Pr744279122od_V_V,B: produc1699571228od_V_V] :
( ( ~ ( member1010391347od_V_V @ A @ B4 )
=> ( A = B ) )
=> ( member1010391347od_V_V @ A @ ( insert135842892od_V_V @ B @ B4 ) ) ) ).
% insertCI
thf(fact_194_Un__iff,axiom,
! [C: product_prod_V_V,A4: set_Product_prod_V_V,B4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ C @ ( sup_su551882943od_V_V @ A4 @ B4 ) )
= ( ( member2015049524od_V_V @ C @ A4 )
| ( member2015049524od_V_V @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_195_Un__iff,axiom,
! [C: v,A4: set_V,B4: set_V] :
( ( member_V2 @ C @ ( sup_sup_set_V2 @ A4 @ B4 ) )
= ( ( member_V2 @ C @ A4 )
| ( member_V2 @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_196_Un__iff,axiom,
! [C: v2,A4: set_V2,B4: set_V2] :
( ( member_V @ C @ ( sup_sup_set_V @ A4 @ B4 ) )
= ( ( member_V @ C @ A4 )
| ( member_V @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_197_Un__iff,axiom,
! [C: produc1699571228od_V_V,A4: set_Pr744279122od_V_V,B4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ C @ ( sup_su693515134od_V_V @ A4 @ B4 ) )
= ( ( member1010391347od_V_V @ C @ A4 )
| ( member1010391347od_V_V @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_198_Un__iff,axiom,
! [C: produc778275879_V_nat,A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ C @ ( sup_su1043134939_V_nat @ A4 @ B4 ) )
= ( ( member1886821968_V_nat @ C @ A4 )
| ( member1886821968_V_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_199_UnCI,axiom,
! [C: product_prod_V_V,B4: set_Product_prod_V_V,A4: set_Product_prod_V_V] :
( ( ~ ( member2015049524od_V_V @ C @ B4 )
=> ( member2015049524od_V_V @ C @ A4 ) )
=> ( member2015049524od_V_V @ C @ ( sup_su551882943od_V_V @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_200_UnCI,axiom,
! [C: v,B4: set_V,A4: set_V] :
( ( ~ ( member_V2 @ C @ B4 )
=> ( member_V2 @ C @ A4 ) )
=> ( member_V2 @ C @ ( sup_sup_set_V2 @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_201_UnCI,axiom,
! [C: v2,B4: set_V2,A4: set_V2] :
( ( ~ ( member_V @ C @ B4 )
=> ( member_V @ C @ A4 ) )
=> ( member_V @ C @ ( sup_sup_set_V @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_202_UnCI,axiom,
! [C: produc1699571228od_V_V,B4: set_Pr744279122od_V_V,A4: set_Pr744279122od_V_V] :
( ( ~ ( member1010391347od_V_V @ C @ B4 )
=> ( member1010391347od_V_V @ C @ A4 ) )
=> ( member1010391347od_V_V @ C @ ( sup_su693515134od_V_V @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_203_UnCI,axiom,
! [C: produc778275879_V_nat,B4: set_Pr1058435079_V_nat,A4: set_Pr1058435079_V_nat] :
( ( ~ ( member1886821968_V_nat @ C @ B4 )
=> ( member1886821968_V_nat @ C @ A4 ) )
=> ( member1886821968_V_nat @ C @ ( sup_su1043134939_V_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_204_singletonI,axiom,
! [A: v2] : ( member_V @ A @ ( insert_V @ A @ bot_bot_set_V ) ) ).
% singletonI
thf(fact_205_singletonI,axiom,
! [A: produc1699571228od_V_V] : ( member1010391347od_V_V @ A @ ( insert135842892od_V_V @ A @ bot_bo667804902od_V_V ) ) ).
% singletonI
thf(fact_206_singletonI,axiom,
! [A: product_prod_V_V] : ( member2015049524od_V_V @ A @ ( insert2024285275od_V_V @ A @ bot_bo1389414743od_V_V ) ) ).
% singletonI
thf(fact_207_singletonI,axiom,
! [A: v] : ( member_V2 @ A @ ( insert_V2 @ A @ bot_bot_set_V2 ) ) ).
% singletonI
thf(fact_208_singletonI,axiom,
! [A: produc778275879_V_nat] : ( member1886821968_V_nat @ A @ ( insert481364471_V_nat @ A @ bot_bo907687539_V_nat ) ) ).
% singletonI
thf(fact_209_Un__empty,axiom,
! [A4: set_Product_prod_V_V,B4: set_Product_prod_V_V] :
( ( ( sup_su551882943od_V_V @ A4 @ B4 )
= bot_bo1389414743od_V_V )
= ( ( A4 = bot_bo1389414743od_V_V )
& ( B4 = bot_bo1389414743od_V_V ) ) ) ).
% Un_empty
thf(fact_210_Un__empty,axiom,
! [A4: set_V,B4: set_V] :
( ( ( sup_sup_set_V2 @ A4 @ B4 )
= bot_bot_set_V2 )
= ( ( A4 = bot_bot_set_V2 )
& ( B4 = bot_bot_set_V2 ) ) ) ).
% Un_empty
thf(fact_211_Un__empty,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ( ( sup_su1043134939_V_nat @ A4 @ B4 )
= bot_bo907687539_V_nat )
= ( ( A4 = bot_bo907687539_V_nat )
& ( B4 = bot_bo907687539_V_nat ) ) ) ).
% Un_empty
thf(fact_212_Un__insert__right,axiom,
! [A4: set_Pr1058435079_V_nat,A: produc778275879_V_nat,B4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A4 @ ( insert481364471_V_nat @ A @ B4 ) )
= ( insert481364471_V_nat @ A @ ( sup_su1043134939_V_nat @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_213_Un__insert__left,axiom,
! [A: produc778275879_V_nat,B4: set_Pr1058435079_V_nat,C4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ ( insert481364471_V_nat @ A @ B4 ) @ C4 )
= ( insert481364471_V_nat @ A @ ( sup_su1043134939_V_nat @ B4 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_214_graph__union__preserves__restrict,axiom,
! [G_1: labele2115946735nt_V_V,G_2: labele2115946735nt_V_V] :
( ( G_1
= ( restri1305980611nt_V_V @ G_1 ) )
=> ( ( G_2
= ( restri1305980611nt_V_V @ G_2 ) )
=> ( ( graph_1141886696nt_V_V @ G_1 @ G_2 )
= ( restri1305980611nt_V_V @ ( graph_1141886696nt_V_V @ G_1 @ G_2 ) ) ) ) ) ).
% graph_union_preserves_restrict
thf(fact_215_graph__union__vertices,axiom,
! [G_1: labele2115946735nt_V_V,G_2: labele2115946735nt_V_V] :
( ( labele1134902411nt_V_V @ ( graph_1141886696nt_V_V @ G_1 @ G_2 ) )
= ( sup_sup_set_V2 @ ( labele1134902411nt_V_V @ G_1 ) @ ( labele1134902411nt_V_V @ G_2 ) ) ) ).
% graph_union_vertices
thf(fact_216_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labele2115946735nt_V_V] :
( ( labele712009229nt_V_V @ ( labele515735386nt_V_V @ Labeled_graph ) @ ( labele1134902411nt_V_V @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_217_extensible__refl,axiom,
! [R2: labele2115946735nt_V_V,G: labele2115946735nt_V_V,F2: set_Product_prod_V_V] :
( ( graph_1808119_V_V_V @ R2 @ G @ F2 )
=> ( extens1533825066_V_V_V @ ( produc533118295nt_V_V @ R2 @ R2 ) @ G @ F2 ) ) ).
% extensible_refl
thf(fact_218_graph__empty__e,axiom,
! [V2: set_V] :
( ( labele712009229nt_V_V @ bot_bo667804902od_V_V @ V2 )
= ( restri1305980611nt_V_V @ ( labele712009229nt_V_V @ bot_bo667804902od_V_V @ V2 ) ) ) ).
% graph_empty_e
thf(fact_219_constant__rules__empty,axiom,
( ( standa1571708471od_V_V @ bot_bo1389414743od_V_V )
= bot_bo1196823539_V_nat ) ).
% constant_rules_empty
thf(fact_220_constant__rules__empty,axiom,
( ( standa294833961ules_V @ bot_bot_set_V2 )
= bot_bo1834032757_V_nat ) ).
% constant_rules_empty
thf(fact_221_constant__rules__empty,axiom,
( ( standa434642387_V_nat @ bot_bo907687539_V_nat )
= bot_bo323535987at_nat ) ).
% constant_rules_empty
thf(fact_222_constant__rules__empty,axiom,
( ( standa1897115807ules_V @ bot_bot_set_V )
= bot_bo907687539_V_nat ) ).
% constant_rules_empty
thf(fact_223_Gr__insert,axiom,
! [X3: labele2128733290_V_nat,F3: set_la14611914_V_nat,F2: labele2128733290_V_nat > labele2128733290_V_nat] :
( ( bNF_Gr17960482_V_nat @ ( insert213439034_V_nat @ X3 @ F3 ) @ F2 )
= ( insert481364471_V_nat @ ( produc698429463_V_nat @ X3 @ ( F2 @ X3 ) ) @ ( bNF_Gr17960482_V_nat @ F3 @ F2 ) ) ) ).
% Gr_insert
thf(fact_224_Gr__insert,axiom,
! [X3: v,F3: set_V,F2: v > v] :
( ( bNF_Gr_V_V2 @ ( insert_V2 @ X3 @ F3 ) @ F2 )
= ( insert2024285275od_V_V @ ( product_Pair_V_V2 @ X3 @ ( F2 @ X3 ) ) @ ( bNF_Gr_V_V2 @ F3 @ F2 ) ) ) ).
% Gr_insert
thf(fact_225_Gr__insert,axiom,
! [X3: standard_Constant_V,F3: set_St1111633946tant_V,F2: standard_Constant_V > product_prod_V_V] :
( ( bNF_Gr320641027od_V_V @ ( insert1818834324tant_V @ X3 @ F3 ) @ F2 )
= ( insert135842892od_V_V @ ( produc107991630od_V_V @ X3 @ ( F2 @ X3 ) ) @ ( bNF_Gr320641027od_V_V @ F3 @ F2 ) ) ) ).
% Gr_insert
thf(fact_226_extensible__refl__concr,axiom,
! [E_1: set_Pr744279122od_V_V,V2: set_V,G: labele2115946735nt_V_V,F2: set_Product_prod_V_V,E_2: set_Pr744279122od_V_V] :
( ( graph_1808119_V_V_V @ ( labele712009229nt_V_V @ E_1 @ V2 ) @ G @ F2 )
=> ( ( extens1533825066_V_V_V @ ( produc533118295nt_V_V @ ( labele712009229nt_V_V @ E_1 @ V2 ) @ ( labele712009229nt_V_V @ E_2 @ V2 ) ) @ G @ F2 )
= ( graph_1808119_V_V_V @ ( labele712009229nt_V_V @ E_2 @ V2 ) @ G @ F2 ) ) ) ).
% extensible_refl_concr
thf(fact_227_graph__single,axiom,
! [A: standard_Constant_V,B: v,C: v] :
( ( labele712009229nt_V_V @ ( insert135842892od_V_V @ ( produc107991630od_V_V @ A @ ( product_Pair_V_V2 @ B @ C ) ) @ bot_bo667804902od_V_V ) @ ( insert_V2 @ B @ ( insert_V2 @ C @ bot_bot_set_V2 ) ) )
= ( restri1305980611nt_V_V @ ( labele712009229nt_V_V @ ( insert135842892od_V_V @ ( produc107991630od_V_V @ A @ ( product_Pair_V_V2 @ B @ C ) ) @ bot_bo667804902od_V_V ) @ ( insert_V2 @ B @ ( insert_V2 @ C @ bot_bot_set_V2 ) ) ) ) ) ).
% graph_single
thf(fact_228_insert__is__Un,axiom,
( insert2024285275od_V_V
= ( ^ [A5: product_prod_V_V] : ( sup_su551882943od_V_V @ ( insert2024285275od_V_V @ A5 @ bot_bo1389414743od_V_V ) ) ) ) ).
% insert_is_Un
thf(fact_229_insert__is__Un,axiom,
( insert_V2
= ( ^ [A5: v] : ( sup_sup_set_V2 @ ( insert_V2 @ A5 @ bot_bot_set_V2 ) ) ) ) ).
% insert_is_Un
thf(fact_230_insert__is__Un,axiom,
( insert481364471_V_nat
= ( ^ [A5: produc778275879_V_nat] : ( sup_su1043134939_V_nat @ ( insert481364471_V_nat @ A5 @ bot_bo907687539_V_nat ) ) ) ) ).
% insert_is_Un
thf(fact_231_Un__singleton__iff,axiom,
! [A4: set_Product_prod_V_V,B4: set_Product_prod_V_V,X3: product_prod_V_V] :
( ( ( sup_su551882943od_V_V @ A4 @ B4 )
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) )
= ( ( ( A4 = bot_bo1389414743od_V_V )
& ( B4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) ) )
| ( ( A4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) )
& ( B4 = bot_bo1389414743od_V_V ) )
| ( ( A4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) )
& ( B4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_232_Un__singleton__iff,axiom,
! [A4: set_V,B4: set_V,X3: v] :
( ( ( sup_sup_set_V2 @ A4 @ B4 )
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) )
= ( ( ( A4 = bot_bot_set_V2 )
& ( B4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) ) )
| ( ( A4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) )
& ( B4 = bot_bot_set_V2 ) )
| ( ( A4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) )
& ( B4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_233_Un__singleton__iff,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat,X3: produc778275879_V_nat] :
( ( ( sup_su1043134939_V_nat @ A4 @ B4 )
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) )
= ( ( ( A4 = bot_bo907687539_V_nat )
& ( B4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) ) )
| ( ( A4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) )
& ( B4 = bot_bo907687539_V_nat ) )
| ( ( A4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) )
& ( B4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_234_singleton__Un__iff,axiom,
! [X3: product_prod_V_V,A4: set_Product_prod_V_V,B4: set_Product_prod_V_V] :
( ( ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V )
= ( sup_su551882943od_V_V @ A4 @ B4 ) )
= ( ( ( A4 = bot_bo1389414743od_V_V )
& ( B4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) ) )
| ( ( A4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) )
& ( B4 = bot_bo1389414743od_V_V ) )
| ( ( A4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) )
& ( B4
= ( insert2024285275od_V_V @ X3 @ bot_bo1389414743od_V_V ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_235_singleton__Un__iff,axiom,
! [X3: v,A4: set_V,B4: set_V] :
( ( ( insert_V2 @ X3 @ bot_bot_set_V2 )
= ( sup_sup_set_V2 @ A4 @ B4 ) )
= ( ( ( A4 = bot_bot_set_V2 )
& ( B4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) ) )
| ( ( A4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) )
& ( B4 = bot_bot_set_V2 ) )
| ( ( A4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) )
& ( B4
= ( insert_V2 @ X3 @ bot_bot_set_V2 ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_236_singleton__Un__iff,axiom,
! [X3: produc778275879_V_nat,A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ( ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat )
= ( sup_su1043134939_V_nat @ A4 @ B4 ) )
= ( ( ( A4 = bot_bo907687539_V_nat )
& ( B4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) ) )
| ( ( A4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) )
& ( B4 = bot_bo907687539_V_nat ) )
| ( ( A4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) )
& ( B4
= ( insert481364471_V_nat @ X3 @ bot_bo907687539_V_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_237_graph__union__def,axiom,
( graph_1141886696nt_V_V
= ( ^ [G_12: labele2115946735nt_V_V,G_22: labele2115946735nt_V_V] : ( labele712009229nt_V_V @ ( sup_su693515134od_V_V @ ( labele515735386nt_V_V @ G_12 ) @ ( labele515735386nt_V_V @ G_22 ) ) @ ( sup_sup_set_V2 @ ( labele1134902411nt_V_V @ G_12 ) @ ( labele1134902411nt_V_V @ G_22 ) ) ) ) ) ).
% graph_union_def
thf(fact_238_Un__empty__left,axiom,
! [B4: set_Product_prod_V_V] :
( ( sup_su551882943od_V_V @ bot_bo1389414743od_V_V @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_239_Un__empty__left,axiom,
! [B4: set_V] :
( ( sup_sup_set_V2 @ bot_bot_set_V2 @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_240_Un__empty__left,axiom,
! [B4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ bot_bo907687539_V_nat @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_241_Un__empty__right,axiom,
! [A4: set_Product_prod_V_V] :
( ( sup_su551882943od_V_V @ A4 @ bot_bo1389414743od_V_V )
= A4 ) ).
% Un_empty_right
thf(fact_242_Un__empty__right,axiom,
! [A4: set_V] :
( ( sup_sup_set_V2 @ A4 @ bot_bot_set_V2 )
= A4 ) ).
% Un_empty_right
thf(fact_243_Un__empty__right,axiom,
! [A4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A4 @ bot_bo907687539_V_nat )
= A4 ) ).
% Un_empty_right
thf(fact_244_singletonD,axiom,
! [B: v2,A: v2] :
( ( member_V @ B @ ( insert_V @ A @ bot_bot_set_V ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_245_singletonD,axiom,
! [B: produc1699571228od_V_V,A: produc1699571228od_V_V] :
( ( member1010391347od_V_V @ B @ ( insert135842892od_V_V @ A @ bot_bo667804902od_V_V ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_246_singletonD,axiom,
! [B: product_prod_V_V,A: product_prod_V_V] :
( ( member2015049524od_V_V @ B @ ( insert2024285275od_V_V @ A @ bot_bo1389414743od_V_V ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_247_singletonD,axiom,
! [B: v,A: v] :
( ( member_V2 @ B @ ( insert_V2 @ A @ bot_bot_set_V2 ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_248_singletonD,axiom,
! [B: produc778275879_V_nat,A: produc778275879_V_nat] :
( ( member1886821968_V_nat @ B @ ( insert481364471_V_nat @ A @ bot_bo907687539_V_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_249_singleton__iff,axiom,
! [B: v2,A: v2] :
( ( member_V @ B @ ( insert_V @ A @ bot_bot_set_V ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_250_singleton__iff,axiom,
! [B: produc1699571228od_V_V,A: produc1699571228od_V_V] :
( ( member1010391347od_V_V @ B @ ( insert135842892od_V_V @ A @ bot_bo667804902od_V_V ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_251_singleton__iff,axiom,
! [B: product_prod_V_V,A: product_prod_V_V] :
( ( member2015049524od_V_V @ B @ ( insert2024285275od_V_V @ A @ bot_bo1389414743od_V_V ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_252_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_V2 @ B @ ( insert_V2 @ A @ bot_bot_set_V2 ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_253_singleton__iff,axiom,
! [B: produc778275879_V_nat,A: produc778275879_V_nat] :
( ( member1886821968_V_nat @ B @ ( insert481364471_V_nat @ A @ bot_bo907687539_V_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_254_doubleton__eq__iff,axiom,
! [A: product_prod_V_V,B: product_prod_V_V,C: product_prod_V_V,D: product_prod_V_V] :
( ( ( insert2024285275od_V_V @ A @ ( insert2024285275od_V_V @ B @ bot_bo1389414743od_V_V ) )
= ( insert2024285275od_V_V @ C @ ( insert2024285275od_V_V @ D @ bot_bo1389414743od_V_V ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_255_doubleton__eq__iff,axiom,
! [A: v,B: v,C: v,D: v] :
( ( ( insert_V2 @ A @ ( insert_V2 @ B @ bot_bot_set_V2 ) )
= ( insert_V2 @ C @ ( insert_V2 @ D @ bot_bot_set_V2 ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_256_doubleton__eq__iff,axiom,
! [A: produc778275879_V_nat,B: produc778275879_V_nat,C: produc778275879_V_nat,D: produc778275879_V_nat] :
( ( ( insert481364471_V_nat @ A @ ( insert481364471_V_nat @ B @ bot_bo907687539_V_nat ) )
= ( insert481364471_V_nat @ C @ ( insert481364471_V_nat @ D @ bot_bo907687539_V_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_257_insert__not__empty,axiom,
! [A: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( insert2024285275od_V_V @ A @ A4 )
!= bot_bo1389414743od_V_V ) ).
% insert_not_empty
thf(fact_258_insert__not__empty,axiom,
! [A: v,A4: set_V] :
( ( insert_V2 @ A @ A4 )
!= bot_bot_set_V2 ) ).
% insert_not_empty
thf(fact_259_insert__not__empty,axiom,
! [A: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( insert481364471_V_nat @ A @ A4 )
!= bot_bo907687539_V_nat ) ).
% insert_not_empty
thf(fact_260_singleton__inject,axiom,
! [A: product_prod_V_V,B: product_prod_V_V] :
( ( ( insert2024285275od_V_V @ A @ bot_bo1389414743od_V_V )
= ( insert2024285275od_V_V @ B @ bot_bo1389414743od_V_V ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_261_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_V2 @ A @ bot_bot_set_V2 )
= ( insert_V2 @ B @ bot_bot_set_V2 ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_262_singleton__inject,axiom,
! [A: produc778275879_V_nat,B: produc778275879_V_nat] :
( ( ( insert481364471_V_nat @ A @ bot_bo907687539_V_nat )
= ( insert481364471_V_nat @ B @ bot_bo907687539_V_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_263_mk__disjoint__insert,axiom,
! [A: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ A @ A4 )
=> ? [B5: set_Pr1058435079_V_nat] :
( ( A4
= ( insert481364471_V_nat @ A @ B5 ) )
& ~ ( member1886821968_V_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_264_mk__disjoint__insert,axiom,
! [A: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ A @ A4 )
=> ? [B5: set_Product_prod_V_V] :
( ( A4
= ( insert2024285275od_V_V @ A @ B5 ) )
& ~ ( member2015049524od_V_V @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_265_mk__disjoint__insert,axiom,
! [A: v,A4: set_V] :
( ( member_V2 @ A @ A4 )
=> ? [B5: set_V] :
( ( A4
= ( insert_V2 @ A @ B5 ) )
& ~ ( member_V2 @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_266_mk__disjoint__insert,axiom,
! [A: v2,A4: set_V2] :
( ( member_V @ A @ A4 )
=> ? [B5: set_V2] :
( ( A4
= ( insert_V @ A @ B5 ) )
& ~ ( member_V @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_267_mk__disjoint__insert,axiom,
! [A: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ A @ A4 )
=> ? [B5: set_Pr744279122od_V_V] :
( ( A4
= ( insert135842892od_V_V @ A @ B5 ) )
& ~ ( member1010391347od_V_V @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_268_Un__left__commute,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat,C4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A4 @ ( sup_su1043134939_V_nat @ B4 @ C4 ) )
= ( sup_su1043134939_V_nat @ B4 @ ( sup_su1043134939_V_nat @ A4 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_269_insert__commute,axiom,
! [X3: produc778275879_V_nat,Y: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( insert481364471_V_nat @ X3 @ ( insert481364471_V_nat @ Y @ A4 ) )
= ( insert481364471_V_nat @ Y @ ( insert481364471_V_nat @ X3 @ A4 ) ) ) ).
% insert_commute
thf(fact_270_Un__left__absorb,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A4 @ ( sup_su1043134939_V_nat @ A4 @ B4 ) )
= ( sup_su1043134939_V_nat @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_271_insert__eq__iff,axiom,
! [A: produc778275879_V_nat,A4: set_Pr1058435079_V_nat,B: produc778275879_V_nat,B4: set_Pr1058435079_V_nat] :
( ~ ( member1886821968_V_nat @ A @ A4 )
=> ( ~ ( member1886821968_V_nat @ B @ B4 )
=> ( ( ( insert481364471_V_nat @ A @ A4 )
= ( insert481364471_V_nat @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_Pr1058435079_V_nat] :
( ( A4
= ( insert481364471_V_nat @ B @ C5 ) )
& ~ ( member1886821968_V_nat @ B @ C5 )
& ( B4
= ( insert481364471_V_nat @ A @ C5 ) )
& ~ ( member1886821968_V_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_272_insert__eq__iff,axiom,
! [A: product_prod_V_V,A4: set_Product_prod_V_V,B: product_prod_V_V,B4: set_Product_prod_V_V] :
( ~ ( member2015049524od_V_V @ A @ A4 )
=> ( ~ ( member2015049524od_V_V @ B @ B4 )
=> ( ( ( insert2024285275od_V_V @ A @ A4 )
= ( insert2024285275od_V_V @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_Product_prod_V_V] :
( ( A4
= ( insert2024285275od_V_V @ B @ C5 ) )
& ~ ( member2015049524od_V_V @ B @ C5 )
& ( B4
= ( insert2024285275od_V_V @ A @ C5 ) )
& ~ ( member2015049524od_V_V @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_273_insert__eq__iff,axiom,
! [A: v,A4: set_V,B: v,B4: set_V] :
( ~ ( member_V2 @ A @ A4 )
=> ( ~ ( member_V2 @ B @ B4 )
=> ( ( ( insert_V2 @ A @ A4 )
= ( insert_V2 @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_V] :
( ( A4
= ( insert_V2 @ B @ C5 ) )
& ~ ( member_V2 @ B @ C5 )
& ( B4
= ( insert_V2 @ A @ C5 ) )
& ~ ( member_V2 @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_274_insert__eq__iff,axiom,
! [A: v2,A4: set_V2,B: v2,B4: set_V2] :
( ~ ( member_V @ A @ A4 )
=> ( ~ ( member_V @ B @ B4 )
=> ( ( ( insert_V @ A @ A4 )
= ( insert_V @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_V2] :
( ( A4
= ( insert_V @ B @ C5 ) )
& ~ ( member_V @ B @ C5 )
& ( B4
= ( insert_V @ A @ C5 ) )
& ~ ( member_V @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_275_insert__eq__iff,axiom,
! [A: produc1699571228od_V_V,A4: set_Pr744279122od_V_V,B: produc1699571228od_V_V,B4: set_Pr744279122od_V_V] :
( ~ ( member1010391347od_V_V @ A @ A4 )
=> ( ~ ( member1010391347od_V_V @ B @ B4 )
=> ( ( ( insert135842892od_V_V @ A @ A4 )
= ( insert135842892od_V_V @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_Pr744279122od_V_V] :
( ( A4
= ( insert135842892od_V_V @ B @ C5 ) )
& ~ ( member1010391347od_V_V @ B @ C5 )
& ( B4
= ( insert135842892od_V_V @ A @ C5 ) )
& ~ ( member1010391347od_V_V @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_276_insert__absorb,axiom,
! [A: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ A @ A4 )
=> ( ( insert481364471_V_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_277_insert__absorb,axiom,
! [A: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ A @ A4 )
=> ( ( insert2024285275od_V_V @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_278_insert__absorb,axiom,
! [A: v,A4: set_V] :
( ( member_V2 @ A @ A4 )
=> ( ( insert_V2 @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_279_insert__absorb,axiom,
! [A: v2,A4: set_V2] :
( ( member_V @ A @ A4 )
=> ( ( insert_V @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_280_insert__absorb,axiom,
! [A: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ A @ A4 )
=> ( ( insert135842892od_V_V @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_281_insert__ident,axiom,
! [X3: produc778275879_V_nat,A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ~ ( member1886821968_V_nat @ X3 @ A4 )
=> ( ~ ( member1886821968_V_nat @ X3 @ B4 )
=> ( ( ( insert481364471_V_nat @ X3 @ A4 )
= ( insert481364471_V_nat @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_282_insert__ident,axiom,
! [X3: product_prod_V_V,A4: set_Product_prod_V_V,B4: set_Product_prod_V_V] :
( ~ ( member2015049524od_V_V @ X3 @ A4 )
=> ( ~ ( member2015049524od_V_V @ X3 @ B4 )
=> ( ( ( insert2024285275od_V_V @ X3 @ A4 )
= ( insert2024285275od_V_V @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_283_insert__ident,axiom,
! [X3: v,A4: set_V,B4: set_V] :
( ~ ( member_V2 @ X3 @ A4 )
=> ( ~ ( member_V2 @ X3 @ B4 )
=> ( ( ( insert_V2 @ X3 @ A4 )
= ( insert_V2 @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_284_insert__ident,axiom,
! [X3: v2,A4: set_V2,B4: set_V2] :
( ~ ( member_V @ X3 @ A4 )
=> ( ~ ( member_V @ X3 @ B4 )
=> ( ( ( insert_V @ X3 @ A4 )
= ( insert_V @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_285_insert__ident,axiom,
! [X3: produc1699571228od_V_V,A4: set_Pr744279122od_V_V,B4: set_Pr744279122od_V_V] :
( ~ ( member1010391347od_V_V @ X3 @ A4 )
=> ( ~ ( member1010391347od_V_V @ X3 @ B4 )
=> ( ( ( insert135842892od_V_V @ X3 @ A4 )
= ( insert135842892od_V_V @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_286_Set_Oset__insert,axiom,
! [X3: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ X3 @ A4 )
=> ~ ! [B5: set_Pr1058435079_V_nat] :
( ( A4
= ( insert481364471_V_nat @ X3 @ B5 ) )
=> ( member1886821968_V_nat @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_287_Set_Oset__insert,axiom,
! [X3: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ X3 @ A4 )
=> ~ ! [B5: set_Product_prod_V_V] :
( ( A4
= ( insert2024285275od_V_V @ X3 @ B5 ) )
=> ( member2015049524od_V_V @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_288_Set_Oset__insert,axiom,
! [X3: v,A4: set_V] :
( ( member_V2 @ X3 @ A4 )
=> ~ ! [B5: set_V] :
( ( A4
= ( insert_V2 @ X3 @ B5 ) )
=> ( member_V2 @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_289_Set_Oset__insert,axiom,
! [X3: v2,A4: set_V2] :
( ( member_V @ X3 @ A4 )
=> ~ ! [B5: set_V2] :
( ( A4
= ( insert_V @ X3 @ B5 ) )
=> ( member_V @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_290_Set_Oset__insert,axiom,
! [X3: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ X3 @ A4 )
=> ~ ! [B5: set_Pr744279122od_V_V] :
( ( A4
= ( insert135842892od_V_V @ X3 @ B5 ) )
=> ( member1010391347od_V_V @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_291_Un__commute,axiom,
( sup_su1043134939_V_nat
= ( ^ [A6: set_Pr1058435079_V_nat,B6: set_Pr1058435079_V_nat] : ( sup_su1043134939_V_nat @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_292_Un__absorb,axiom,
! [A4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_293_insertI2,axiom,
! [A: produc778275879_V_nat,B4: set_Pr1058435079_V_nat,B: produc778275879_V_nat] :
( ( member1886821968_V_nat @ A @ B4 )
=> ( member1886821968_V_nat @ A @ ( insert481364471_V_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_294_insertI2,axiom,
! [A: product_prod_V_V,B4: set_Product_prod_V_V,B: product_prod_V_V] :
( ( member2015049524od_V_V @ A @ B4 )
=> ( member2015049524od_V_V @ A @ ( insert2024285275od_V_V @ B @ B4 ) ) ) ).
% insertI2
thf(fact_295_insertI2,axiom,
! [A: v,B4: set_V,B: v] :
( ( member_V2 @ A @ B4 )
=> ( member_V2 @ A @ ( insert_V2 @ B @ B4 ) ) ) ).
% insertI2
thf(fact_296_insertI2,axiom,
! [A: v2,B4: set_V2,B: v2] :
( ( member_V @ A @ B4 )
=> ( member_V @ A @ ( insert_V @ B @ B4 ) ) ) ).
% insertI2
thf(fact_297_insertI2,axiom,
! [A: produc1699571228od_V_V,B4: set_Pr744279122od_V_V,B: produc1699571228od_V_V] :
( ( member1010391347od_V_V @ A @ B4 )
=> ( member1010391347od_V_V @ A @ ( insert135842892od_V_V @ B @ B4 ) ) ) ).
% insertI2
thf(fact_298_insertI1,axiom,
! [A: produc778275879_V_nat,B4: set_Pr1058435079_V_nat] : ( member1886821968_V_nat @ A @ ( insert481364471_V_nat @ A @ B4 ) ) ).
% insertI1
thf(fact_299_insertI1,axiom,
! [A: product_prod_V_V,B4: set_Product_prod_V_V] : ( member2015049524od_V_V @ A @ ( insert2024285275od_V_V @ A @ B4 ) ) ).
% insertI1
thf(fact_300_insertI1,axiom,
! [A: v,B4: set_V] : ( member_V2 @ A @ ( insert_V2 @ A @ B4 ) ) ).
% insertI1
thf(fact_301_insertI1,axiom,
! [A: v2,B4: set_V2] : ( member_V @ A @ ( insert_V @ A @ B4 ) ) ).
% insertI1
thf(fact_302_insertI1,axiom,
! [A: produc1699571228od_V_V,B4: set_Pr744279122od_V_V] : ( member1010391347od_V_V @ A @ ( insert135842892od_V_V @ A @ B4 ) ) ).
% insertI1
thf(fact_303_Un__assoc,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat,C4: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ ( sup_su1043134939_V_nat @ A4 @ B4 ) @ C4 )
= ( sup_su1043134939_V_nat @ A4 @ ( sup_su1043134939_V_nat @ B4 @ C4 ) ) ) ).
% Un_assoc
thf(fact_304_insertE,axiom,
! [A: produc778275879_V_nat,B: produc778275879_V_nat,A4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ A @ ( insert481364471_V_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member1886821968_V_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_305_insertE,axiom,
! [A: product_prod_V_V,B: product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ A @ ( insert2024285275od_V_V @ B @ A4 ) )
=> ( ( A != B )
=> ( member2015049524od_V_V @ A @ A4 ) ) ) ).
% insertE
thf(fact_306_insertE,axiom,
! [A: v,B: v,A4: set_V] :
( ( member_V2 @ A @ ( insert_V2 @ B @ A4 ) )
=> ( ( A != B )
=> ( member_V2 @ A @ A4 ) ) ) ).
% insertE
thf(fact_307_insertE,axiom,
! [A: v2,B: v2,A4: set_V2] :
( ( member_V @ A @ ( insert_V @ B @ A4 ) )
=> ( ( A != B )
=> ( member_V @ A @ A4 ) ) ) ).
% insertE
thf(fact_308_insertE,axiom,
! [A: produc1699571228od_V_V,B: produc1699571228od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ A @ ( insert135842892od_V_V @ B @ A4 ) )
=> ( ( A != B )
=> ( member1010391347od_V_V @ A @ A4 ) ) ) ).
% insertE
thf(fact_309_ball__Un,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat,P2: produc778275879_V_nat > $o] :
( ( ! [X6: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X6 @ ( sup_su1043134939_V_nat @ A4 @ B4 ) )
=> ( P2 @ X6 ) ) )
= ( ! [X6: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X6 @ A4 )
=> ( P2 @ X6 ) )
& ! [X6: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X6 @ B4 )
=> ( P2 @ X6 ) ) ) ) ).
% ball_Un
thf(fact_310_bex__Un,axiom,
! [A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat,P2: produc778275879_V_nat > $o] :
( ( ? [X6: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X6 @ ( sup_su1043134939_V_nat @ A4 @ B4 ) )
& ( P2 @ X6 ) ) )
= ( ? [X6: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X6 @ A4 )
& ( P2 @ X6 ) )
| ? [X6: produc778275879_V_nat] :
( ( member1886821968_V_nat @ X6 @ B4 )
& ( P2 @ X6 ) ) ) ) ).
% bex_Un
thf(fact_311_UnI2,axiom,
! [C: product_prod_V_V,B4: set_Product_prod_V_V,A4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ C @ B4 )
=> ( member2015049524od_V_V @ C @ ( sup_su551882943od_V_V @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_312_UnI2,axiom,
! [C: v,B4: set_V,A4: set_V] :
( ( member_V2 @ C @ B4 )
=> ( member_V2 @ C @ ( sup_sup_set_V2 @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_313_UnI2,axiom,
! [C: v2,B4: set_V2,A4: set_V2] :
( ( member_V @ C @ B4 )
=> ( member_V @ C @ ( sup_sup_set_V @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_314_UnI2,axiom,
! [C: produc1699571228od_V_V,B4: set_Pr744279122od_V_V,A4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ C @ B4 )
=> ( member1010391347od_V_V @ C @ ( sup_su693515134od_V_V @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_315_UnI2,axiom,
! [C: produc778275879_V_nat,B4: set_Pr1058435079_V_nat,A4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ C @ B4 )
=> ( member1886821968_V_nat @ C @ ( sup_su1043134939_V_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_316_UnI1,axiom,
! [C: product_prod_V_V,A4: set_Product_prod_V_V,B4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ C @ A4 )
=> ( member2015049524od_V_V @ C @ ( sup_su551882943od_V_V @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_317_UnI1,axiom,
! [C: v,A4: set_V,B4: set_V] :
( ( member_V2 @ C @ A4 )
=> ( member_V2 @ C @ ( sup_sup_set_V2 @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_318_UnI1,axiom,
! [C: v2,A4: set_V2,B4: set_V2] :
( ( member_V @ C @ A4 )
=> ( member_V @ C @ ( sup_sup_set_V @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_319_UnI1,axiom,
! [C: produc1699571228od_V_V,A4: set_Pr744279122od_V_V,B4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ C @ A4 )
=> ( member1010391347od_V_V @ C @ ( sup_su693515134od_V_V @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_320_UnI1,axiom,
! [C: produc778275879_V_nat,A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ C @ A4 )
=> ( member1886821968_V_nat @ C @ ( sup_su1043134939_V_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_321_UnE,axiom,
! [C: product_prod_V_V,A4: set_Product_prod_V_V,B4: set_Product_prod_V_V] :
( ( member2015049524od_V_V @ C @ ( sup_su551882943od_V_V @ A4 @ B4 ) )
=> ( ~ ( member2015049524od_V_V @ C @ A4 )
=> ( member2015049524od_V_V @ C @ B4 ) ) ) ).
% UnE
thf(fact_322_UnE,axiom,
! [C: v,A4: set_V,B4: set_V] :
( ( member_V2 @ C @ ( sup_sup_set_V2 @ A4 @ B4 ) )
=> ( ~ ( member_V2 @ C @ A4 )
=> ( member_V2 @ C @ B4 ) ) ) ).
% UnE
thf(fact_323_UnE,axiom,
! [C: v2,A4: set_V2,B4: set_V2] :
( ( member_V @ C @ ( sup_sup_set_V @ A4 @ B4 ) )
=> ( ~ ( member_V @ C @ A4 )
=> ( member_V @ C @ B4 ) ) ) ).
% UnE
thf(fact_324_UnE,axiom,
! [C: produc1699571228od_V_V,A4: set_Pr744279122od_V_V,B4: set_Pr744279122od_V_V] :
( ( member1010391347od_V_V @ C @ ( sup_su693515134od_V_V @ A4 @ B4 ) )
=> ( ~ ( member1010391347od_V_V @ C @ A4 )
=> ( member1010391347od_V_V @ C @ B4 ) ) ) ).
% UnE
thf(fact_325_UnE,axiom,
! [C: produc778275879_V_nat,A4: set_Pr1058435079_V_nat,B4: set_Pr1058435079_V_nat] :
( ( member1886821968_V_nat @ C @ ( sup_su1043134939_V_nat @ A4 @ B4 ) )
=> ( ~ ( member1886821968_V_nat @ C @ A4 )
=> ( member1886821968_V_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_326_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr744279122od_V_V,X2: set_V] :
( ( labele1134902411nt_V_V @ ( labele712009229nt_V_V @ X1 @ X2 ) )
= X2 ) ).
% labeled_graph.sel(2)
thf(fact_327_labeled__graph_Osel_I1_J,axiom,
! [X1: set_Pr744279122od_V_V,X2: set_V] :
( ( labele515735386nt_V_V @ ( labele712009229nt_V_V @ X1 @ X2 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_328_graph__homomorphism__semantics,axiom,
! [A4: labele2115946735nt_V_V,B4: labele2115946735nt_V_V,F2: set_Product_prod_V_V,A: v,B: v,E: allego859987871tant_V,A2: v,B2: v] :
( ( graph_1808119_V_V_V @ A4 @ B4 @ F2 )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( semant993842370nt_V_V @ A4 @ E ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ A2 ) @ F2 )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ B @ B2 ) @ F2 )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A2 @ B2 ) @ ( semant993842370nt_V_V @ B4 @ E ) ) ) ) ) ) ).
% graph_homomorphism_semantics
thf(fact_329_graph__homomorphism__nonempty,axiom,
! [A4: labele2115946735nt_V_V,B4: labele2115946735nt_V_V,F2: set_Product_prod_V_V,E: allego859987871tant_V] :
( ( graph_1808119_V_V_V @ A4 @ B4 @ F2 )
=> ( ( ( semant993842370nt_V_V @ A4 @ E )
!= bot_bo1389414743od_V_V )
=> ( ( semant993842370nt_V_V @ B4 @ E )
!= bot_bo1389414743od_V_V ) ) ) ).
% graph_homomorphism_nonempty
thf(fact_330_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labele2115946735nt_V_V] :
( Labeled_graph
= ( labele712009229nt_V_V @ ( labele515735386nt_V_V @ Labeled_graph ) @ ( labele1134902411nt_V_V @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_331_standard__rules__def,axiom,
( standa157301464ules_V
= ( ^ [C5: set_V2,L3: set_St1111633946tant_V] : ( sup_su1043134939_V_nat @ ( sup_su1043134939_V_nat @ ( standa1897115807ules_V @ C5 ) @ ( standa1568205529ules_V @ L3 ) ) @ ( insert481364471_V_nat @ ( standa214871990tant_V @ standard_S_Top_V ) @ ( insert481364471_V_nat @ standa1319953089tant_V @ bot_bo907687539_V_nat ) ) ) ) ) ).
% standard_rules_def
thf(fact_332_semantics__in__vertices_I2_J,axiom,
! [A4: labele2115946735nt_V_V,A: v,B: v,E: allego859987871tant_V] :
( ( A4
= ( restri1305980611nt_V_V @ A4 ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( semant993842370nt_V_V @ A4 @ E ) )
=> ( member_V2 @ B @ ( labele1134902411nt_V_V @ A4 ) ) ) ) ).
% semantics_in_vertices(2)
thf(fact_333_semantics__in__vertices_I1_J,axiom,
! [A4: labele2115946735nt_V_V,A: v,B: v,E: allego859987871tant_V] :
( ( A4
= ( restri1305980611nt_V_V @ A4 ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( semant993842370nt_V_V @ A4 @ E ) )
=> ( member_V2 @ A @ ( labele1134902411nt_V_V @ A4 ) ) ) ) ).
% semantics_in_vertices(1)
thf(fact_334_subgraph__semantics,axiom,
! [A4: labele2115946735nt_V_V,B4: labele2115946735nt_V_V,A: v,B: v,E: allego859987871tant_V] :
( ( graph_1808119_V_V_V @ A4 @ B4 @ ( id_on_V2 @ ( labele1134902411nt_V_V @ A4 ) ) )
=> ( ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( semant993842370nt_V_V @ A4 @ E ) )
=> ( member2015049524od_V_V @ ( product_Pair_V_V2 @ A @ B ) @ ( semant993842370nt_V_V @ B4 @ E ) ) ) ) ).
% subgraph_semantics
thf(fact_335_maintainedD,axiom,
! [A4: labele2128733290_V_nat,B4: labele2128733290_V_nat,G: labele2115946735nt_V_V,F2: set_Pr53436518_nat_V] :
( ( mainta197426964_nat_V @ ( produc698429463_V_nat @ A4 @ B4 ) @ G )
=> ( ( graph_1541575154_nat_V @ A4 @ G @ F2 )
=> ( extens1072773157_nat_V @ ( produc698429463_V_nat @ A4 @ B4 ) @ G @ F2 ) ) ) ).
% maintainedD
thf(fact_336_maintainedD,axiom,
! [A4: labele2115946735nt_V_V,B4: labele2115946735nt_V_V,G: labele2115946735nt_V_V,F2: set_Product_prod_V_V] :
( ( mainta1699210777_V_V_V @ ( produc533118295nt_V_V @ A4 @ B4 ) @ G )
=> ( ( graph_1808119_V_V_V @ A4 @ G @ F2 )
=> ( extens1533825066_V_V_V @ ( produc533118295nt_V_V @ A4 @ B4 ) @ G @ F2 ) ) ) ).
% maintainedD
thf(fact_337_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_V_V] :
( ( sup_su551882943od_V_V @ A @ bot_bo1389414743od_V_V )
= A ) ).
% sup_bot.right_neutral
thf(fact_338_sup__bot_Oright__neutral,axiom,
! [A: set_V] :
( ( sup_sup_set_V2 @ A @ bot_bot_set_V2 )
= A ) ).
% sup_bot.right_neutral
thf(fact_339_sup__bot_Oright__neutral,axiom,
! [A: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A @ bot_bo907687539_V_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_340_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_V_V,B: set_Product_prod_V_V] :
( ( bot_bo1389414743od_V_V
= ( sup_su551882943od_V_V @ A @ B ) )
= ( ( A = bot_bo1389414743od_V_V )
& ( B = bot_bo1389414743od_V_V ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_341_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_V,B: set_V] :
( ( bot_bot_set_V2
= ( sup_sup_set_V2 @ A @ B ) )
= ( ( A = bot_bot_set_V2 )
& ( B = bot_bot_set_V2 ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_342_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr1058435079_V_nat,B: set_Pr1058435079_V_nat] :
( ( bot_bo907687539_V_nat
= ( sup_su1043134939_V_nat @ A @ B ) )
= ( ( A = bot_bo907687539_V_nat )
& ( B = bot_bo907687539_V_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_343_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_V_V] :
( ( sup_su551882943od_V_V @ bot_bo1389414743od_V_V @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_344_sup__bot_Oleft__neutral,axiom,
! [A: set_V] :
( ( sup_sup_set_V2 @ bot_bot_set_V2 @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_345_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ bot_bo907687539_V_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_346_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_V_V,B: set_Product_prod_V_V] :
( ( ( sup_su551882943od_V_V @ A @ B )
= bot_bo1389414743od_V_V )
= ( ( A = bot_bo1389414743od_V_V )
& ( B = bot_bo1389414743od_V_V ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_347_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_V,B: set_V] :
( ( ( sup_sup_set_V2 @ A @ B )
= bot_bot_set_V2 )
= ( ( A = bot_bot_set_V2 )
& ( B = bot_bot_set_V2 ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_348_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr1058435079_V_nat,B: set_Pr1058435079_V_nat] :
( ( ( sup_su1043134939_V_nat @ A @ B )
= bot_bo907687539_V_nat )
= ( ( A = bot_bo907687539_V_nat )
& ( B = bot_bo907687539_V_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_349_sup_Oright__idem,axiom,
! [A: set_Pr1058435079_V_nat,B: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ ( sup_su1043134939_V_nat @ A @ B ) @ B )
= ( sup_su1043134939_V_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_350_sup__left__idem,axiom,
! [X3: set_Pr1058435079_V_nat,Y: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ X3 @ ( sup_su1043134939_V_nat @ X3 @ Y ) )
= ( sup_su1043134939_V_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_351_sup_Oleft__idem,axiom,
! [A: set_Pr1058435079_V_nat,B: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A @ ( sup_su1043134939_V_nat @ A @ B ) )
= ( sup_su1043134939_V_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_352_sup__idem,axiom,
! [X3: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_353_sup_Oidem,axiom,
! [A: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_354_sup__bot__left,axiom,
! [X3: set_Pr1058435079_V_nat] :
( ( sup_su1043134939_V_nat @ bot_bo907687539_V_nat @ X3 )
= X3 ) ).
% sup_bot_left
% Conjectures (1)
thf(conj_0,conjecture,
member2015049524od_V_V @ ( product_Pair_V_V2 @ ( m @ xa ) @ ( m @ xa ) ) @ ( getRel1432786916nt_V_V @ ( standard_S_Const_V @ y ) @ g ) ).
%------------------------------------------------------------------------------