TPTP Problem File: ITP177^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP177^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_340__5390244_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_340__5390244_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 563 ( 156 unt; 203 typ; 0 def)
% Number of atoms : 988 ( 324 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3599 ( 142 ~; 9 |; 62 &;2979 @)
% ( 0 <=>; 407 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 321 ( 321 >; 0 *; 0 +; 0 <<)
% Number of symbols : 154 ( 151 usr; 17 con; 0-3 aty)
% Number of variables : 1074 ( 24 ^;1023 !; 27 ?;1074 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:07.514
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J_J,type,
set_Pr197580003_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J,type,
set_Pr924198087_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
produc398057191_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
set_Pr1954438265od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
produc1169496899od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J_J,type,
set_Pr1190604087od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
produc1084374401od_b_b: $tType ).
thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
labele1322729112_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
set_Pr667377223od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc1485083751od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
set_Pr2109276827od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J,type,
set_Pr1646010159_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mtf__b_J_J,type,
set_Pr812188767_nat_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
produc970027067od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mtf__b_J,type,
produc845793663_nat_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
set_Pr1987088711_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
produc1871334759_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mtf__b_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mtf__b_J_J,type,
produc1398946881nt_a_b: $tType ).
thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
labele1644675410od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
set_Pr1071216121od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
set_Pr2134561957od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
set_Pr1021918435od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
produc1052326083od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc1990339951od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
produc644633773od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
set_Pr2060523537od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J_J,type,
set_Pr1906739389_b_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
produc1605346267od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J,type,
produc276146823_b_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J_J,type,
set_Pr154086431tant_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
set_Pr1324126435od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
produc90515391tant_a: $tType ).
thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
labele1835558936od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
set_Pr2007082183od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc1213276845od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
set_la1083530965_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc1168037607od_b_b: $tType ).
thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
labele935650037_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J_J,type,
set_Pr1649131859tant_a: $tType ).
thf(ty_n_t__LabeledGraphSemantics__Oallegorical____term_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
allego510293162tant_a: $tType ).
thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mtf__b_J,type,
labele1159362096nt_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
set_Pr621705391od_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J_J,type,
set_Pr357419743_b_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc255402447od_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J,type,
produc2112523263_b_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
set_St761939237tant_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
set_Pr2106913242_nat_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
set_Product_prod_b_b: $tType ).
thf(ty_n_t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standard_Constant_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
product_prod_b_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (151)
thf(sy_c_BNF__Def_OGr_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
bNF_Gr1272216980od_b_b: set_St761939237tant_a > ( standard_Constant_a > product_prod_b_b ) > set_Pr1324126435od_b_b ).
thf(sy_c_BNF__Def_OGr_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
bNF_Gr996500706_a_nat: set_la1083530965_a_nat > ( labele935650037_a_nat > labele935650037_a_nat ) > set_Pr1987088711_a_nat ).
thf(sy_c_BNF__Def_OGr_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b,type,
bNF_Gr492676974_b_b_b: set_Pr1324126435od_b_b > ( produc1213276845od_b_b > b ) > set_Pr1906739389_b_b_b ).
thf(sy_c_BNF__Def_OGr_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b,type,
bNF_Gr521784954_b_b_b: set_Product_prod_b_b > ( product_prod_b_b > b ) > set_Pr357419743_b_b_b ).
thf(sy_c_BNF__Def_OGr_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
bNF_Gr230160332tant_a: set_b > ( b > standard_Constant_a ) > set_Pr1649131859tant_a ).
thf(sy_c_BNF__Def_OGr_001tf__b_001tf__b,type,
bNF_Gr_b_b: set_b > ( b > b ) > set_Product_prod_b_b ).
thf(sy_c_Equiv__Relations_Ocongruent_001tf__b_001t__Set__Oset_Itf__b_J,type,
equiv_1821124534_set_b: set_Product_prod_b_b > ( b > set_b ) > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
equiv_2048442231od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
equiv_291114781_a_nat: set_Pr1987088711_a_nat > set_Pr924198087_a_nat > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
equiv_1125628061od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001tf__b,type,
equiv_equiv_b: set_b > set_Product_prod_b_b > $o ).
thf(sy_c_Hilbert__Choice_OEps_001tf__b,type,
hilbert_Eps_b: ( b > $o ) > b ).
thf(sy_c_If_001tf__b,type,
if_b: $o > b > b > b ).
thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt_001tf__a,type,
standard_S_Idt_a: standard_Constant_a ).
thf(sy_c_LabeledGraphSemantics_OgetRel_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
getRel59882567od_b_b: standard_Constant_a > labele1644675410od_b_b > set_Pr2109276827od_b_b ).
thf(sy_c_LabeledGraphSemantics_OgetRel_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
getRel118493133od_b_b: standard_Constant_a > labele1835558936od_b_b > set_Pr2007082183od_b_b ).
thf(sy_c_LabeledGraphSemantics_OgetRel_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
getRel904497637nt_a_b: standard_Constant_a > labele1159362096nt_a_b > set_Product_prod_b_b ).
thf(sy_c_LabeledGraphSemantics_Osemantics_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
semant55076487nt_a_b: labele1159362096nt_a_b > allego510293162tant_a > set_Product_prod_b_b ).
thf(sy_c_LabeledGraphs_Oedge__preserving_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
edge_p1817988046tant_a: set_Pr1906739389_b_b_b > set_Pr1190604087od_b_b > set_Pr1324126435od_b_b > $o ).
thf(sy_c_LabeledGraphs_Oedge__preserving_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
edge_p1969196346tant_a: set_Pr357419743_b_b_b > set_Pr1021918435od_b_b > set_Pr1324126435od_b_b > $o ).
thf(sy_c_LabeledGraphs_Oedge__preserving_001tf__b_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
edge_p1384198690tant_a: set_Product_prod_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b > $o ).
thf(sy_c_LabeledGraphs_Ograph__homomorphism_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Nat__Onat_001tf__b,type,
graph_714568023_nat_b: labele935650037_a_nat > labele1159362096nt_a_b > set_Pr2106913242_nat_b > $o ).
thf(sy_c_LabeledGraphs_Ograph__homomorphism_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001tf__b,type,
graph_726202286_nat_b: labele1322729112_a_nat > labele1159362096nt_a_b > set_Pr812188767_nat_b > $o ).
thf(sy_c_LabeledGraphs_Ograph__homomorphism_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b_001tf__b,type,
graph_222606102_a_b_b: labele1159362096nt_a_b > labele1159362096nt_a_b > set_Product_prod_b_b > $o ).
thf(sy_c_LabeledGraphs_Ograph__union_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
graph_1309249505nt_a_b: labele1159362096nt_a_b > labele1159362096nt_a_b > labele1159362096nt_a_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_OLG_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
labele27098724_a_nat: set_Pr197580003_a_nat > set_Pr1987088711_a_nat > labele1322729112_a_nat ).
thf(sy_c_LabeledGraphs_Olabeled__graph_OLG_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
labele1230159100nt_a_b: set_Pr1324126435od_b_b > set_b > labele1159362096nt_a_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
labele2032816061od_b_b: labele1644675410od_b_b > set_Pr1190604087od_b_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
labele372159959od_b_b: labele1835558936od_b_b > set_Pr1021918435od_b_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Oedges_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
labele1741081071nt_a_b: labele1159362096nt_a_b > set_Pr1324126435od_b_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
labele595103470od_b_b: labele1644675410od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
labele442485990od_b_b: labele1835558936od_b_b > set_Product_prod_b_b ).
thf(sy_c_LabeledGraphs_Olabeled__graph_Overtices_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
labele1424214014nt_a_b: labele1159362096nt_a_b > set_b ).
thf(sy_c_LabeledGraphs_Omap__graph_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
map_gr1495881666tant_a: set_Pr1906739389_b_b_b > labele1644675410od_b_b > labele1159362096nt_a_b ).
thf(sy_c_LabeledGraphs_Omap__graph_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
map_gr1314489926tant_a: set_Pr357419743_b_b_b > labele1835558936od_b_b > labele1159362096nt_a_b ).
thf(sy_c_LabeledGraphs_Omap__graph_001tf__b_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
map_gr926947118tant_a: set_Product_prod_b_b > labele1159362096nt_a_b > labele1159362096nt_a_b ).
thf(sy_c_LabeledGraphs_Orestrict_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b,type,
restri446606278nt_a_b: labele1159362096nt_a_b > labele1159362096nt_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
bot_bo1160111033tant_a: set_St761939237tant_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
bot_bo2122869057_a_nat: set_la1083530965_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J_J,type,
bot_bo1653310327_a_nat: set_Pr197580003_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
bot_bo1664927607od_b_b: set_Pr1324126435od_b_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
bot_bo1836341171_a_nat: set_Pr1987088711_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J,type,
bot_bo1973379891_a_nat: set_Pr924198087_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mtf__b_J_J,type,
bot_bo1113554635_nat_b: set_Pr812188767_nat_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
bot_bo1343651123od_b_b: set_Product_prod_b_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
ord_le123089219od_b_b: set_Pr1324126435od_b_b > set_Pr1324126435od_b_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
ord_le1718765799_a_nat: set_Pr1987088711_a_nat > set_Pr1987088711_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
ord_le1484132922_nat_b: set_Pr2106913242_nat_b > set_Pr2106913242_nat_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J_J,type,
ord_le107617383_a_nat: set_Pr924198087_a_nat > set_Pr924198087_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
ord_le1036320359od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Product__Type_OPair_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
produc342647tant_a: standard_Constant_a > standard_Constant_a > produc90515391tant_a ).
thf(sy_c_Product__Type_OPair_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
produc1788032435od_b_b: standard_Constant_a > produc970027067od_b_b > produc1084374401od_b_b ).
thf(sy_c_Product__Type_OPair_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc618266719od_b_b: standard_Constant_a > produc1168037607od_b_b > produc644633773od_b_b ).
thf(sy_c_Product__Type_OPair_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
produc1432590431od_b_b: standard_Constant_a > product_prod_b_b > produc1213276845od_b_b ).
thf(sy_c_Product__Type_OPair_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
produc1676969687_a_nat: labele935650037_a_nat > labele935650037_a_nat > produc1871334759_a_nat ).
thf(sy_c_Product__Type_OPair_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mtf__b_J_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mtf__b_J,type,
produc1569511545nt_a_b: labele1159362096nt_a_b > labele1159362096nt_a_b > produc1398946881nt_a_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc449289715od_b_b: produc1213276845od_b_b > produc1213276845od_b_b > produc970027067od_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
produc1597690145od_b_b: produc1213276845od_b_b > product_prod_b_b > produc1990339951od_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b,type,
produc1580777273_b_b_b: produc1213276845od_b_b > b > produc276146823_b_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc401731773od_b_b: produc1871334759_a_nat > produc1213276845od_b_b > produc1169496899od_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
produc1677124439_a_nat: produc1871334759_a_nat > produc1871334759_a_nat > produc398057191_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
produc590959831od_b_b: produc1871334759_a_nat > product_prod_b_b > produc1485083751od_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001tf__b,type,
produc2059954415_nat_b: produc1871334759_a_nat > b > produc845793663_nat_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc732676669od_b_b: product_prod_b_b > produc1213276845od_b_b > produc1052326083od_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
produc546497367od_b_b: product_prod_b_b > product_prod_b_b > produc1168037607od_b_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b,type,
produc1001682799_b_b_b: product_prod_b_b > b > produc2112523263_b_b_b ).
thf(sy_c_Product__Type_OPair_001tf__b_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
produc800495189od_b_b: b > produc1213276845od_b_b > produc1605346267od_b_b ).
thf(sy_c_Product__Type_OPair_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
produc1257047359od_b_b: b > product_prod_b_b > produc255402447od_b_b ).
thf(sy_c_Product__Type_OPair_001tf__b_001tf__b,type,
product_Pair_b_b: b > b > product_prod_b_b ).
thf(sy_c_Relation_OId__on_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
id_on_689842066_a_nat: set_la1083530965_a_nat > set_Pr1987088711_a_nat ).
thf(sy_c_Relation_OId__on_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
id_on_250271696od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b ).
thf(sy_c_Relation_OId__on_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
id_on_1651096324_a_nat: set_Pr1987088711_a_nat > set_Pr924198087_a_nat ).
thf(sy_c_Relation_OId__on_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
id_on_2019020932od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b ).
thf(sy_c_Relation_OId__on_001tf__b,type,
id_on_b: set_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1916422435od_b_b: set_Pr1324126435od_b_b > set_St761939237tant_a > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
image_1971191571_a_nat: set_Pr1987088711_a_nat > set_la1083530965_a_nat > set_la1083530965_a_nat ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_763305007od_b_b: set_Pr2109276827od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1397930469od_b_b: set_Pr2134561957od_b_b > set_Pr1324126435od_b_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b,type,
image_1764625405_b_b_b: set_Pr1906739389_b_b_b > set_Pr1324126435od_b_b > set_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_480774529od_b_b: set_Pr1954438265od_b_b > set_Pr1987088711_a_nat > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
image_1168831379_a_nat: set_Pr924198087_a_nat > set_Pr1987088711_a_nat > set_Pr1987088711_a_nat ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1214223635od_b_b: set_Pr667377223od_b_b > set_Pr1987088711_a_nat > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_001tf__b,type,
image_924992811_nat_b: set_Pr812188767_nat_b > set_Pr1987088711_a_nat > set_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_532916993od_b_b: set_Pr1071216121od_b_b > set_Product_prod_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1928024979od_b_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b,type,
image_921732011_b_b_b: set_Pr357419743_b_b_b > set_Product_prod_b_b > set_b ).
thf(sy_c_Relation_OImage_001tf__b_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_984343321od_b_b: set_Pr2060523537od_b_b > set_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001tf__b_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
image_1749766139_a_nat: set_Pr1646010159_a_nat > set_b > set_Pr1987088711_a_nat ).
thf(sy_c_Relation_OImage_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1177096571od_b_b: set_Pr621705391od_b_b > set_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001tf__b_001tf__b,type,
image_b_b: set_Product_prod_b_b > set_b > set_b ).
thf(sy_c_Relation_Orefl__on_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
refl_o1031343860_a_nat: set_la1083530965_a_nat > set_Pr1987088711_a_nat > $o ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
refl_o1921098222od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
refl_o1213627494_a_nat: set_Pr1987088711_a_nat > set_Pr924198087_a_nat > $o ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
refl_o2134473190od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_Relation_Orefl__on_001tf__b,type,
refl_on_b: set_b > set_Product_prod_b_b > $o ).
thf(sy_c_Relation_Orelcomp_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
relcom883816262od_b_b: set_Pr154086431tant_a > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_Orelcomp_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
relcom1823941168od_b_b: set_Pr1324126435od_b_b > set_Pr2007082183od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_Orelcomp_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
relcom1338300020_a_nat: set_Pr1987088711_a_nat > set_Pr1987088711_a_nat > set_Pr1987088711_a_nat ).
thf(sy_c_Relation_Orelcomp_001tf__b_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
relcom112561144od_b_b: set_Pr1649131859tant_a > set_Pr1324126435od_b_b > set_Pr621705391od_b_b ).
thf(sy_c_Relation_Orelcomp_001tf__b_001tf__b_001tf__b,type,
relcomp_b_b_b: set_Product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(sy_c_RulesAndChains_Oagree__on_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b_001tf__b,type,
agree_221379389_a_b_b: labele1159362096nt_a_b > set_Product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_RulesAndChains_Oconsequence__graph_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Nat__Onat_001tf__b,type,
conseq956760372_nat_b: set_Pr1987088711_a_nat > labele1159362096nt_a_b > $o ).
thf(sy_c_RulesAndChains_Oextensible_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Nat__Onat_001tf__b,type,
extens1554515172_nat_b: produc1871334759_a_nat > labele1159362096nt_a_b > set_Pr2106913242_nat_b > $o ).
thf(sy_c_RulesAndChains_Oextensible_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b_001tf__b,type,
extens1779443913_a_b_b: produc1398946881nt_a_b > labele1159362096nt_a_b > set_Product_prod_b_b > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Nat__Onat_001tf__b,type,
mainta1604381365_nat_b: produc1871334759_a_nat > labele1159362096nt_a_b > $o ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001tf__b_001tf__b,type,
mainta374363448_a_b_b: produc1398946881nt_a_b > labele1159362096nt_a_b > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
collec279084418od_b_b: ( produc1213276845od_b_b > $o ) > set_Pr1324126435od_b_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
collec357096914_a_nat: ( produc1871334759_a_nat > $o ) > set_Pr1987088711_a_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
collec1481886546od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1683732397od_b_b: ( b > product_prod_b_b ) > set_b > set_Product_prod_b_b ).
thf(sy_c_Set_Oinsert_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
insert1909710879tant_a: standard_Constant_a > set_St761939237tant_a > set_St761939237tant_a ).
thf(sy_c_Set_Oinsert_001t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J,type,
insert1167839429_a_nat: labele935650037_a_nat > set_la1083530965_a_nat > set_la1083530965_a_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
insert2037698781od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
insert1574423351_a_nat: produc1871334759_a_nat > set_Pr1987088711_a_nat > set_Pr1987088711_a_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
insert1810136247_a_nat: produc398057191_a_nat > set_Pr924198087_a_nat > set_Pr924198087_a_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
insert1952693431od_b_b: product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oidentity__rules_001tf__a,type,
standa1568205540ules_a: set_St761939237tant_a > set_Pr1987088711_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oreflexivity__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa63370785tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Osymmetry__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa997693288tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Otransitive__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa1795879409tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_member_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
member1632892294tant_a: standard_Constant_a > set_St761939237tant_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
member1214736488tant_a: produc90515391tant_a > set_Pr154086431tant_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
member147868824od_b_b: produc1084374401od_b_b > set_Pr1190604087od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1744485444od_b_b: produc644633773od_b_b > set_Pr1021918435od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member1516365892od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J,type,
member832397200_a_nat: produc1871334759_a_nat > set_Pr1987088711_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1086024932od_b_b: produc970027067od_b_b > set_Pr2109276827od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member942707974od_b_b: produc1990339951od_b_b > set_Pr2134561957od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J,type,
member1417789726_b_b_b: produc276146823_b_b_b > set_Pr1906739389_b_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1533761242od_b_b: produc1169496899od_b_b > set_Pr1954438265od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_J,type,
member584645392_a_nat: produc398057191_a_nat > set_Pr924198087_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member301892752od_b_b: produc1485083751od_b_b > set_Pr667377223od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Nat__Onat_J_J_Mtf__b_J,type,
member1544800936_nat_b: produc845793663_nat_b > set_Pr812188767_nat_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member4694106od_b_b: produc1052326083od_b_b > set_Pr1071216121od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J,type,
member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member599505522od_b_b: produc1605346267od_b_b > set_Pr2060523537od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_G,type,
g: labele1159362096nt_a_b ).
thf(sy_v_L,type,
l: set_St761939237tant_a ).
thf(sy_v_P____,type,
p: b > b > $o ).
thf(sy_v_f____,type,
f: b > b ).
thf(sy_v_l____,type,
l2: standard_Constant_a ).
thf(sy_v_x____,type,
x: b ).
thf(sy_v_y____,type,
y: b ).
% Relevant facts (356)
thf(fact_0__092_060open_062_092_060And_062xa_O_A_092_060lbrakk_062f_Axa_A_092_060in_062_Avertices_AG_059_Axa_A_092_060notin_062_Avertices_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
! [Xa: b] :
( ( member_b @ ( f @ Xa ) @ ( labele1424214014nt_a_b @ g ) )
=> ( member_b @ Xa @ ( labele1424214014nt_a_b @ g ) ) ) ).
% \<open>\<And>xa. \<lbrakk>f xa \<in> vertices G; xa \<notin> vertices G\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_1_fv_I2_J,axiom,
member_b @ ( f @ y ) @ ( labele1424214014nt_a_b @ g ) ).
% fv(2)
thf(fact_2_fv_I1_J,axiom,
member_b @ ( f @ x ) @ ( labele1424214014nt_a_b @ g ) ).
% fv(1)
thf(fact_3_gu2,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
=> ( member_b @ ( f @ X ) @ ( labele1424214014nt_a_b @ g ) ) ) ).
% gu2
thf(fact_4_assms_I1_J,axiom,
( g
= ( restri446606278nt_a_b @ g ) ) ).
% assms(1)
thf(fact_5_r1,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
= ( p @ X @ X ) ) ).
% r1
thf(fact_6_assms_I2_J,axiom,
! [X2: produc1871334759_a_nat] :
( ( member832397200_a_nat @ X2 @ ( standa1568205540ules_a @ l ) )
=> ( mainta1604381365_nat_b @ X2 @ g ) ) ).
% assms(2)
thf(fact_7_gr,axiom,
member1285940496od_b_b @ ( product_Pair_b_b @ ( f @ x ) @ ( f @ y ) ) @ ( getRel904497637nt_a_b @ l2 @ g ) ).
% gr
thf(fact_8_a,axiom,
member1516365892od_b_b @ ( produc1432590431od_b_b @ l2 @ ( product_Pair_b_b @ ( f @ x ) @ ( f @ y ) ) ) @ ( labele1741081071nt_a_b @ g ) ).
% a
thf(fact_9_local_Orefl,axiom,
refl_on_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% local.refl
thf(fact_10_equiv,axiom,
equiv_equiv_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% equiv
thf(fact_11__092_060open_062graph__union_A_Imap__graph__fn_AG_Af_J_AG_A_061_AG_092_060close_062,axiom,
( ( graph_1309249505nt_a_b @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ g ) @ f ) @ g ) @ g )
= g ) ).
% \<open>graph_union (map_graph_fn G f) G = G\<close>
thf(fact_12_gu1,axiom,
! [L: standard_Constant_a,X: b,Y: b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ L @ ( product_Pair_b_b @ X @ Y ) ) @ ( labele1741081071nt_a_b @ g ) )
=> ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
=> ( ( member_b @ Y @ ( labele1424214014nt_a_b @ g ) )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ L @ ( product_Pair_b_b @ ( f @ X ) @ ( f @ Y ) ) ) @ ( labele1741081071nt_a_b @ g ) ) ) ) ) ).
% gu1
thf(fact_13__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ X ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% \<open>\<And>x. x \<in> vertices G \<Longrightarrow> (x, x) \<in> getRel S_Idt G\<close>
thf(fact_14_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr1324126435od_b_b,X22: set_b] :
( ( labele1424214014nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
= X22 ) ).
% labeled_graph.sel(2)
thf(fact_15_subg,axiom,
graph_222606102_a_b_b @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ g ) @ f ) @ g ) @ g @ ( id_on_b @ ( labele1424214014nt_a_b @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ g ) @ f ) @ g ) ) ) ).
% subg
thf(fact_16_vertices__restrict,axiom,
! [G: labele1159362096nt_a_b] :
( ( labele1424214014nt_a_b @ ( restri446606278nt_a_b @ G ) )
= ( labele1424214014nt_a_b @ G ) ) ).
% vertices_restrict
thf(fact_17_restrict__idemp,axiom,
! [X: labele1159362096nt_a_b] :
( ( restri446606278nt_a_b @ ( restri446606278nt_a_b @ X ) )
= ( restri446606278nt_a_b @ X ) ) ).
% restrict_idemp
thf(fact_18_graph__union__idemp_I3_J,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b] :
( ( graph_1309249505nt_a_b @ A @ ( graph_1309249505nt_a_b @ B @ A ) )
= ( graph_1309249505nt_a_b @ B @ A ) ) ).
% graph_union_idemp(3)
thf(fact_19_graph__union__idemp_I2_J,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b] :
( ( graph_1309249505nt_a_b @ A @ ( graph_1309249505nt_a_b @ A @ B ) )
= ( graph_1309249505nt_a_b @ A @ B ) ) ).
% graph_union_idemp(2)
thf(fact_20_graph__union__idemp_I1_J,axiom,
! [A: labele1159362096nt_a_b] :
( ( graph_1309249505nt_a_b @ A @ A )
= A ) ).
% graph_union_idemp(1)
thf(fact_21_map__graph__preserves__restricted,axiom,
! [G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( map_gr926947118tant_a @ F @ G )
= ( restri446606278nt_a_b @ ( map_gr926947118tant_a @ F @ G ) ) ) ) ).
% map_graph_preserves_restricted
thf(fact_22_graph__union__preserves__restrict,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( G_1
= ( restri446606278nt_a_b @ G_1 ) )
=> ( ( G_2
= ( restri446606278nt_a_b @ G_2 ) )
=> ( ( graph_1309249505nt_a_b @ G_1 @ G_2 )
= ( restri446606278nt_a_b @ ( graph_1309249505nt_a_b @ G_1 @ G_2 ) ) ) ) ) ).
% graph_union_preserves_restrict
thf(fact_23_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labele1159362096nt_a_b] :
( ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_24_subgraph__refl,axiom,
! [G: labele1159362096nt_a_b] :
( ( graph_222606102_a_b_b @ G @ G @ ( id_on_b @ ( labele1424214014nt_a_b @ G ) ) )
= ( G
= ( restri446606278nt_a_b @ G ) ) ) ).
% subgraph_refl
thf(fact_25_subgraph__restrict,axiom,
! [G: labele1159362096nt_a_b] :
( ( graph_222606102_a_b_b @ G @ ( restri446606278nt_a_b @ G ) @ ( id_on_b @ ( labele1424214014nt_a_b @ G ) ) )
= ( G
= ( restri446606278nt_a_b @ G ) ) ) ).
% subgraph_restrict
thf(fact_26_graph__homomorphism__Id,axiom,
! [A2: labele1159362096nt_a_b] : ( graph_222606102_a_b_b @ ( restri446606278nt_a_b @ A2 ) @ ( restri446606278nt_a_b @ A2 ) @ ( id_on_b @ ( labele1424214014nt_a_b @ A2 ) ) ) ).
% graph_homomorphism_Id
thf(fact_27_map__graph__preserves__subgraph,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ A @ B @ ( id_on_b @ ( labele1424214014nt_a_b @ A ) ) )
=> ( graph_222606102_a_b_b @ ( map_gr926947118tant_a @ F @ A ) @ ( map_gr926947118tant_a @ F @ B ) @ ( id_on_b @ ( labele1424214014nt_a_b @ ( map_gr926947118tant_a @ F @ A ) ) ) ) ) ).
% map_graph_preserves_subgraph
thf(fact_28_map__graph__fn__id_I2_J,axiom,
! [X3: labele1159362096nt_a_b] :
( ( map_gr926947118tant_a @ ( id_on_b @ ( labele1424214014nt_a_b @ X3 ) ) @ X3 )
= ( restri446606278nt_a_b @ X3 ) ) ).
% map_graph_fn_id(2)
thf(fact_29_map__graph__fn__graphI,axiom,
! [G: labele1159362096nt_a_b,F: b > b] :
( ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G )
= ( restri446606278nt_a_b @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G ) ) ) ).
% map_graph_fn_graphI
thf(fact_30_graph__homo,axiom,
! [G: labele1159362096nt_a_b,F: b > b] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( graph_222606102_a_b_b @ G @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G ) @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) ) ) ).
% graph_homo
thf(fact_31_labeled__graph_Osel_I1_J,axiom,
! [X1: set_Pr1324126435od_b_b,X22: set_b] :
( ( labele1741081071nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_32_subgraph__def,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( graph_222606102_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_1 ) ) )
= ( ( G_1
= ( restri446606278nt_a_b @ G_1 ) )
& ( G_2
= ( restri446606278nt_a_b @ G_2 ) )
& ( ( graph_1309249505nt_a_b @ G_1 @ G_2 )
= G_2 ) ) ) ).
% subgraph_def
thf(fact_33_subgraph__trans,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b,G_3: labele1159362096nt_a_b] :
( ( graph_222606102_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_1 ) ) )
=> ( ( graph_222606102_a_b_b @ G_2 @ G_3 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_2 ) ) )
=> ( graph_222606102_a_b_b @ G_1 @ G_3 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_1 ) ) ) ) ) ).
% subgraph_trans
thf(fact_34_map__graph__fn__eqI,axiom,
! [G: labele1159362096nt_a_b,F: b > b,G2: b > b] :
( ! [X4: b] :
( ( member_b @ X4 @ ( labele1424214014nt_a_b @ G ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G )
= ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ G2 ) @ G ) ) ) ).
% map_graph_fn_eqI
thf(fact_35_subgraph__preserves__hom,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,X3: labele1159362096nt_a_b,H: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ A @ B @ ( id_on_b @ ( labele1424214014nt_a_b @ A ) ) )
=> ( ( graph_222606102_a_b_b @ X3 @ A @ H )
=> ( graph_222606102_a_b_b @ X3 @ B @ H ) ) ) ).
% subgraph_preserves_hom
thf(fact_36_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labele1159362096nt_a_b] :
( Labeled_graph
= ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_37_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labele1159362096nt_a_b,Labeled_graph2: labele1159362096nt_a_b] :
( ( ( ( labele1741081071nt_a_b @ Labeled_graph )
= ( labele1741081071nt_a_b @ Labeled_graph2 ) )
& ( ( labele1424214014nt_a_b @ Labeled_graph )
= ( labele1424214014nt_a_b @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_38_getRel__hom,axiom,
! [Y: product_prod_b_b,Z: product_prod_b_b,L: standard_Constant_a,G: labele1835558936od_b_b,F: product_prod_b_b > b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y @ Z ) @ ( getRel118493133od_b_b @ L @ G ) )
=> ( ( member1285940496od_b_b @ Y @ ( labele442485990od_b_b @ G ) )
=> ( ( member1285940496od_b_b @ Z @ ( labele442485990od_b_b @ G ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ ( F @ Y ) @ ( F @ Z ) ) @ ( getRel904497637nt_a_b @ L @ ( map_gr1314489926tant_a @ ( bNF_Gr521784954_b_b_b @ ( labele442485990od_b_b @ G ) @ F ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_39_getRel__hom,axiom,
! [Y: produc1213276845od_b_b,Z: produc1213276845od_b_b,L: standard_Constant_a,G: labele1644675410od_b_b,F: produc1213276845od_b_b > b] :
( ( member1086024932od_b_b @ ( produc449289715od_b_b @ Y @ Z ) @ ( getRel59882567od_b_b @ L @ G ) )
=> ( ( member1516365892od_b_b @ Y @ ( labele595103470od_b_b @ G ) )
=> ( ( member1516365892od_b_b @ Z @ ( labele595103470od_b_b @ G ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ ( F @ Y ) @ ( F @ Z ) ) @ ( getRel904497637nt_a_b @ L @ ( map_gr1495881666tant_a @ ( bNF_Gr492676974_b_b_b @ ( labele595103470od_b_b @ G ) @ F ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_40_getRel__hom,axiom,
! [Y: b,Z: b,L: standard_Constant_a,G: labele1159362096nt_a_b,F: b > b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( ( member_b @ Y @ ( labele1424214014nt_a_b @ G ) )
=> ( ( member_b @ Z @ ( labele1424214014nt_a_b @ G ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ ( F @ Y ) @ ( F @ Z ) ) @ ( getRel904497637nt_a_b @ L @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G ) ) ) ) ) ) ).
% getRel_hom
thf(fact_41_getRel__map__fn,axiom,
! [A22: product_prod_b_b,G: labele1835558936od_b_b,B2: product_prod_b_b,L: standard_Constant_a,F: product_prod_b_b > b,A2: b,B3: b] :
( ( member1285940496od_b_b @ A22 @ ( labele442485990od_b_b @ G ) )
=> ( ( member1285940496od_b_b @ B2 @ ( labele442485990od_b_b @ G ) )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A22 @ B2 ) @ ( getRel118493133od_b_b @ L @ G ) )
=> ( ( ( F @ A22 )
= A2 )
=> ( ( ( F @ B2 )
= B3 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( getRel904497637nt_a_b @ L @ ( map_gr1314489926tant_a @ ( bNF_Gr521784954_b_b_b @ ( labele442485990od_b_b @ G ) @ F ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_42_getRel__map__fn,axiom,
! [A22: produc1213276845od_b_b,G: labele1644675410od_b_b,B2: produc1213276845od_b_b,L: standard_Constant_a,F: produc1213276845od_b_b > b,A2: b,B3: b] :
( ( member1516365892od_b_b @ A22 @ ( labele595103470od_b_b @ G ) )
=> ( ( member1516365892od_b_b @ B2 @ ( labele595103470od_b_b @ G ) )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A22 @ B2 ) @ ( getRel59882567od_b_b @ L @ G ) )
=> ( ( ( F @ A22 )
= A2 )
=> ( ( ( F @ B2 )
= B3 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( getRel904497637nt_a_b @ L @ ( map_gr1495881666tant_a @ ( bNF_Gr492676974_b_b_b @ ( labele595103470od_b_b @ G ) @ F ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_43_getRel__map__fn,axiom,
! [A22: b,G: labele1159362096nt_a_b,B2: b,L: standard_Constant_a,F: b > b,A2: b,B3: b] :
( ( member_b @ A22 @ ( labele1424214014nt_a_b @ G ) )
=> ( ( member_b @ B2 @ ( labele1424214014nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A22 @ B2 ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( ( ( F @ A22 )
= A2 )
=> ( ( ( F @ B2 )
= B3 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( getRel904497637nt_a_b @ L @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G ) ) ) ) ) ) ) ) ).
% getRel_map_fn
thf(fact_44_idt__eq,axiom,
! [X: b,Y: b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ standard_S_Idt_a @ ( product_Pair_b_b @ X @ Y ) ) @ ( labele1741081071nt_a_b @ g ) )
=> ( ( hilbert_Eps_b @ ( p @ X ) )
= ( hilbert_Eps_b @ ( p @ Y ) ) ) ) ).
% idt_eq
thf(fact_45_vert__P,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ ( hilbert_Eps_b @ ( p @ X ) ) ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% vert_P
thf(fact_46_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A2: product_prod_b_b,P: product_prod_b_b > $o] :
( ( member1285940496od_b_b @ A2 @ ( collec1481886546od_b_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A2: produc1213276845od_b_b,P: produc1213276845od_b_b > $o] :
( ( member1516365892od_b_b @ A2 @ ( collec279084418od_b_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X5: b] : ( member_b @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A: set_Product_prod_b_b] :
( ( collec1481886546od_b_b
@ ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A: set_Pr1324126435od_b_b] :
( ( collec279084418od_b_b
@ ^ [X5: produc1213276845od_b_b] : ( member1516365892od_b_b @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_52_getRel__subgraph,axiom,
! [Y: b,Z: b,L: standard_Constant_a,G: labele1159362096nt_a_b,G3: labele1159362096nt_a_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( ( graph_222606102_a_b_b @ G @ G3 @ ( id_on_b @ ( labele1424214014nt_a_b @ G ) ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ ( getRel904497637nt_a_b @ L @ G3 ) ) ) ) ).
% getRel_subgraph
thf(fact_53_in__Gr,axiom,
! [X: b,Y: b,A: set_b,F: b > b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( bNF_Gr_b_b @ A @ F ) )
= ( ( member_b @ X @ A )
& ( ( F @ X )
= Y ) ) ) ).
% in_Gr
thf(fact_54_in__Gr,axiom,
! [X: standard_Constant_a,Y: product_prod_b_b,A: set_St761939237tant_a,F: standard_Constant_a > product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X @ Y ) @ ( bNF_Gr1272216980od_b_b @ A @ F ) )
= ( ( member1632892294tant_a @ X @ A )
& ( ( F @ X )
= Y ) ) ) ).
% in_Gr
thf(fact_55_Gr__not__in,axiom,
! [X: b,F2: set_b,F: b > b,Y: b] :
( ( ~ ( member_b @ X @ F2 )
| ( ( F @ X )
!= Y ) )
=> ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( bNF_Gr_b_b @ F2 @ F ) ) ) ).
% Gr_not_in
thf(fact_56_Gr__not__in,axiom,
! [X: standard_Constant_a,F2: set_St761939237tant_a,F: standard_Constant_a > product_prod_b_b,Y: product_prod_b_b] :
( ( ~ ( member1632892294tant_a @ X @ F2 )
| ( ( F @ X )
!= Y ) )
=> ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X @ Y ) @ ( bNF_Gr1272216980od_b_b @ F2 @ F ) ) ) ).
% Gr_not_in
thf(fact_57_Id__onI,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A2 @ A )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ A2 ) @ ( id_on_2019020932od_b_b @ A ) ) ) ).
% Id_onI
thf(fact_58_Id__onI,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A2 @ A )
=> ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ A2 ) @ ( id_on_250271696od_b_b @ A ) ) ) ).
% Id_onI
thf(fact_59_Id__onI,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ A2 ) @ ( id_on_b @ A ) ) ) ).
% Id_onI
thf(fact_60_getRel__dom_I2_J,axiom,
! [G: labele1159362096nt_a_b,A2: b,B3: b,L: standard_Constant_a] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( member_b @ B3 @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% getRel_dom(2)
thf(fact_61_getRel__dom_I1_J,axiom,
! [G: labele1159362096nt_a_b,A2: b,B3: b,L: standard_Constant_a] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( member_b @ A2 @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% getRel_dom(1)
thf(fact_62_Id__on__iff,axiom,
! [X: product_prod_b_b,Y: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y ) @ ( id_on_2019020932od_b_b @ A ) )
= ( ( X = Y )
& ( member1285940496od_b_b @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_63_Id__on__iff,axiom,
! [X: produc1213276845od_b_b,Y: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y ) @ ( id_on_250271696od_b_b @ A ) )
= ( ( X = Y )
& ( member1516365892od_b_b @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_64_Id__on__iff,axiom,
! [X: b,Y: b,A: set_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( id_on_b @ A ) )
= ( ( X = Y )
& ( member_b @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_65_Id__on__eqI,axiom,
! [A2: product_prod_b_b,B3: product_prod_b_b,A: set_Product_prod_b_b] :
( ( A2 = B3 )
=> ( ( member1285940496od_b_b @ A2 @ A )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ B3 ) @ ( id_on_2019020932od_b_b @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_66_Id__on__eqI,axiom,
! [A2: produc1213276845od_b_b,B3: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( A2 = B3 )
=> ( ( member1516365892od_b_b @ A2 @ A )
=> ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ B3 ) @ ( id_on_250271696od_b_b @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_67_Id__on__eqI,axiom,
! [A2: b,B3: b,A: set_b] :
( ( A2 = B3 )
=> ( ( member_b @ A2 @ A )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( id_on_b @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_68_Id__onE,axiom,
! [C: produc1168037607od_b_b,A: set_Product_prod_b_b] :
( ( member1652792080od_b_b @ C @ ( id_on_2019020932od_b_b @ A ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A )
=> ( C
!= ( produc546497367od_b_b @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_69_Id__onE,axiom,
! [C: produc970027067od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1086024932od_b_b @ C @ ( id_on_250271696od_b_b @ A ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X4 @ A )
=> ( C
!= ( produc449289715od_b_b @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_70_Id__onE,axiom,
! [C: product_prod_b_b,A: set_b] :
( ( member1285940496od_b_b @ C @ ( id_on_b @ A ) )
=> ~ ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( C
!= ( product_Pair_b_b @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_71_refl__onD2,axiom,
! [A: set_Product_prod_b_b,R: set_Pr2007082183od_b_b,X: product_prod_b_b,Y: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A @ R )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y ) @ R )
=> ( member1285940496od_b_b @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_72_refl__onD2,axiom,
! [A: set_Pr1324126435od_b_b,R: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A @ R )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y ) @ R )
=> ( member1516365892od_b_b @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_73_refl__onD2,axiom,
! [A: set_b,R: set_Product_prod_b_b,X: b,Y: b] :
( ( refl_on_b @ A @ R )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
=> ( member_b @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_74_refl__onD1,axiom,
! [A: set_Product_prod_b_b,R: set_Pr2007082183od_b_b,X: product_prod_b_b,Y: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A @ R )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y ) @ R )
=> ( member1285940496od_b_b @ X @ A ) ) ) ).
% refl_onD1
thf(fact_75_refl__onD1,axiom,
! [A: set_Pr1324126435od_b_b,R: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A @ R )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y ) @ R )
=> ( member1516365892od_b_b @ X @ A ) ) ) ).
% refl_onD1
thf(fact_76_refl__onD1,axiom,
! [A: set_b,R: set_Product_prod_b_b,X: b,Y: b] :
( ( refl_on_b @ A @ R )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
=> ( member_b @ X @ A ) ) ) ).
% refl_onD1
thf(fact_77_refl__onD,axiom,
! [A: set_Product_prod_b_b,R: set_Pr2007082183od_b_b,A2: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A @ R )
=> ( ( member1285940496od_b_b @ A2 @ A )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_78_refl__onD,axiom,
! [A: set_Pr1324126435od_b_b,R: set_Pr2109276827od_b_b,A2: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A @ R )
=> ( ( member1516365892od_b_b @ A2 @ A )
=> ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_79_refl__onD,axiom,
! [A: set_b,R: set_Product_prod_b_b,A2: b] :
( ( refl_on_b @ A @ R )
=> ( ( member_b @ A2 @ A )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_80_refl__on__Id__on,axiom,
! [A: set_b] : ( refl_on_b @ A @ ( id_on_b @ A ) ) ).
% refl_on_Id_on
thf(fact_81_getRel__homR,axiom,
! [Y: b,Z: b,L: standard_Constant_a,G: labele1159362096nt_a_b,U: b,F: set_Product_prod_b_b,V: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ U ) @ F )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Z @ V ) @ F )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ U @ V ) @ ( getRel904497637nt_a_b @ L @ ( map_gr926947118tant_a @ F @ G ) ) ) ) ) ) ).
% getRel_homR
thf(fact_82_maintained__refl,axiom,
! [R2: labele935650037_a_nat,G: labele1159362096nt_a_b] : ( mainta1604381365_nat_b @ ( produc1676969687_a_nat @ R2 @ R2 ) @ G ) ).
% maintained_refl
thf(fact_83_edge__preserving__on__graphI,axiom,
! [X3: labele1835558936od_b_b,F: product_prod_b_b > b,Y2: set_Pr1324126435od_b_b] :
( ! [L2: standard_Constant_a,X4: product_prod_b_b,Y3: product_prod_b_b] :
( ( member1744485444od_b_b @ ( produc618266719od_b_b @ L2 @ ( produc546497367od_b_b @ X4 @ Y3 ) ) @ ( labele372159959od_b_b @ X3 ) )
=> ( ( member1285940496od_b_b @ X4 @ ( labele442485990od_b_b @ X3 ) )
=> ( ( member1285940496od_b_b @ Y3 @ ( labele442485990od_b_b @ X3 ) )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ L2 @ ( product_Pair_b_b @ ( F @ X4 ) @ ( F @ Y3 ) ) ) @ Y2 ) ) ) )
=> ( edge_p1969196346tant_a @ ( bNF_Gr521784954_b_b_b @ ( labele442485990od_b_b @ X3 ) @ F ) @ ( labele372159959od_b_b @ X3 ) @ Y2 ) ) ).
% edge_preserving_on_graphI
thf(fact_84_edge__preserving__on__graphI,axiom,
! [X3: labele1644675410od_b_b,F: produc1213276845od_b_b > b,Y2: set_Pr1324126435od_b_b] :
( ! [L2: standard_Constant_a,X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
( ( member147868824od_b_b @ ( produc1788032435od_b_b @ L2 @ ( produc449289715od_b_b @ X4 @ Y3 ) ) @ ( labele2032816061od_b_b @ X3 ) )
=> ( ( member1516365892od_b_b @ X4 @ ( labele595103470od_b_b @ X3 ) )
=> ( ( member1516365892od_b_b @ Y3 @ ( labele595103470od_b_b @ X3 ) )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ L2 @ ( product_Pair_b_b @ ( F @ X4 ) @ ( F @ Y3 ) ) ) @ Y2 ) ) ) )
=> ( edge_p1817988046tant_a @ ( bNF_Gr492676974_b_b_b @ ( labele595103470od_b_b @ X3 ) @ F ) @ ( labele2032816061od_b_b @ X3 ) @ Y2 ) ) ).
% edge_preserving_on_graphI
thf(fact_85_edge__preserving__on__graphI,axiom,
! [X3: labele1159362096nt_a_b,F: b > b,Y2: set_Pr1324126435od_b_b] :
( ! [L2: standard_Constant_a,X4: b,Y3: b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ L2 @ ( product_Pair_b_b @ X4 @ Y3 ) ) @ ( labele1741081071nt_a_b @ X3 ) )
=> ( ( member_b @ X4 @ ( labele1424214014nt_a_b @ X3 ) )
=> ( ( member_b @ Y3 @ ( labele1424214014nt_a_b @ X3 ) )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ L2 @ ( product_Pair_b_b @ ( F @ X4 ) @ ( F @ Y3 ) ) ) @ Y2 ) ) ) )
=> ( edge_p1384198690tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ X3 ) @ F ) @ ( labele1741081071nt_a_b @ X3 ) @ Y2 ) ) ).
% edge_preserving_on_graphI
thf(fact_86_reflexivity__rule,axiom,
! [G: labele1159362096nt_a_b,L: standard_Constant_a] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( mainta1604381365_nat_b @ ( standa63370785tant_a @ L ) @ G )
=> ( refl_on_b @ ( labele1424214014nt_a_b @ G ) @ ( getRel904497637nt_a_b @ L @ G ) ) ) ) ).
% reflexivity_rule
thf(fact_87_old_Oprod_Oinject,axiom,
! [A2: b,B3: b,A3: b,B4: b] :
( ( ( product_Pair_b_b @ A2 @ B3 )
= ( product_Pair_b_b @ A3 @ B4 ) )
= ( ( A2 = A3 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_88_old_Oprod_Oinject,axiom,
! [A2: standard_Constant_a,B3: product_prod_b_b,A3: standard_Constant_a,B4: product_prod_b_b] :
( ( ( produc1432590431od_b_b @ A2 @ B3 )
= ( produc1432590431od_b_b @ A3 @ B4 ) )
= ( ( A2 = A3 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_89_prod_Oinject,axiom,
! [X1: b,X22: b,Y1: b,Y22: b] :
( ( ( product_Pair_b_b @ X1 @ X22 )
= ( product_Pair_b_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_90_prod_Oinject,axiom,
! [X1: standard_Constant_a,X22: product_prod_b_b,Y1: standard_Constant_a,Y22: product_prod_b_b] :
( ( ( produc1432590431od_b_b @ X1 @ X22 )
= ( produc1432590431od_b_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_91_refl__on__domain,axiom,
! [A: set_Product_prod_b_b,R: set_Pr2007082183od_b_b,A2: product_prod_b_b,B3: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A @ R )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1285940496od_b_b @ A2 @ A )
& ( member1285940496od_b_b @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_92_refl__on__domain,axiom,
! [A: set_Pr1324126435od_b_b,R: set_Pr2109276827od_b_b,A2: produc1213276845od_b_b,B3: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A @ R )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1516365892od_b_b @ A2 @ A )
& ( member1516365892od_b_b @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_93_refl__on__domain,axiom,
! [A: set_b,R: set_Product_prod_b_b,A2: b,B3: b] :
( ( refl_on_b @ A @ R )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R )
=> ( ( member_b @ A2 @ A )
& ( member_b @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_94_GrD2,axiom,
! [X: b,Fx: b,A: set_b,F: b > b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Fx ) @ ( bNF_Gr_b_b @ A @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_95_GrD2,axiom,
! [X: standard_Constant_a,Fx: product_prod_b_b,A: set_St761939237tant_a,F: standard_Constant_a > product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X @ Fx ) @ ( bNF_Gr1272216980od_b_b @ A @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_96_GrD1,axiom,
! [X: b,Fx: b,A: set_b,F: b > b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Fx ) @ ( bNF_Gr_b_b @ A @ F ) )
=> ( member_b @ X @ A ) ) ).
% GrD1
thf(fact_97_GrD1,axiom,
! [X: standard_Constant_a,Fx: product_prod_b_b,A: set_St761939237tant_a,F: standard_Constant_a > product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X @ Fx ) @ ( bNF_Gr1272216980od_b_b @ A @ F ) )
=> ( member1632892294tant_a @ X @ A ) ) ).
% GrD1
thf(fact_98_map__graph__edge__preserving,axiom,
! [F: set_Product_prod_b_b,G: labele1159362096nt_a_b] : ( edge_p1384198690tant_a @ F @ ( labele1741081071nt_a_b @ G ) @ ( labele1741081071nt_a_b @ ( map_gr926947118tant_a @ F @ G ) ) ) ).
% map_graph_edge_preserving
thf(fact_99_edge__preserving__atomic,axiom,
! [H1: set_Product_prod_b_b,E1: set_Pr1324126435od_b_b,E2: set_Pr1324126435od_b_b,V1: b,V12: b,V2: b,V22: b,K: standard_Constant_a] :
( ( edge_p1384198690tant_a @ H1 @ E1 @ E2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V12 ) @ H1 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V22 ) @ H1 )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ K @ ( product_Pair_b_b @ V1 @ V2 ) ) @ E1 )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ K @ ( product_Pair_b_b @ V12 @ V22 ) ) @ E2 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_100_surj__pair,axiom,
! [P2: product_prod_b_b] :
? [X4: b,Y3: b] :
( P2
= ( product_Pair_b_b @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_101_surj__pair,axiom,
! [P2: produc1213276845od_b_b] :
? [X4: standard_Constant_a,Y3: product_prod_b_b] :
( P2
= ( produc1432590431od_b_b @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_102_prod__cases,axiom,
! [P: product_prod_b_b > $o,P2: product_prod_b_b] :
( ! [A4: b,B5: b] : ( P @ ( product_Pair_b_b @ A4 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_103_prod__cases,axiom,
! [P: produc1213276845od_b_b > $o,P2: produc1213276845od_b_b] :
( ! [A4: standard_Constant_a,B5: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A4 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_104_Pair__inject,axiom,
! [A2: b,B3: b,A3: b,B4: b] :
( ( ( product_Pair_b_b @ A2 @ B3 )
= ( product_Pair_b_b @ A3 @ B4 ) )
=> ~ ( ( A2 = A3 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_105_Pair__inject,axiom,
! [A2: standard_Constant_a,B3: product_prod_b_b,A3: standard_Constant_a,B4: product_prod_b_b] :
( ( ( produc1432590431od_b_b @ A2 @ B3 )
= ( produc1432590431od_b_b @ A3 @ B4 ) )
=> ~ ( ( A2 = A3 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_106_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_b_b] :
~ ! [A4: b,B5: b] :
( Y
!= ( product_Pair_b_b @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_107_old_Oprod_Oexhaust,axiom,
! [Y: produc1213276845od_b_b] :
~ ! [A4: standard_Constant_a,B5: product_prod_b_b] :
( Y
!= ( produc1432590431od_b_b @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_108_old_Oprod_Oinducts,axiom,
! [P: product_prod_b_b > $o,Prod: product_prod_b_b] :
( ! [A4: b,B5: b] : ( P @ ( product_Pair_b_b @ A4 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_109_old_Oprod_Oinducts,axiom,
! [P: produc1213276845od_b_b > $o,Prod: produc1213276845od_b_b] :
( ! [A4: standard_Constant_a,B5: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A4 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_110_prod__cases3,axiom,
! [Y: produc1213276845od_b_b] :
~ ! [A4: standard_Constant_a,B5: b,C2: b] :
( Y
!= ( produc1432590431od_b_b @ A4 @ ( product_Pair_b_b @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_111_prod__induct3,axiom,
! [P: produc1213276845od_b_b > $o,X: produc1213276845od_b_b] :
( ! [A4: standard_Constant_a,B5: b,C2: b] : ( P @ ( produc1432590431od_b_b @ A4 @ ( product_Pair_b_b @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_112_map__graph__in,axiom,
! [G: labele1159362096nt_a_b,A2: b,B3: b,E: allego510293162tant_a,F: b > b] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( semant55076487nt_a_b @ G @ E ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ ( F @ A2 ) @ ( F @ B3 ) ) @ ( semant55076487nt_a_b @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ F ) @ G ) @ E ) ) ) ) ).
% map_graph_in
thf(fact_113_maintainedI,axiom,
! [A: labele935650037_a_nat,G: labele1159362096nt_a_b,B: labele935650037_a_nat] :
( ! [F3: set_Pr2106913242_nat_b] :
( ( graph_714568023_nat_b @ A @ G @ F3 )
=> ( extens1554515172_nat_b @ ( produc1676969687_a_nat @ A @ B ) @ G @ F3 ) )
=> ( mainta1604381365_nat_b @ ( produc1676969687_a_nat @ A @ B ) @ G ) ) ).
% maintainedI
thf(fact_114_maintainedI,axiom,
! [A: labele1159362096nt_a_b,G: labele1159362096nt_a_b,B: labele1159362096nt_a_b] :
( ! [F3: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ A @ G @ F3 )
=> ( extens1779443913_a_b_b @ ( produc1569511545nt_a_b @ A @ B ) @ G @ F3 ) )
=> ( mainta374363448_a_b_b @ ( produc1569511545nt_a_b @ A @ B ) @ G ) ) ).
% maintainedI
thf(fact_115_extensible__refl__concr,axiom,
! [E_1: set_Pr1324126435od_b_b,V: set_b,G: labele1159362096nt_a_b,F: set_Product_prod_b_b,E_2: set_Pr1324126435od_b_b] :
( ( graph_222606102_a_b_b @ ( labele1230159100nt_a_b @ E_1 @ V ) @ G @ F )
=> ( ( extens1779443913_a_b_b @ ( produc1569511545nt_a_b @ ( labele1230159100nt_a_b @ E_1 @ V ) @ ( labele1230159100nt_a_b @ E_2 @ V ) ) @ G @ F )
= ( graph_222606102_a_b_b @ ( labele1230159100nt_a_b @ E_2 @ V ) @ G @ F ) ) ) ).
% extensible_refl_concr
thf(fact_116_extensible__refl,axiom,
! [R2: labele1159362096nt_a_b,G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ R2 @ G @ F )
=> ( extens1779443913_a_b_b @ ( produc1569511545nt_a_b @ R2 @ R2 ) @ G @ F ) ) ).
% extensible_refl
thf(fact_117_graph__homomorphism__semantics,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,F: set_Product_prod_b_b,A2: b,B3: b,E: allego510293162tant_a,A3: b,B4: b] :
( ( graph_222606102_a_b_b @ A @ B @ F )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( semant55076487nt_a_b @ A @ E ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ A3 ) @ F )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B3 @ B4 ) @ F )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B4 ) @ ( semant55076487nt_a_b @ B @ E ) ) ) ) ) ) ).
% graph_homomorphism_semantics
thf(fact_118_consequence__graphD_I1_J,axiom,
! [Rs: set_Pr1987088711_a_nat,G: labele1159362096nt_a_b,R2: produc1871334759_a_nat] :
( ( conseq956760372_nat_b @ Rs @ G )
=> ( ( member832397200_a_nat @ R2 @ Rs )
=> ( mainta1604381365_nat_b @ R2 @ G ) ) ) ).
% consequence_graphD(1)
thf(fact_119_semantics__in__vertices_I1_J,axiom,
! [A: labele1159362096nt_a_b,A2: b,B3: b,E: allego510293162tant_a] :
( ( A
= ( restri446606278nt_a_b @ A ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( semant55076487nt_a_b @ A @ E ) )
=> ( member_b @ A2 @ ( labele1424214014nt_a_b @ A ) ) ) ) ).
% semantics_in_vertices(1)
thf(fact_120_semantics__in__vertices_I2_J,axiom,
! [A: labele1159362096nt_a_b,A2: b,B3: b,E: allego510293162tant_a] :
( ( A
= ( restri446606278nt_a_b @ A ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( semant55076487nt_a_b @ A @ E ) )
=> ( member_b @ B3 @ ( labele1424214014nt_a_b @ A ) ) ) ) ).
% semantics_in_vertices(2)
thf(fact_121_subgraph__semantics,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,A2: b,B3: b,E: allego510293162tant_a] :
( ( graph_222606102_a_b_b @ A @ B @ ( id_on_b @ ( labele1424214014nt_a_b @ A ) ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( semant55076487nt_a_b @ A @ E ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ ( semant55076487nt_a_b @ B @ E ) ) ) ) ).
% subgraph_semantics
thf(fact_122_maintainedD,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,G: labele1159362096nt_a_b,F: set_Pr2106913242_nat_b] :
( ( mainta1604381365_nat_b @ ( produc1676969687_a_nat @ A @ B ) @ G )
=> ( ( graph_714568023_nat_b @ A @ G @ F )
=> ( extens1554515172_nat_b @ ( produc1676969687_a_nat @ A @ B ) @ G @ F ) ) ) ).
% maintainedD
thf(fact_123_maintainedD,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( mainta374363448_a_b_b @ ( produc1569511545nt_a_b @ A @ B ) @ G )
=> ( ( graph_222606102_a_b_b @ A @ G @ F )
=> ( extens1779443913_a_b_b @ ( produc1569511545nt_a_b @ A @ B ) @ G @ F ) ) ) ).
% maintainedD
thf(fact_124_extensibleI,axiom,
! [R22: labele1159362096nt_a_b,G: labele1159362096nt_a_b,G2: set_Product_prod_b_b,R1: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ R22 @ G @ G2 )
=> ( ( agree_221379389_a_b_b @ R1 @ F @ G2 )
=> ( extens1779443913_a_b_b @ ( produc1569511545nt_a_b @ R1 @ R22 ) @ G @ F ) ) ) ).
% extensibleI
thf(fact_125_subgraph__subset_I2_J,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( graph_222606102_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_1 ) ) )
=> ( ord_le123089219od_b_b @ ( labele1741081071nt_a_b @ ( restri446606278nt_a_b @ G_1 ) ) @ ( labele1741081071nt_a_b @ G_2 ) ) ) ).
% subgraph_subset(2)
thf(fact_126_graph__homo__union__id_I1_J,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ ( graph_1309249505nt_a_b @ A @ B ) @ G @ F )
=> ( ( A
= ( restri446606278nt_a_b @ A ) )
=> ( graph_222606102_a_b_b @ A @ G @ ( relcomp_b_b_b @ ( id_on_b @ ( labele1424214014nt_a_b @ A ) ) @ F ) ) ) ) ).
% graph_homo_union_id(1)
thf(fact_127_graph__homo__union__id_I2_J,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ ( graph_1309249505nt_a_b @ A @ B ) @ G @ F )
=> ( ( B
= ( restri446606278nt_a_b @ B ) )
=> ( graph_222606102_a_b_b @ B @ G @ ( relcomp_b_b_b @ ( id_on_b @ ( labele1424214014nt_a_b @ B ) ) @ F ) ) ) ) ).
% graph_homo_union_id(2)
thf(fact_128__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
( f
= ( ^ [X5: b] :
( if_b
@ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X5 @ bot_bot_set_b ) )
= bot_bot_set_b )
@ X5
@ ( hilbert_Eps_b @ ( p @ X5 ) ) ) ) ) ).
% \<open>f \<equiv> \<lambda>x. if getRel S_Idt G `` {x} = {} then x else Eps (P x)\<close>
thf(fact_129_ImageI,axiom,
! [A2: b,B3: product_prod_b_b,R: set_Pr621705391od_b_b,A: set_b] :
( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A2 @ B3 ) @ R )
=> ( ( member_b @ A2 @ A )
=> ( member1285940496od_b_b @ B3 @ ( image_1177096571od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_130_ImageI,axiom,
! [A2: b,B3: produc1213276845od_b_b,R: set_Pr2060523537od_b_b,A: set_b] :
( ( member599505522od_b_b @ ( produc800495189od_b_b @ A2 @ B3 ) @ R )
=> ( ( member_b @ A2 @ A )
=> ( member1516365892od_b_b @ B3 @ ( image_984343321od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_131_ImageI,axiom,
! [A2: product_prod_b_b,B3: b,R: set_Pr357419743_b_b_b,A: set_Product_prod_b_b] :
( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A2 @ B3 ) @ R )
=> ( ( member1285940496od_b_b @ A2 @ A )
=> ( member_b @ B3 @ ( image_921732011_b_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_132_ImageI,axiom,
! [A2: product_prod_b_b,B3: product_prod_b_b,R: set_Pr2007082183od_b_b,A: set_Product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1285940496od_b_b @ A2 @ A )
=> ( member1285940496od_b_b @ B3 @ ( image_1928024979od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_133_ImageI,axiom,
! [A2: product_prod_b_b,B3: produc1213276845od_b_b,R: set_Pr1071216121od_b_b,A: set_Product_prod_b_b] :
( ( member4694106od_b_b @ ( produc732676669od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1285940496od_b_b @ A2 @ A )
=> ( member1516365892od_b_b @ B3 @ ( image_532916993od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_134_ImageI,axiom,
! [A2: produc1213276845od_b_b,B3: b,R: set_Pr1906739389_b_b_b,A: set_Pr1324126435od_b_b] :
( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A2 @ B3 ) @ R )
=> ( ( member1516365892od_b_b @ A2 @ A )
=> ( member_b @ B3 @ ( image_1764625405_b_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_135_ImageI,axiom,
! [A2: produc1213276845od_b_b,B3: product_prod_b_b,R: set_Pr2134561957od_b_b,A: set_Pr1324126435od_b_b] :
( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1516365892od_b_b @ A2 @ A )
=> ( member1285940496od_b_b @ B3 @ ( image_1397930469od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_136_ImageI,axiom,
! [A2: produc1213276845od_b_b,B3: produc1213276845od_b_b,R: set_Pr2109276827od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1516365892od_b_b @ A2 @ A )
=> ( member1516365892od_b_b @ B3 @ ( image_763305007od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_137_ImageI,axiom,
! [A2: b,B3: b,R: set_Product_prod_b_b,A: set_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R )
=> ( ( member_b @ A2 @ A )
=> ( member_b @ B3 @ ( image_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_138_ImageI,axiom,
! [A2: standard_Constant_a,B3: product_prod_b_b,R: set_Pr1324126435od_b_b,A: set_St761939237tant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1632892294tant_a @ A2 @ A )
=> ( member1285940496od_b_b @ B3 @ ( image_1916422435od_b_b @ R @ A ) ) ) ) ).
% ImageI
thf(fact_139_Image__empty2,axiom,
! [R2: set_Product_prod_b_b] :
( ( image_b_b @ R2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Image_empty2
thf(fact_140_Image__empty2,axiom,
! [R2: set_Pr1646010159_a_nat] :
( ( image_1749766139_a_nat @ R2 @ bot_bot_set_b )
= bot_bo1836341171_a_nat ) ).
% Image_empty2
thf(fact_141_Image__empty2,axiom,
! [R2: set_Pr812188767_nat_b] :
( ( image_924992811_nat_b @ R2 @ bot_bo1836341171_a_nat )
= bot_bot_set_b ) ).
% Image_empty2
thf(fact_142_Image__empty2,axiom,
! [R2: set_Pr924198087_a_nat] :
( ( image_1168831379_a_nat @ R2 @ bot_bo1836341171_a_nat )
= bot_bo1836341171_a_nat ) ).
% Image_empty2
thf(fact_143_Image__empty1,axiom,
! [X3: set_b] :
( ( image_b_b @ bot_bo1343651123od_b_b @ X3 )
= bot_bot_set_b ) ).
% Image_empty1
thf(fact_144_Image__empty1,axiom,
! [X3: set_la1083530965_a_nat] :
( ( image_1971191571_a_nat @ bot_bo1836341171_a_nat @ X3 )
= bot_bo2122869057_a_nat ) ).
% Image_empty1
thf(fact_145_Id__on__empty,axiom,
( ( id_on_689842066_a_nat @ bot_bo2122869057_a_nat )
= bot_bo1836341171_a_nat ) ).
% Id_on_empty
thf(fact_146_Id__on__empty,axiom,
( ( id_on_b @ bot_bot_set_b )
= bot_bo1343651123od_b_b ) ).
% Id_on_empty
thf(fact_147_Id__on__empty,axiom,
( ( id_on_1651096324_a_nat @ bot_bo1836341171_a_nat )
= bot_bo1973379891_a_nat ) ).
% Id_on_empty
thf(fact_148_Gr__empty,axiom,
! [F: labele935650037_a_nat > labele935650037_a_nat] :
( ( bNF_Gr996500706_a_nat @ bot_bo2122869057_a_nat @ F )
= bot_bo1836341171_a_nat ) ).
% Gr_empty
thf(fact_149_Gr__empty,axiom,
! [F: b > b] :
( ( bNF_Gr_b_b @ bot_bot_set_b @ F )
= bot_bo1343651123od_b_b ) ).
% Gr_empty
thf(fact_150_graph__homomorphism__composes,axiom,
! [A2: labele1159362096nt_a_b,B3: labele1159362096nt_a_b,X: set_Product_prod_b_b,C: labele1159362096nt_a_b,Y: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ A2 @ B3 @ X )
=> ( ( graph_222606102_a_b_b @ B3 @ C @ Y )
=> ( graph_222606102_a_b_b @ A2 @ C @ ( relcomp_b_b_b @ X @ Y ) ) ) ) ).
% graph_homomorphism_composes
thf(fact_151_P__eq,axiom,
! [X: b] :
( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
!= bot_bot_set_b )
=> ( ( hilbert_Eps_b @ ( p @ ( hilbert_Eps_b @ ( p @ X ) ) ) )
= ( hilbert_Eps_b @ ( p @ X ) ) ) ) ).
% P_eq
thf(fact_152_Gr__insert,axiom,
! [X: labele935650037_a_nat,F2: set_la1083530965_a_nat,F: labele935650037_a_nat > labele935650037_a_nat] :
( ( bNF_Gr996500706_a_nat @ ( insert1167839429_a_nat @ X @ F2 ) @ F )
= ( insert1574423351_a_nat @ ( produc1676969687_a_nat @ X @ ( F @ X ) ) @ ( bNF_Gr996500706_a_nat @ F2 @ F ) ) ) ).
% Gr_insert
thf(fact_153_Gr__insert,axiom,
! [X: b,F2: set_b,F: b > b] :
( ( bNF_Gr_b_b @ ( insert_b @ X @ F2 ) @ F )
= ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ ( F @ X ) ) @ ( bNF_Gr_b_b @ F2 @ F ) ) ) ).
% Gr_insert
thf(fact_154_Gr__insert,axiom,
! [X: standard_Constant_a,F2: set_St761939237tant_a,F: standard_Constant_a > product_prod_b_b] :
( ( bNF_Gr1272216980od_b_b @ ( insert1909710879tant_a @ X @ F2 ) @ F )
= ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ X @ ( F @ X ) ) @ ( bNF_Gr1272216980od_b_b @ F2 @ F ) ) ) ).
% Gr_insert
thf(fact_155_map__graph__selectors_I1_J,axiom,
! [F: set_Product_prod_b_b,G: labele1159362096nt_a_b] :
( ( labele1424214014nt_a_b @ ( map_gr926947118tant_a @ F @ G ) )
= ( image_b_b @ F @ ( labele1424214014nt_a_b @ G ) ) ) ).
% map_graph_selectors(1)
thf(fact_156_Image__singleton__iff,axiom,
! [B3: product_prod_b_b,R: set_Pr1324126435od_b_b,A2: standard_Constant_a] :
( ( member1285940496od_b_b @ B3 @ ( image_1916422435od_b_b @ R @ ( insert1909710879tant_a @ A2 @ bot_bo1160111033tant_a ) ) )
= ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_157_Image__singleton__iff,axiom,
! [B3: product_prod_b_b,R: set_Pr621705391od_b_b,A2: b] :
( ( member1285940496od_b_b @ B3 @ ( image_1177096571od_b_b @ R @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( member641857272od_b_b @ ( produc1257047359od_b_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_158_Image__singleton__iff,axiom,
! [B3: produc1213276845od_b_b,R: set_Pr2060523537od_b_b,A2: b] :
( ( member1516365892od_b_b @ B3 @ ( image_984343321od_b_b @ R @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( member599505522od_b_b @ ( produc800495189od_b_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_159_Image__singleton__iff,axiom,
! [B3: b,R: set_Product_prod_b_b,A2: b] :
( ( member_b @ B3 @ ( image_b_b @ R @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_160_Image__singleton__iff,axiom,
! [B3: b,R: set_Pr812188767_nat_b,A2: produc1871334759_a_nat] :
( ( member_b @ B3 @ ( image_924992811_nat_b @ R @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) )
= ( member1544800936_nat_b @ ( produc2059954415_nat_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_161_Image__singleton__iff,axiom,
! [B3: product_prod_b_b,R: set_Pr667377223od_b_b,A2: produc1871334759_a_nat] :
( ( member1285940496od_b_b @ B3 @ ( image_1214223635od_b_b @ R @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) )
= ( member301892752od_b_b @ ( produc590959831od_b_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_162_Image__singleton__iff,axiom,
! [B3: produc1213276845od_b_b,R: set_Pr1954438265od_b_b,A2: produc1871334759_a_nat] :
( ( member1516365892od_b_b @ B3 @ ( image_480774529od_b_b @ R @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) )
= ( member1533761242od_b_b @ ( produc401731773od_b_b @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_163_in__on__graph,axiom,
! [X: b,G: labele1159362096nt_a_b,A2: b > b,Y: b,B3: set_Product_prod_b_b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ ( A2 @ X ) @ Y ) @ B3 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( relcomp_b_b_b @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ G ) @ A2 ) @ B3 ) ) ) ) ).
% in_on_graph
thf(fact_164_in__on__graph,axiom,
! [X: b,G: labele1159362096nt_a_b,A2: b > standard_Constant_a,Y: product_prod_b_b,B3: set_Pr1324126435od_b_b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ G ) )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ ( A2 @ X ) @ Y ) @ B3 )
=> ( member641857272od_b_b @ ( produc1257047359od_b_b @ X @ Y ) @ ( relcom112561144od_b_b @ ( bNF_Gr230160332tant_a @ ( labele1424214014nt_a_b @ G ) @ A2 ) @ B3 ) ) ) ) ).
% in_on_graph
thf(fact_165_graph__homomorphism__empty,axiom,
! [G: labele1159362096nt_a_b,F: set_Pr812188767_nat_b] :
( ( graph_726202286_nat_b @ ( labele27098724_a_nat @ bot_bo1653310327_a_nat @ bot_bo1836341171_a_nat ) @ G @ F )
= ( ( F = bot_bo1113554635_nat_b )
& ( G
= ( restri446606278nt_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_166_graph__homomorphism__empty,axiom,
! [G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ bot_bot_set_b ) @ G @ F )
= ( ( F = bot_bo1343651123od_b_b )
& ( G
= ( restri446606278nt_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_167_graph__unionI,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( ord_le123089219od_b_b @ ( labele1741081071nt_a_b @ G_1 ) @ ( labele1741081071nt_a_b @ G_2 ) )
=> ( ( ord_less_eq_set_b @ ( labele1424214014nt_a_b @ G_1 ) @ ( labele1424214014nt_a_b @ G_2 ) )
=> ( ( graph_1309249505nt_a_b @ G_1 @ G_2 )
= G_2 ) ) ) ).
% graph_unionI
thf(fact_168_graph__single,axiom,
! [A2: standard_Constant_a,B3: b,C: b] :
( ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A2 @ ( product_Pair_b_b @ B3 @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B3 @ ( insert_b @ C @ bot_bot_set_b ) ) )
= ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A2 @ ( product_Pair_b_b @ B3 @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B3 @ ( insert_b @ C @ bot_bot_set_b ) ) ) ) ) ).
% graph_single
thf(fact_169_r2,axiom,
! [X: b] :
( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
= bot_bot_set_b )
= ( ~ ( member_b @ X @ ( labele1424214014nt_a_b @ g ) ) ) ) ).
% r2
thf(fact_170_refl__on__singleton,axiom,
! [X: labele935650037_a_nat] : ( refl_o1031343860_a_nat @ ( insert1167839429_a_nat @ X @ bot_bo2122869057_a_nat ) @ ( insert1574423351_a_nat @ ( produc1676969687_a_nat @ X @ X ) @ bot_bo1836341171_a_nat ) ) ).
% refl_on_singleton
thf(fact_171_refl__on__singleton,axiom,
! [X: b] : ( refl_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% refl_on_singleton
thf(fact_172_refl__on__singleton,axiom,
! [X: produc1871334759_a_nat] : ( refl_o1213627494_a_nat @ ( insert1574423351_a_nat @ X @ bot_bo1836341171_a_nat ) @ ( insert1810136247_a_nat @ ( produc1677124439_a_nat @ X @ X ) @ bot_bo1973379891_a_nat ) ) ).
% refl_on_singleton
thf(fact_173_rev__ImageI,axiom,
! [A2: b,A: set_b,B3: product_prod_b_b,R: set_Pr621705391od_b_b] :
( ( member_b @ A2 @ A )
=> ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A2 @ B3 ) @ R )
=> ( member1285940496od_b_b @ B3 @ ( image_1177096571od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_174_rev__ImageI,axiom,
! [A2: b,A: set_b,B3: produc1213276845od_b_b,R: set_Pr2060523537od_b_b] :
( ( member_b @ A2 @ A )
=> ( ( member599505522od_b_b @ ( produc800495189od_b_b @ A2 @ B3 ) @ R )
=> ( member1516365892od_b_b @ B3 @ ( image_984343321od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_175_rev__ImageI,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b,B3: b,R: set_Pr357419743_b_b_b] :
( ( member1285940496od_b_b @ A2 @ A )
=> ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A2 @ B3 ) @ R )
=> ( member_b @ B3 @ ( image_921732011_b_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_176_rev__ImageI,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b,B3: product_prod_b_b,R: set_Pr2007082183od_b_b] :
( ( member1285940496od_b_b @ A2 @ A )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ B3 ) @ R )
=> ( member1285940496od_b_b @ B3 @ ( image_1928024979od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_177_rev__ImageI,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b,B3: produc1213276845od_b_b,R: set_Pr1071216121od_b_b] :
( ( member1285940496od_b_b @ A2 @ A )
=> ( ( member4694106od_b_b @ ( produc732676669od_b_b @ A2 @ B3 ) @ R )
=> ( member1516365892od_b_b @ B3 @ ( image_532916993od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_178_rev__ImageI,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b,B3: b,R: set_Pr1906739389_b_b_b] :
( ( member1516365892od_b_b @ A2 @ A )
=> ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A2 @ B3 ) @ R )
=> ( member_b @ B3 @ ( image_1764625405_b_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_179_rev__ImageI,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b,B3: product_prod_b_b,R: set_Pr2134561957od_b_b] :
( ( member1516365892od_b_b @ A2 @ A )
=> ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A2 @ B3 ) @ R )
=> ( member1285940496od_b_b @ B3 @ ( image_1397930469od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_180_rev__ImageI,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b,B3: produc1213276845od_b_b,R: set_Pr2109276827od_b_b] :
( ( member1516365892od_b_b @ A2 @ A )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ B3 ) @ R )
=> ( member1516365892od_b_b @ B3 @ ( image_763305007od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_181_rev__ImageI,axiom,
! [A2: b,A: set_b,B3: b,R: set_Product_prod_b_b] :
( ( member_b @ A2 @ A )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R )
=> ( member_b @ B3 @ ( image_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_182_rev__ImageI,axiom,
! [A2: standard_Constant_a,A: set_St761939237tant_a,B3: product_prod_b_b,R: set_Pr1324126435od_b_b] :
( ( member1632892294tant_a @ A2 @ A )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ B3 ) @ R )
=> ( member1285940496od_b_b @ B3 @ ( image_1916422435od_b_b @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_183_Image__iff,axiom,
! [B3: b,R: set_Product_prod_b_b,A: set_b] :
( ( member_b @ B3 @ ( image_b_b @ R @ A ) )
= ( ? [X5: b] :
( ( member_b @ X5 @ A )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ B3 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_184_Image__iff,axiom,
! [B3: product_prod_b_b,R: set_Pr1324126435od_b_b,A: set_St761939237tant_a] :
( ( member1285940496od_b_b @ B3 @ ( image_1916422435od_b_b @ R @ A ) )
= ( ? [X5: standard_Constant_a] :
( ( member1632892294tant_a @ X5 @ A )
& ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X5 @ B3 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_185_ImageE,axiom,
! [B3: b,R: set_Pr357419743_b_b_b,A: set_Product_prod_b_b] :
( ( member_b @ B3 @ ( image_921732011_b_b_b @ R @ A ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1285940496od_b_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_186_ImageE,axiom,
! [B3: b,R: set_Pr1906739389_b_b_b,A: set_Pr1324126435od_b_b] :
( ( member_b @ B3 @ ( image_1764625405_b_b_b @ R @ A ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1516365892od_b_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_187_ImageE,axiom,
! [B3: product_prod_b_b,R: set_Pr621705391od_b_b,A: set_b] :
( ( member1285940496od_b_b @ B3 @ ( image_1177096571od_b_b @ R @ A ) )
=> ~ ! [X4: b] :
( ( member641857272od_b_b @ ( produc1257047359od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_188_ImageE,axiom,
! [B3: product_prod_b_b,R: set_Pr2007082183od_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ B3 @ ( image_1928024979od_b_b @ R @ A ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1285940496od_b_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_189_ImageE,axiom,
! [B3: product_prod_b_b,R: set_Pr2134561957od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1285940496od_b_b @ B3 @ ( image_1397930469od_b_b @ R @ A ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member942707974od_b_b @ ( produc1597690145od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1516365892od_b_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_190_ImageE,axiom,
! [B3: produc1213276845od_b_b,R: set_Pr2060523537od_b_b,A: set_b] :
( ( member1516365892od_b_b @ B3 @ ( image_984343321od_b_b @ R @ A ) )
=> ~ ! [X4: b] :
( ( member599505522od_b_b @ ( produc800495189od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_191_ImageE,axiom,
! [B3: produc1213276845od_b_b,R: set_Pr1071216121od_b_b,A: set_Product_prod_b_b] :
( ( member1516365892od_b_b @ B3 @ ( image_532916993od_b_b @ R @ A ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member4694106od_b_b @ ( produc732676669od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1285940496od_b_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_192_ImageE,axiom,
! [B3: produc1213276845od_b_b,R: set_Pr2109276827od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ B3 @ ( image_763305007od_b_b @ R @ A ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1516365892od_b_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_193_ImageE,axiom,
! [B3: b,R: set_Product_prod_b_b,A: set_b] :
( ( member_b @ B3 @ ( image_b_b @ R @ A ) )
=> ~ ! [X4: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ B3 ) @ R )
=> ~ ( member_b @ X4 @ A ) ) ) ).
% ImageE
thf(fact_194_ImageE,axiom,
! [B3: product_prod_b_b,R: set_Pr1324126435od_b_b,A: set_St761939237tant_a] :
( ( member1285940496od_b_b @ B3 @ ( image_1916422435od_b_b @ R @ A ) )
=> ~ ! [X4: standard_Constant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ B3 ) @ R )
=> ~ ( member1632892294tant_a @ X4 @ A ) ) ) ).
% ImageE
thf(fact_195_relcomp_OrelcompI,axiom,
! [A2: standard_Constant_a,B3: standard_Constant_a,R: set_Pr154086431tant_a,C: product_prod_b_b,S: set_Pr1324126435od_b_b] :
( ( member1214736488tant_a @ ( produc342647tant_a @ A2 @ B3 ) @ R )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B3 @ C ) @ S )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ C ) @ ( relcom883816262od_b_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_196_relcomp_OrelcompI,axiom,
! [A2: b,B3: b,R: set_Product_prod_b_b,C: b,S: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B3 @ C ) @ S )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ C ) @ ( relcomp_b_b_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_197_relcomp_OrelcompI,axiom,
! [A2: standard_Constant_a,B3: product_prod_b_b,R: set_Pr1324126435od_b_b,C: product_prod_b_b,S: set_Pr2007082183od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ B3 ) @ R )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ B3 @ C ) @ S )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ C ) @ ( relcom1823941168od_b_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_198_relcomp_Oinducts,axiom,
! [X1: b,X22: b,R: set_Product_prod_b_b,S: set_Product_prod_b_b,P: b > b > $o] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X22 ) @ ( relcomp_b_b_b @ R @ S ) )
=> ( ! [A4: b,B5: b,C2: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A4 @ B5 ) @ R )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B5 @ C2 ) @ S )
=> ( P @ A4 @ C2 ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_199_relcomp_Oinducts,axiom,
! [X1: standard_Constant_a,X22: product_prod_b_b,R: set_Pr154086431tant_a,S: set_Pr1324126435od_b_b,P: standard_Constant_a > product_prod_b_b > $o] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X1 @ X22 ) @ ( relcom883816262od_b_b @ R @ S ) )
=> ( ! [A4: standard_Constant_a,B5: standard_Constant_a,C2: product_prod_b_b] :
( ( member1214736488tant_a @ ( produc342647tant_a @ A4 @ B5 ) @ R )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B5 @ C2 ) @ S )
=> ( P @ A4 @ C2 ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_200_relcomp_Oinducts,axiom,
! [X1: standard_Constant_a,X22: product_prod_b_b,R: set_Pr1324126435od_b_b,S: set_Pr2007082183od_b_b,P: standard_Constant_a > product_prod_b_b > $o] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X1 @ X22 ) @ ( relcom1823941168od_b_b @ R @ S ) )
=> ( ! [A4: standard_Constant_a,B5: product_prod_b_b,C2: product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A4 @ B5 ) @ R )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ B5 @ C2 ) @ S )
=> ( P @ A4 @ C2 ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_201_relcomp_Osimps,axiom,
! [A1: b,A22: b,R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( relcomp_b_b_b @ R @ S ) )
= ( ? [A5: b,B6: b,C3: b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ A5 @ B6 ) @ R )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ B6 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_202_relcomp_Osimps,axiom,
! [A1: standard_Constant_a,A22: product_prod_b_b,R: set_Pr154086431tant_a,S: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( relcom883816262od_b_b @ R @ S ) )
= ( ? [A5: standard_Constant_a,B6: standard_Constant_a,C3: product_prod_b_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member1214736488tant_a @ ( produc342647tant_a @ A5 @ B6 ) @ R )
& ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B6 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_203_relcomp_Osimps,axiom,
! [A1: standard_Constant_a,A22: product_prod_b_b,R: set_Pr1324126435od_b_b,S: set_Pr2007082183od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( relcom1823941168od_b_b @ R @ S ) )
= ( ? [A5: standard_Constant_a,B6: product_prod_b_b,C3: product_prod_b_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A5 @ B6 ) @ R )
& ( member1652792080od_b_b @ ( produc546497367od_b_b @ B6 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_204_relcomp_Ocases,axiom,
! [A1: b,A22: b,R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( relcomp_b_b_b @ R @ S ) )
=> ~ ! [B5: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ B5 ) @ R )
=> ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ B5 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_205_relcomp_Ocases,axiom,
! [A1: standard_Constant_a,A22: product_prod_b_b,R: set_Pr154086431tant_a,S: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( relcom883816262od_b_b @ R @ S ) )
=> ~ ! [B5: standard_Constant_a] :
( ( member1214736488tant_a @ ( produc342647tant_a @ A1 @ B5 ) @ R )
=> ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B5 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_206_relcomp_Ocases,axiom,
! [A1: standard_Constant_a,A22: product_prod_b_b,R: set_Pr1324126435od_b_b,S: set_Pr2007082183od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( relcom1823941168od_b_b @ R @ S ) )
=> ~ ! [B5: product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ B5 ) @ R )
=> ~ ( member1652792080od_b_b @ ( produc546497367od_b_b @ B5 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_207_relcompEpair,axiom,
! [A2: b,C: b,R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ C ) @ ( relcomp_b_b_b @ R @ S ) )
=> ~ ! [B5: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B5 ) @ R )
=> ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ B5 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_208_relcompEpair,axiom,
! [A2: standard_Constant_a,C: product_prod_b_b,R: set_Pr154086431tant_a,S: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ C ) @ ( relcom883816262od_b_b @ R @ S ) )
=> ~ ! [B5: standard_Constant_a] :
( ( member1214736488tant_a @ ( produc342647tant_a @ A2 @ B5 ) @ R )
=> ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B5 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_209_relcompEpair,axiom,
! [A2: standard_Constant_a,C: product_prod_b_b,R: set_Pr1324126435od_b_b,S: set_Pr2007082183od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ C ) @ ( relcom1823941168od_b_b @ R @ S ) )
=> ~ ! [B5: product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A2 @ B5 ) @ R )
=> ~ ( member1652792080od_b_b @ ( produc546497367od_b_b @ B5 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_210_relcompE,axiom,
! [Xz: product_prod_b_b,R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ Xz @ ( relcomp_b_b_b @ R @ S ) )
=> ~ ! [X4: b,Y3: b,Z2: b] :
( ( Xz
= ( product_Pair_b_b @ X4 @ Z2 ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R )
=> ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_211_relcompE,axiom,
! [Xz: produc1213276845od_b_b,R: set_Pr154086431tant_a,S: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ Xz @ ( relcom883816262od_b_b @ R @ S ) )
=> ~ ! [X4: standard_Constant_a,Y3: standard_Constant_a,Z2: product_prod_b_b] :
( ( Xz
= ( produc1432590431od_b_b @ X4 @ Z2 ) )
=> ( ( member1214736488tant_a @ ( produc342647tant_a @ X4 @ Y3 ) @ R )
=> ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_212_relcompE,axiom,
! [Xz: produc1213276845od_b_b,R: set_Pr1324126435od_b_b,S: set_Pr2007082183od_b_b] :
( ( member1516365892od_b_b @ Xz @ ( relcom1823941168od_b_b @ R @ S ) )
=> ~ ! [X4: standard_Constant_a,Y3: product_prod_b_b,Z2: product_prod_b_b] :
( ( Xz
= ( produc1432590431od_b_b @ X4 @ Z2 ) )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R )
=> ~ ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_213_relcomp__Image,axiom,
! [X3: set_Product_prod_b_b,Y2: set_Product_prod_b_b,Z3: set_b] :
( ( image_b_b @ ( relcomp_b_b_b @ X3 @ Y2 ) @ Z3 )
= ( image_b_b @ Y2 @ ( image_b_b @ X3 @ Z3 ) ) ) ).
% relcomp_Image
thf(fact_214_Image__mono,axiom,
! [R3: set_Product_prod_b_b,R: set_Product_prod_b_b,A6: set_b,A: set_b] :
( ( ord_le1036320359od_b_b @ R3 @ R )
=> ( ( ord_less_eq_set_b @ A6 @ A )
=> ( ord_less_eq_set_b @ ( image_b_b @ R3 @ A6 ) @ ( image_b_b @ R @ A ) ) ) ) ).
% Image_mono
thf(fact_215_agree__on__def,axiom,
( agree_221379389_a_b_b
= ( ^ [G4: labele1159362096nt_a_b,F_1: set_Product_prod_b_b,F_2: set_Product_prod_b_b] :
! [X5: b] :
( ( member_b @ X5 @ ( labele1424214014nt_a_b @ G4 ) )
=> ( ( image_b_b @ F_1 @ ( insert_b @ X5 @ bot_bot_set_b ) )
= ( image_b_b @ F_2 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) ) ) ) ).
% agree_on_def
thf(fact_216_refl__on__empty,axiom,
refl_o1031343860_a_nat @ bot_bo2122869057_a_nat @ bot_bo1836341171_a_nat ).
% refl_on_empty
thf(fact_217_refl__on__empty,axiom,
refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% refl_on_empty
thf(fact_218_refl__on__empty,axiom,
refl_o1213627494_a_nat @ bot_bo1836341171_a_nat @ bot_bo1973379891_a_nat ).
% refl_on_empty
thf(fact_219_subrelI,axiom,
! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ! [X4: b,Y3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ S ) )
=> ( ord_le1036320359od_b_b @ R @ S ) ) ).
% subrelI
thf(fact_220_subrelI,axiom,
! [R: set_Pr1324126435od_b_b,S: set_Pr1324126435od_b_b] :
( ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ S ) )
=> ( ord_le123089219od_b_b @ R @ S ) ) ).
% subrelI
thf(fact_221_graph__union__iff,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( ( graph_1309249505nt_a_b @ G_1 @ G_2 )
= G_2 )
= ( ( ord_le123089219od_b_b @ ( labele1741081071nt_a_b @ G_1 ) @ ( labele1741081071nt_a_b @ G_2 ) )
& ( ord_less_eq_set_b @ ( labele1424214014nt_a_b @ G_1 ) @ ( labele1424214014nt_a_b @ G_2 ) ) ) ) ).
% graph_union_iff
thf(fact_222_restrict__subsD,axiom,
! [G: labele1159362096nt_a_b] :
( ( ord_le123089219od_b_b @ ( labele1741081071nt_a_b @ G ) @ ( labele1741081071nt_a_b @ ( restri446606278nt_a_b @ G ) ) )
=> ( G
= ( restri446606278nt_a_b @ G ) ) ) ).
% restrict_subsD
thf(fact_223_subgraph__def2,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( G_1
= ( restri446606278nt_a_b @ G_1 ) )
=> ( ( G_2
= ( restri446606278nt_a_b @ G_2 ) )
=> ( ( graph_222606102_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_1 ) ) )
= ( ( ord_less_eq_set_b @ ( labele1424214014nt_a_b @ G_1 ) @ ( labele1424214014nt_a_b @ G_2 ) )
& ( ord_le123089219od_b_b @ ( labele1741081071nt_a_b @ G_1 ) @ ( labele1741081071nt_a_b @ G_2 ) ) ) ) ) ) ).
% subgraph_def2
thf(fact_224_subgraph__subset_I1_J,axiom,
! [G_1: labele1159362096nt_a_b,G_2: labele1159362096nt_a_b] :
( ( graph_222606102_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labele1424214014nt_a_b @ G_1 ) ) )
=> ( ord_less_eq_set_b @ ( labele1424214014nt_a_b @ G_1 ) @ ( labele1424214014nt_a_b @ G_2 ) ) ) ).
% subgraph_subset(1)
thf(fact_225_maintainedD2,axiom,
! [A: labele935650037_a_nat,B: labele935650037_a_nat,G: labele1159362096nt_a_b,F: set_Pr2106913242_nat_b] :
( ( mainta1604381365_nat_b @ ( produc1676969687_a_nat @ A @ B ) @ G )
=> ( ( graph_714568023_nat_b @ A @ G @ F )
=> ~ ! [G5: set_Pr2106913242_nat_b] :
( ( graph_714568023_nat_b @ B @ G @ G5 )
=> ~ ( ord_le1484132922_nat_b @ F @ G5 ) ) ) ) ).
% maintainedD2
thf(fact_226_maintainedD2,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( mainta374363448_a_b_b @ ( produc1569511545nt_a_b @ A @ B ) @ G )
=> ( ( graph_222606102_a_b_b @ A @ G @ F )
=> ~ ! [G5: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ B @ G @ G5 )
=> ~ ( ord_le1036320359od_b_b @ F @ G5 ) ) ) ) ).
% maintainedD2
thf(fact_227_Id__on__vertices__identity_I2_J,axiom,
! [A2: labele1159362096nt_a_b,B3: labele1159362096nt_a_b,F: set_Product_prod_b_b,Aa: b,Ba: b] :
( ( graph_222606102_a_b_b @ A2 @ B3 @ F )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Aa @ Ba ) @ F )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ Aa @ Ba ) @ ( relcomp_b_b_b @ F @ ( id_on_b @ ( labele1424214014nt_a_b @ B3 ) ) ) ) ) ) ).
% Id_on_vertices_identity(2)
thf(fact_228_Id__on__vertices__identity_I1_J,axiom,
! [A2: labele1159362096nt_a_b,B3: labele1159362096nt_a_b,F: set_Product_prod_b_b,Aa: b,Ba: b] :
( ( graph_222606102_a_b_b @ A2 @ B3 @ F )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Aa @ Ba ) @ F )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ Aa @ Ba ) @ ( relcomp_b_b_b @ ( id_on_b @ ( labele1424214014nt_a_b @ A2 ) ) @ F ) ) ) ) ).
% Id_on_vertices_identity(1)
thf(fact_229_graph__homomorphism__on__graph,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,R2: set_Product_prod_b_b,F: b > b] :
( ( graph_222606102_a_b_b @ A @ B @ R2 )
=> ( graph_222606102_a_b_b @ A @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ B ) @ F ) @ B ) @ ( relcomp_b_b_b @ R2 @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ B ) @ F ) ) ) ) ).
% graph_homomorphism_on_graph
thf(fact_230_singleton__insert__inj__eq,axiom,
! [B3: b,A2: b,A: set_b] :
( ( ( insert_b @ B3 @ bot_bot_set_b )
= ( insert_b @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B3 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_231_singleton__insert__inj__eq,axiom,
! [B3: produc1871334759_a_nat,A2: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat )
= ( insert1574423351_a_nat @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le1718765799_a_nat @ A @ ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_232_singleton__insert__inj__eq_H,axiom,
! [A2: b,A: set_b,B3: b] :
( ( ( insert_b @ A2 @ A )
= ( insert_b @ B3 @ bot_bot_set_b ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B3 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_233_singleton__insert__inj__eq_H,axiom,
! [A2: produc1871334759_a_nat,A: set_Pr1987088711_a_nat,B3: produc1871334759_a_nat] :
( ( ( insert1574423351_a_nat @ A2 @ A )
= ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) )
= ( ( A2 = B3 )
& ( ord_le1718765799_a_nat @ A @ ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_234_equiv__class__subset,axiom,
! [A: set_b,R: set_Product_prod_b_b,A2: b,B3: b] :
( ( equiv_equiv_b @ A @ R )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R )
=> ( ord_less_eq_set_b @ ( image_b_b @ R @ ( insert_b @ A2 @ bot_bot_set_b ) ) @ ( image_b_b @ R @ ( insert_b @ B3 @ bot_bot_set_b ) ) ) ) ) ).
% equiv_class_subset
thf(fact_235_equiv__class__subset,axiom,
! [A: set_Pr1987088711_a_nat,R: set_Pr924198087_a_nat,A2: produc1871334759_a_nat,B3: produc1871334759_a_nat] :
( ( equiv_291114781_a_nat @ A @ R )
=> ( ( member584645392_a_nat @ ( produc1677124439_a_nat @ A2 @ B3 ) @ R )
=> ( ord_le1718765799_a_nat @ ( image_1168831379_a_nat @ R @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) @ ( image_1168831379_a_nat @ R @ ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) ) ) ) ) ).
% equiv_class_subset
thf(fact_236_subset__equiv__class,axiom,
! [A: set_Product_prod_b_b,R: set_Pr2007082183od_b_b,B3: product_prod_b_b,A2: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A @ R )
=> ( ( ord_le1036320359od_b_b @ ( image_1928024979od_b_b @ R @ ( insert1952693431od_b_b @ B3 @ bot_bo1343651123od_b_b ) ) @ ( image_1928024979od_b_b @ R @ ( insert1952693431od_b_b @ A2 @ bot_bo1343651123od_b_b ) ) )
=> ( ( member1285940496od_b_b @ B3 @ A )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A2 @ B3 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_237_subset__equiv__class,axiom,
! [A: set_Pr1324126435od_b_b,R: set_Pr2109276827od_b_b,B3: produc1213276845od_b_b,A2: produc1213276845od_b_b] :
( ( equiv_2048442231od_b_b @ A @ R )
=> ( ( ord_le123089219od_b_b @ ( image_763305007od_b_b @ R @ ( insert2037698781od_b_b @ B3 @ bot_bo1664927607od_b_b ) ) @ ( image_763305007od_b_b @ R @ ( insert2037698781od_b_b @ A2 @ bot_bo1664927607od_b_b ) ) )
=> ( ( member1516365892od_b_b @ B3 @ A )
=> ( member1086024932od_b_b @ ( produc449289715od_b_b @ A2 @ B3 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_238_subset__equiv__class,axiom,
! [A: set_b,R: set_Product_prod_b_b,B3: b,A2: b] :
( ( equiv_equiv_b @ A @ R )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ R @ ( insert_b @ B3 @ bot_bot_set_b ) ) @ ( image_b_b @ R @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
=> ( ( member_b @ B3 @ A )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A2 @ B3 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_239_subset__equiv__class,axiom,
! [A: set_Pr1987088711_a_nat,R: set_Pr924198087_a_nat,B3: produc1871334759_a_nat,A2: produc1871334759_a_nat] :
( ( equiv_291114781_a_nat @ A @ R )
=> ( ( ord_le1718765799_a_nat @ ( image_1168831379_a_nat @ R @ ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) ) @ ( image_1168831379_a_nat @ R @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) )
=> ( ( member832397200_a_nat @ B3 @ A )
=> ( member584645392_a_nat @ ( produc1677124439_a_nat @ A2 @ B3 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_240_refines__equiv__class__eq2,axiom,
! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b,A: set_b,A2: b] :
( ( ord_le1036320359od_b_b @ R2 @ S2 )
=> ( ( equiv_equiv_b @ A @ R2 )
=> ( ( equiv_equiv_b @ A @ S2 )
=> ( ( image_b_b @ S2 @ ( image_b_b @ R2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( image_b_b @ S2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_241_refines__equiv__class__eq2,axiom,
! [R2: set_Pr924198087_a_nat,S2: set_Pr924198087_a_nat,A: set_Pr1987088711_a_nat,A2: produc1871334759_a_nat] :
( ( ord_le107617383_a_nat @ R2 @ S2 )
=> ( ( equiv_291114781_a_nat @ A @ R2 )
=> ( ( equiv_291114781_a_nat @ A @ S2 )
=> ( ( image_1168831379_a_nat @ S2 @ ( image_1168831379_a_nat @ R2 @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) )
= ( image_1168831379_a_nat @ S2 @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_242_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X5: b] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_243_empty__Collect__eq,axiom,
! [P: produc1871334759_a_nat > $o] :
( ( bot_bo1836341171_a_nat
= ( collec357096914_a_nat @ P ) )
= ( ! [X5: produc1871334759_a_nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_244_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X5: b] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_245_Collect__empty__eq,axiom,
! [P: produc1871334759_a_nat > $o] :
( ( ( collec357096914_a_nat @ P )
= bot_bo1836341171_a_nat )
= ( ! [X5: produc1871334759_a_nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_246_all__not__in__conv,axiom,
! [A: set_Product_prod_b_b] :
( ( ! [X5: product_prod_b_b] :
~ ( member1285940496od_b_b @ X5 @ A ) )
= ( A = bot_bo1343651123od_b_b ) ) ).
% all_not_in_conv
thf(fact_247_all__not__in__conv,axiom,
! [A: set_Pr1324126435od_b_b] :
( ( ! [X5: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ X5 @ A ) )
= ( A = bot_bo1664927607od_b_b ) ) ).
% all_not_in_conv
thf(fact_248_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X5: b] :
~ ( member_b @ X5 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_249_all__not__in__conv,axiom,
! [A: set_Pr1987088711_a_nat] :
( ( ! [X5: produc1871334759_a_nat] :
~ ( member832397200_a_nat @ X5 @ A ) )
= ( A = bot_bo1836341171_a_nat ) ) ).
% all_not_in_conv
thf(fact_250_empty__iff,axiom,
! [C: product_prod_b_b] :
~ ( member1285940496od_b_b @ C @ bot_bo1343651123od_b_b ) ).
% empty_iff
thf(fact_251_empty__iff,axiom,
! [C: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ C @ bot_bo1664927607od_b_b ) ).
% empty_iff
thf(fact_252_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_253_empty__iff,axiom,
! [C: produc1871334759_a_nat] :
~ ( member832397200_a_nat @ C @ bot_bo1836341171_a_nat ) ).
% empty_iff
thf(fact_254_subsetI,axiom,
! [A: set_b,B: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( member_b @ X4 @ B ) )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% subsetI
thf(fact_255_subsetI,axiom,
! [A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
( ! [X4: product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A )
=> ( member1285940496od_b_b @ X4 @ B ) )
=> ( ord_le1036320359od_b_b @ A @ B ) ) ).
% subsetI
thf(fact_256_subsetI,axiom,
! [A: set_Pr1324126435od_b_b,B: set_Pr1324126435od_b_b] :
( ! [X4: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X4 @ A )
=> ( member1516365892od_b_b @ X4 @ B ) )
=> ( ord_le123089219od_b_b @ A @ B ) ) ).
% subsetI
thf(fact_257_insert__absorb2,axiom,
! [X: b,A: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A ) )
= ( insert_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_258_insert__absorb2,axiom,
! [X: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( insert1574423351_a_nat @ X @ ( insert1574423351_a_nat @ X @ A ) )
= ( insert1574423351_a_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_259_insert__iff,axiom,
! [A2: produc1871334759_a_nat,B3: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ A2 @ ( insert1574423351_a_nat @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member832397200_a_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_260_insert__iff,axiom,
! [A2: b,B3: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_261_insert__iff,axiom,
! [A2: product_prod_b_b,B3: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A2 @ ( insert1952693431od_b_b @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member1285940496od_b_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_262_insert__iff,axiom,
! [A2: produc1213276845od_b_b,B3: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A2 @ ( insert2037698781od_b_b @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member1516365892od_b_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_263_insertCI,axiom,
! [A2: produc1871334759_a_nat,B: set_Pr1987088711_a_nat,B3: produc1871334759_a_nat] :
( ( ~ ( member832397200_a_nat @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member832397200_a_nat @ A2 @ ( insert1574423351_a_nat @ B3 @ B ) ) ) ).
% insertCI
thf(fact_264_insertCI,axiom,
! [A2: b,B: set_b,B3: b] :
( ( ~ ( member_b @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_b @ A2 @ ( insert_b @ B3 @ B ) ) ) ).
% insertCI
thf(fact_265_insertCI,axiom,
! [A2: product_prod_b_b,B: set_Product_prod_b_b,B3: product_prod_b_b] :
( ( ~ ( member1285940496od_b_b @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member1285940496od_b_b @ A2 @ ( insert1952693431od_b_b @ B3 @ B ) ) ) ).
% insertCI
thf(fact_266_insertCI,axiom,
! [A2: produc1213276845od_b_b,B: set_Pr1324126435od_b_b,B3: produc1213276845od_b_b] :
( ( ~ ( member1516365892od_b_b @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member1516365892od_b_b @ A2 @ ( insert2037698781od_b_b @ B3 @ B ) ) ) ).
% insertCI
thf(fact_267_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_268_empty__subsetI,axiom,
! [A: set_Pr1987088711_a_nat] : ( ord_le1718765799_a_nat @ bot_bo1836341171_a_nat @ A ) ).
% empty_subsetI
thf(fact_269_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_270_subset__empty,axiom,
! [A: set_Pr1987088711_a_nat] :
( ( ord_le1718765799_a_nat @ A @ bot_bo1836341171_a_nat )
= ( A = bot_bo1836341171_a_nat ) ) ).
% subset_empty
thf(fact_271_singletonI,axiom,
! [A2: product_prod_b_b] : ( member1285940496od_b_b @ A2 @ ( insert1952693431od_b_b @ A2 @ bot_bo1343651123od_b_b ) ) ).
% singletonI
thf(fact_272_singletonI,axiom,
! [A2: produc1213276845od_b_b] : ( member1516365892od_b_b @ A2 @ ( insert2037698781od_b_b @ A2 @ bot_bo1664927607od_b_b ) ) ).
% singletonI
thf(fact_273_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_274_singletonI,axiom,
! [A2: produc1871334759_a_nat] : ( member832397200_a_nat @ A2 @ ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat ) ) ).
% singletonI
thf(fact_275_insert__subset,axiom,
! [X: produc1871334759_a_nat,A: set_Pr1987088711_a_nat,B: set_Pr1987088711_a_nat] :
( ( ord_le1718765799_a_nat @ ( insert1574423351_a_nat @ X @ A ) @ B )
= ( ( member832397200_a_nat @ X @ B )
& ( ord_le1718765799_a_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_276_insert__subset,axiom,
! [X: b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B )
= ( ( member_b @ X @ B )
& ( ord_less_eq_set_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_277_insert__subset,axiom,
! [X: product_prod_b_b,A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X @ A ) @ B )
= ( ( member1285940496od_b_b @ X @ B )
& ( ord_le1036320359od_b_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_278_insert__subset,axiom,
! [X: produc1213276845od_b_b,A: set_Pr1324126435od_b_b,B: set_Pr1324126435od_b_b] :
( ( ord_le123089219od_b_b @ ( insert2037698781od_b_b @ X @ A ) @ B )
= ( ( member1516365892od_b_b @ X @ B )
& ( ord_le123089219od_b_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_279_relcomp__empty1,axiom,
! [R2: set_Pr1987088711_a_nat] :
( ( relcom1338300020_a_nat @ bot_bo1836341171_a_nat @ R2 )
= bot_bo1836341171_a_nat ) ).
% relcomp_empty1
thf(fact_280_relcomp__empty2,axiom,
! [R2: set_Pr1987088711_a_nat] :
( ( relcom1338300020_a_nat @ R2 @ bot_bo1836341171_a_nat )
= bot_bo1836341171_a_nat ) ).
% relcomp_empty2
thf(fact_281_graph__empty__e,axiom,
! [V: set_b] :
( ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V )
= ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V ) ) ) ).
% graph_empty_e
thf(fact_282_graph__homomorphism__nonempty,axiom,
! [A: labele1159362096nt_a_b,B: labele1159362096nt_a_b,F: set_Product_prod_b_b,E: allego510293162tant_a] :
( ( graph_222606102_a_b_b @ A @ B @ F )
=> ( ( ( semant55076487nt_a_b @ A @ E )
!= bot_bo1343651123od_b_b )
=> ( ( semant55076487nt_a_b @ B @ E )
!= bot_bo1343651123od_b_b ) ) ) ).
% graph_homomorphism_nonempty
thf(fact_283_ex__in__conv,axiom,
! [A: set_Product_prod_b_b] :
( ( ? [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A ) )
= ( A != bot_bo1343651123od_b_b ) ) ).
% ex_in_conv
thf(fact_284_ex__in__conv,axiom,
! [A: set_Pr1324126435od_b_b] :
( ( ? [X5: produc1213276845od_b_b] : ( member1516365892od_b_b @ X5 @ A ) )
= ( A != bot_bo1664927607od_b_b ) ) ).
% ex_in_conv
thf(fact_285_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X5: b] : ( member_b @ X5 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_286_ex__in__conv,axiom,
! [A: set_Pr1987088711_a_nat] :
( ( ? [X5: produc1871334759_a_nat] : ( member832397200_a_nat @ X5 @ A ) )
= ( A != bot_bo1836341171_a_nat ) ) ).
% ex_in_conv
thf(fact_287_equals0I,axiom,
! [A: set_Product_prod_b_b] :
( ! [Y3: product_prod_b_b] :
~ ( member1285940496od_b_b @ Y3 @ A )
=> ( A = bot_bo1343651123od_b_b ) ) ).
% equals0I
thf(fact_288_equals0I,axiom,
! [A: set_Pr1324126435od_b_b] :
( ! [Y3: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ Y3 @ A )
=> ( A = bot_bo1664927607od_b_b ) ) ).
% equals0I
thf(fact_289_equals0I,axiom,
! [A: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_290_equals0I,axiom,
! [A: set_Pr1987088711_a_nat] :
( ! [Y3: produc1871334759_a_nat] :
~ ( member832397200_a_nat @ Y3 @ A )
=> ( A = bot_bo1836341171_a_nat ) ) ).
% equals0I
thf(fact_291_equals0D,axiom,
! [A: set_Product_prod_b_b,A2: product_prod_b_b] :
( ( A = bot_bo1343651123od_b_b )
=> ~ ( member1285940496od_b_b @ A2 @ A ) ) ).
% equals0D
thf(fact_292_equals0D,axiom,
! [A: set_Pr1324126435od_b_b,A2: produc1213276845od_b_b] :
( ( A = bot_bo1664927607od_b_b )
=> ~ ( member1516365892od_b_b @ A2 @ A ) ) ).
% equals0D
thf(fact_293_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_294_equals0D,axiom,
! [A: set_Pr1987088711_a_nat,A2: produc1871334759_a_nat] :
( ( A = bot_bo1836341171_a_nat )
=> ~ ( member832397200_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_295_emptyE,axiom,
! [A2: product_prod_b_b] :
~ ( member1285940496od_b_b @ A2 @ bot_bo1343651123od_b_b ) ).
% emptyE
thf(fact_296_emptyE,axiom,
! [A2: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ A2 @ bot_bo1664927607od_b_b ) ).
% emptyE
thf(fact_297_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_298_emptyE,axiom,
! [A2: produc1871334759_a_nat] :
~ ( member832397200_a_nat @ A2 @ bot_bo1836341171_a_nat ) ).
% emptyE
thf(fact_299_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A7: set_b,B7: set_b] :
! [T: b] :
( ( member_b @ T @ A7 )
=> ( member_b @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_300_subset__iff,axiom,
( ord_le1036320359od_b_b
= ( ^ [A7: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
! [T: product_prod_b_b] :
( ( member1285940496od_b_b @ T @ A7 )
=> ( member1285940496od_b_b @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_301_subset__iff,axiom,
( ord_le123089219od_b_b
= ( ^ [A7: set_Pr1324126435od_b_b,B7: set_Pr1324126435od_b_b] :
! [T: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ T @ A7 )
=> ( member1516365892od_b_b @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_302_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A7: set_b,B7: set_b] :
! [X5: b] :
( ( member_b @ X5 @ A7 )
=> ( member_b @ X5 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_303_subset__eq,axiom,
( ord_le1036320359od_b_b
= ( ^ [A7: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
! [X5: product_prod_b_b] :
( ( member1285940496od_b_b @ X5 @ A7 )
=> ( member1285940496od_b_b @ X5 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_304_subset__eq,axiom,
( ord_le123089219od_b_b
= ( ^ [A7: set_Pr1324126435od_b_b,B7: set_Pr1324126435od_b_b] :
! [X5: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X5 @ A7 )
=> ( member1516365892od_b_b @ X5 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_305_subsetD,axiom,
! [A: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% subsetD
thf(fact_306_subsetD,axiom,
! [A: set_Product_prod_b_b,B: set_Product_prod_b_b,C: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A @ B )
=> ( ( member1285940496od_b_b @ C @ A )
=> ( member1285940496od_b_b @ C @ B ) ) ) ).
% subsetD
thf(fact_307_subsetD,axiom,
! [A: set_Pr1324126435od_b_b,B: set_Pr1324126435od_b_b,C: produc1213276845od_b_b] :
( ( ord_le123089219od_b_b @ A @ B )
=> ( ( member1516365892od_b_b @ C @ A )
=> ( member1516365892od_b_b @ C @ B ) ) ) ).
% subsetD
thf(fact_308_in__mono,axiom,
! [A: set_b,B: set_b,X: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ X @ A )
=> ( member_b @ X @ B ) ) ) ).
% in_mono
thf(fact_309_in__mono,axiom,
! [A: set_Product_prod_b_b,B: set_Product_prod_b_b,X: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A @ B )
=> ( ( member1285940496od_b_b @ X @ A )
=> ( member1285940496od_b_b @ X @ B ) ) ) ).
% in_mono
thf(fact_310_in__mono,axiom,
! [A: set_Pr1324126435od_b_b,B: set_Pr1324126435od_b_b,X: produc1213276845od_b_b] :
( ( ord_le123089219od_b_b @ A @ B )
=> ( ( member1516365892od_b_b @ X @ A )
=> ( member1516365892od_b_b @ X @ B ) ) ) ).
% in_mono
thf(fact_311_mk__disjoint__insert,axiom,
! [A2: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ A2 @ A )
=> ? [B8: set_Pr1987088711_a_nat] :
( ( A
= ( insert1574423351_a_nat @ A2 @ B8 ) )
& ~ ( member832397200_a_nat @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_312_mk__disjoint__insert,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ? [B8: set_b] :
( ( A
= ( insert_b @ A2 @ B8 ) )
& ~ ( member_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_313_mk__disjoint__insert,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A2 @ A )
=> ? [B8: set_Product_prod_b_b] :
( ( A
= ( insert1952693431od_b_b @ A2 @ B8 ) )
& ~ ( member1285940496od_b_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_314_mk__disjoint__insert,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A2 @ A )
=> ? [B8: set_Pr1324126435od_b_b] :
( ( A
= ( insert2037698781od_b_b @ A2 @ B8 ) )
& ~ ( member1516365892od_b_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_315_insert__commute,axiom,
! [X: b,Y: b,A: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A ) )
= ( insert_b @ Y @ ( insert_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_316_insert__commute,axiom,
! [X: produc1871334759_a_nat,Y: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( insert1574423351_a_nat @ X @ ( insert1574423351_a_nat @ Y @ A ) )
= ( insert1574423351_a_nat @ Y @ ( insert1574423351_a_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_317_insert__eq__iff,axiom,
! [A2: produc1871334759_a_nat,A: set_Pr1987088711_a_nat,B3: produc1871334759_a_nat,B: set_Pr1987088711_a_nat] :
( ~ ( member832397200_a_nat @ A2 @ A )
=> ( ~ ( member832397200_a_nat @ B3 @ B )
=> ( ( ( insert1574423351_a_nat @ A2 @ A )
= ( insert1574423351_a_nat @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_Pr1987088711_a_nat] :
( ( A
= ( insert1574423351_a_nat @ B3 @ C4 ) )
& ~ ( member832397200_a_nat @ B3 @ C4 )
& ( B
= ( insert1574423351_a_nat @ A2 @ C4 ) )
& ~ ( member832397200_a_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_318_insert__eq__iff,axiom,
! [A2: b,A: set_b,B3: b,B: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ~ ( member_b @ B3 @ B )
=> ( ( ( insert_b @ A2 @ A )
= ( insert_b @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_b] :
( ( A
= ( insert_b @ B3 @ C4 ) )
& ~ ( member_b @ B3 @ C4 )
& ( B
= ( insert_b @ A2 @ C4 ) )
& ~ ( member_b @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_319_insert__eq__iff,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b,B3: product_prod_b_b,B: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ A2 @ A )
=> ( ~ ( member1285940496od_b_b @ B3 @ B )
=> ( ( ( insert1952693431od_b_b @ A2 @ A )
= ( insert1952693431od_b_b @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_Product_prod_b_b] :
( ( A
= ( insert1952693431od_b_b @ B3 @ C4 ) )
& ~ ( member1285940496od_b_b @ B3 @ C4 )
& ( B
= ( insert1952693431od_b_b @ A2 @ C4 ) )
& ~ ( member1285940496od_b_b @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_320_insert__eq__iff,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b,B3: produc1213276845od_b_b,B: set_Pr1324126435od_b_b] :
( ~ ( member1516365892od_b_b @ A2 @ A )
=> ( ~ ( member1516365892od_b_b @ B3 @ B )
=> ( ( ( insert2037698781od_b_b @ A2 @ A )
= ( insert2037698781od_b_b @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_Pr1324126435od_b_b] :
( ( A
= ( insert2037698781od_b_b @ B3 @ C4 ) )
& ~ ( member1516365892od_b_b @ B3 @ C4 )
& ( B
= ( insert2037698781od_b_b @ A2 @ C4 ) )
& ~ ( member1516365892od_b_b @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_321_insert__absorb,axiom,
! [A2: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ A2 @ A )
=> ( ( insert1574423351_a_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_322_insert__absorb,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_323_insert__absorb,axiom,
! [A2: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A2 @ A )
=> ( ( insert1952693431od_b_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_324_insert__absorb,axiom,
! [A2: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A2 @ A )
=> ( ( insert2037698781od_b_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_325_insert__ident,axiom,
! [X: produc1871334759_a_nat,A: set_Pr1987088711_a_nat,B: set_Pr1987088711_a_nat] :
( ~ ( member832397200_a_nat @ X @ A )
=> ( ~ ( member832397200_a_nat @ X @ B )
=> ( ( ( insert1574423351_a_nat @ X @ A )
= ( insert1574423351_a_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_326_insert__ident,axiom,
! [X: b,A: set_b,B: set_b] :
( ~ ( member_b @ X @ A )
=> ( ~ ( member_b @ X @ B )
=> ( ( ( insert_b @ X @ A )
= ( insert_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_327_insert__ident,axiom,
! [X: product_prod_b_b,A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ X @ A )
=> ( ~ ( member1285940496od_b_b @ X @ B )
=> ( ( ( insert1952693431od_b_b @ X @ A )
= ( insert1952693431od_b_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_328_insert__ident,axiom,
! [X: produc1213276845od_b_b,A: set_Pr1324126435od_b_b,B: set_Pr1324126435od_b_b] :
( ~ ( member1516365892od_b_b @ X @ A )
=> ( ~ ( member1516365892od_b_b @ X @ B )
=> ( ( ( insert2037698781od_b_b @ X @ A )
= ( insert2037698781od_b_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_329_Set_Oset__insert,axiom,
! [X: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ X @ A )
=> ~ ! [B8: set_Pr1987088711_a_nat] :
( ( A
= ( insert1574423351_a_nat @ X @ B8 ) )
=> ( member832397200_a_nat @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_330_Set_Oset__insert,axiom,
! [X: b,A: set_b] :
( ( member_b @ X @ A )
=> ~ ! [B8: set_b] :
( ( A
= ( insert_b @ X @ B8 ) )
=> ( member_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_331_Set_Oset__insert,axiom,
! [X: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ X @ A )
=> ~ ! [B8: set_Product_prod_b_b] :
( ( A
= ( insert1952693431od_b_b @ X @ B8 ) )
=> ( member1285940496od_b_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_332_Set_Oset__insert,axiom,
! [X: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ X @ A )
=> ~ ! [B8: set_Pr1324126435od_b_b] :
( ( A
= ( insert2037698781od_b_b @ X @ B8 ) )
=> ( member1516365892od_b_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_333_insertI2,axiom,
! [A2: produc1871334759_a_nat,B: set_Pr1987088711_a_nat,B3: produc1871334759_a_nat] :
( ( member832397200_a_nat @ A2 @ B )
=> ( member832397200_a_nat @ A2 @ ( insert1574423351_a_nat @ B3 @ B ) ) ) ).
% insertI2
thf(fact_334_insertI2,axiom,
! [A2: b,B: set_b,B3: b] :
( ( member_b @ A2 @ B )
=> ( member_b @ A2 @ ( insert_b @ B3 @ B ) ) ) ).
% insertI2
thf(fact_335_insertI2,axiom,
! [A2: product_prod_b_b,B: set_Product_prod_b_b,B3: product_prod_b_b] :
( ( member1285940496od_b_b @ A2 @ B )
=> ( member1285940496od_b_b @ A2 @ ( insert1952693431od_b_b @ B3 @ B ) ) ) ).
% insertI2
thf(fact_336_insertI2,axiom,
! [A2: produc1213276845od_b_b,B: set_Pr1324126435od_b_b,B3: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ A2 @ B )
=> ( member1516365892od_b_b @ A2 @ ( insert2037698781od_b_b @ B3 @ B ) ) ) ).
% insertI2
thf(fact_337_insertI1,axiom,
! [A2: produc1871334759_a_nat,B: set_Pr1987088711_a_nat] : ( member832397200_a_nat @ A2 @ ( insert1574423351_a_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_338_insertI1,axiom,
! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B ) ) ).
% insertI1
thf(fact_339_insertI1,axiom,
! [A2: product_prod_b_b,B: set_Product_prod_b_b] : ( member1285940496od_b_b @ A2 @ ( insert1952693431od_b_b @ A2 @ B ) ) ).
% insertI1
thf(fact_340_insertI1,axiom,
! [A2: produc1213276845od_b_b,B: set_Pr1324126435od_b_b] : ( member1516365892od_b_b @ A2 @ ( insert2037698781od_b_b @ A2 @ B ) ) ).
% insertI1
thf(fact_341_insertE,axiom,
! [A2: produc1871334759_a_nat,B3: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( member832397200_a_nat @ A2 @ ( insert1574423351_a_nat @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member832397200_a_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_342_insertE,axiom,
! [A2: b,B3: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_343_insertE,axiom,
! [A2: product_prod_b_b,B3: product_prod_b_b,A: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A2 @ ( insert1952693431od_b_b @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member1285940496od_b_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_344_insertE,axiom,
! [A2: produc1213276845od_b_b,B3: produc1213276845od_b_b,A: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A2 @ ( insert2037698781od_b_b @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member1516365892od_b_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_345_singleton__inject,axiom,
! [A2: b,B3: b] :
( ( ( insert_b @ A2 @ bot_bot_set_b )
= ( insert_b @ B3 @ bot_bot_set_b ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_346_singleton__inject,axiom,
! [A2: produc1871334759_a_nat,B3: produc1871334759_a_nat] :
( ( ( insert1574423351_a_nat @ A2 @ bot_bo1836341171_a_nat )
= ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_347_insert__not__empty,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ A )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_348_insert__not__empty,axiom,
! [A2: produc1871334759_a_nat,A: set_Pr1987088711_a_nat] :
( ( insert1574423351_a_nat @ A2 @ A )
!= bot_bo1836341171_a_nat ) ).
% insert_not_empty
thf(fact_349_doubleton__eq__iff,axiom,
! [A2: b,B3: b,C: b,D: b] :
( ( ( insert_b @ A2 @ ( insert_b @ B3 @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_350_doubleton__eq__iff,axiom,
! [A2: produc1871334759_a_nat,B3: produc1871334759_a_nat,C: produc1871334759_a_nat,D: produc1871334759_a_nat] :
( ( ( insert1574423351_a_nat @ A2 @ ( insert1574423351_a_nat @ B3 @ bot_bo1836341171_a_nat ) )
= ( insert1574423351_a_nat @ C @ ( insert1574423351_a_nat @ D @ bot_bo1836341171_a_nat ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_351_ma,axiom,
! [X2: produc1871334759_a_nat] :
( ( member832397200_a_nat @ X2 @ ( insert1574423351_a_nat @ ( standa63370785tant_a @ standard_S_Idt_a ) @ ( insert1574423351_a_nat @ ( standa1795879409tant_a @ standard_S_Idt_a ) @ ( insert1574423351_a_nat @ ( standa997693288tant_a @ standard_S_Idt_a ) @ bot_bo1836341171_a_nat ) ) ) )
=> ( mainta1604381365_nat_b @ X2 @ g ) ) ).
% ma
thf(fact_352_f,axiom,
( f
= ( ^ [X5: b] :
( if_b
@ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X5 @ bot_bot_set_b ) )
= bot_bot_set_b )
@ X5
@ ( hilbert_Eps_b @ ( p @ X5 ) ) ) ) ) ).
% f
thf(fact_353_P,axiom,
( p
= ( ^ [X5: b,Y4: b] : ( member_b @ Y4 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) ) ) ).
% P
thf(fact_354_congr_I1_J,axiom,
! [X: b,Y: b,L: standard_Constant_a] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( getRel904497637nt_a_b @ L @ g ) )
=> ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
@ ^ [V3: b] : ( image_b_b @ ( getRel904497637nt_a_b @ L @ g ) @ ( insert_b @ V3 @ bot_bot_set_b ) ) ) ) ).
% congr(1)
thf(fact_355_ident,axiom,
( ( getRel904497637nt_a_b @ standard_S_Idt_a @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ g ) @ f ) @ g ) )
= ( image_1683732397od_b_b
@ ^ [X5: b] : ( product_Pair_b_b @ X5 @ X5 )
@ ( labele1424214014nt_a_b @ ( map_gr926947118tant_a @ ( bNF_Gr_b_b @ ( labele1424214014nt_a_b @ g ) @ f ) @ g ) ) ) ) ).
% ident
% Helper facts (3)
thf(help_If_3_1_If_001tf__b_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X: b,Y: b] :
( ( if_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X: b,Y: b] :
( ( if_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_b @ x @ ( labele1424214014nt_a_b @ g ) ).
%------------------------------------------------------------------------------