SET007 Axioms: SET007+756.ax
%------------------------------------------------------------------------------
% File : SET007+756 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Underlying Principle of Dijkstra's Shortest Path Algorithm
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : graph_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 117 ( 3 unt; 0 def)
% Number of atoms : 974 ( 94 equ)
% Maximal formula atoms : 29 ( 8 avg)
% Number of connectives : 920 ( 63 ~; 11 |; 424 &)
% ( 25 <=>; 397 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-6 aty)
% Number of functors : 55 ( 55 usr; 10 con; 0-4 aty)
% Number of variables : 433 ( 419 !; 14 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_graph_5,axiom,
( ~ v1_xboole_0(k8_graph_5)
& v1_membered(k8_graph_5)
& v2_membered(k8_graph_5) ) ).
fof(fc2_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> v1_finset_1(k7_graph_5(A)) ) ).
fof(fc3_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> v1_finset_1(k6_graph_5(A,B)) ) ).
fof(fc4_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,u1_graph_1(A)) )
=> v1_finset_1(k4_graph_5(A,B,C)) ) ).
fof(fc5_graph_5,axiom,
! [A,B,C,D] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,u1_graph_1(A)) )
=> v1_finset_1(k5_graph_5(A,B,C,D)) ) ).
fof(t1_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A) )
=> r1_xreal_0(k4_card_1(k2_relat_1(A)),k4_card_1(k1_relat_1(A))) ) ).
fof(t2_graph_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ~ ( r1_tarski(k2_relat_1(B),k2_relat_1(C))
& r2_hidden(A,k1_relat_1(B))
& ! [D] :
~ ( r2_hidden(D,k1_relat_1(C))
& k1_funct_1(B,A) = k1_funct_1(C,D) ) ) ) ) ).
fof(t5_graph_5,axiom,
! [A] :
( v1_finset_1(A)
=> ( ~ ( k4_card_1(A) != np__0
& A = k1_xboole_0 )
& ~ ( A != k1_xboole_0
& k4_card_1(A) = np__0 ) ) ) ).
fof(t6_graph_5,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( k4_card_1(A) = k1_nat_1(B,np__1)
& ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( A = k2_xboole_0(D,k1_tarski(C))
& k4_card_1(D) = B ) ) ) ) ) ) ).
fof(t7_graph_5,axiom,
! [A] :
( v1_finset_1(A)
=> ~ ( k4_card_1(A) = np__1
& ! [B] :
( m1_subset_1(B,A)
=> A != k1_tarski(B) ) ) ) ).
fof(t8_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( A != k1_xboole_0
<=> r1_xreal_0(np__1,k3_finseq_1(A)) ) ) ).
fof(t9_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& ~ r1_xreal_0(C,B)
& r1_xreal_0(C,k3_finseq_1(A))
& k1_funct_1(A,B) = k1_funct_1(A,C) ) ) )
<=> v2_funct_1(A) ) ) ).
fof(t10_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& ~ r1_xreal_0(C,B)
& r1_xreal_0(C,k3_finseq_1(A))
& k1_funct_1(A,B) = k1_funct_1(A,C) ) ) )
<=> k4_card_1(k2_relat_1(A)) = k3_finseq_1(A) ) ) ).
fof(t11_graph_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ( r2_hidden(A,k4_finseq_1(C))
=> ( r2_hidden(k1_funct_1(u3_graph_1(B),k1_funct_1(C,A)),u1_graph_1(B))
& r2_hidden(k1_funct_1(u4_graph_1(B),k1_funct_1(C,A)),u1_graph_1(B)) ) ) ) ) ) ).
fof(t12_graph_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ~ ( v2_funct_1(k7_finseq_1(B,k9_finseq_1(A)))
& r1_tarski(k2_relat_1(k7_finseq_1(B,k9_finseq_1(A))),k2_relat_1(C))
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ~ ( C = k7_finseq_1(k7_finseq_1(D,k9_finseq_1(A)),E)
& r1_tarski(k2_relat_1(B),k2_relat_1(k7_finseq_1(D,E))) ) ) ) ) ) ) ).
fof(t13_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( m2_graph_1(k7_finseq_1(A,B),C)
=> ( m2_graph_1(A,C)
& m2_graph_1(B,C) ) ) ) ) ) ).
fof(t14_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( v8_graph_1(k7_finseq_1(A,B),C)
& m2_graph_1(k7_finseq_1(A,B),C) )
=> ( v8_graph_1(A,C)
& m2_graph_1(A,C)
& v8_graph_1(B,C)
& m2_graph_1(B,C) ) ) ) ) ) ).
fof(t15_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( k1_funct_1(u4_graph_1(A),k1_funct_1(B,k3_finseq_1(B))) = k1_funct_1(u3_graph_1(A),k1_funct_1(C,np__1))
=> ( v8_graph_1(k8_finseq_1(u2_graph_1(A),B,C),A)
& m2_graph_1(k8_finseq_1(u2_graph_1(A),B,C),A) ) ) ) ) ) ).
fof(t16_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v8_graph_1(k1_xboole_0,A)
& v1_graph_4(k1_xboole_0,A)
& m2_graph_1(k1_xboole_0,A) ) ) ).
fof(t17_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( v8_graph_1(k7_finseq_1(A,B),C)
& v1_graph_4(k7_finseq_1(A,B),C)
& m2_graph_1(k7_finseq_1(A,B),C) )
=> ( v8_graph_1(A,C)
& v1_graph_4(A,C)
& m2_graph_1(A,C)
& v8_graph_1(B,C)
& v1_graph_4(B,C)
& m2_graph_1(B,C) ) ) ) ) ) ).
fof(t18_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ( k3_finseq_1(B) = np__1
=> ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) ) ) ) ) ).
fof(t19_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(A))
=> ( ( r1_xreal_0(np__1,k3_finseq_1(B))
& k3_finseq_1(C) = np__1
& k1_funct_1(u3_graph_1(A),k1_funct_1(C,np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,k3_finseq_1(B)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B))
& k1_funct_1(u4_graph_1(A),k1_funct_1(B,D)) = k1_funct_1(u4_graph_1(A),k1_funct_1(C,np__1)) ) ) )
=> ( k1_funct_1(u3_graph_1(A),k1_funct_1(B,np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,k3_finseq_1(B)))
| ( v8_graph_1(k8_finseq_1(u2_graph_1(A),B,C),A)
& v1_graph_4(k8_finseq_1(u2_graph_1(A),B,C),A)
& m2_graph_1(k8_finseq_1(u2_graph_1(A),B,C),A) ) ) ) ) ) ) ).
fof(t20_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) )
=> v2_funct_1(B) ) ) ).
fof(d1_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_graph_1(A))
=> k1_graph_5(A,B) = k2_tarski(k1_funct_1(u3_graph_1(A),B),k1_funct_1(u4_graph_1(A),B)) ) ) ).
fof(t21_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& v1_graph_4(D,A)
& m2_graph_1(D,A) )
=> ( ( D = k8_finseq_1(u2_graph_1(A),B,C)
& r1_xreal_0(np__1,k3_finseq_1(B))
& r1_xreal_0(np__1,k3_finseq_1(C)) )
=> ( k1_funct_1(u3_graph_1(A),k1_funct_1(D,np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(D,k3_finseq_1(D)))
| ( ~ r2_hidden(k1_funct_1(u3_graph_1(A),k1_funct_1(D,np__1)),k2_graph_5(A,C))
& ~ r2_hidden(k1_funct_1(u4_graph_1(A),k1_funct_1(D,k3_finseq_1(D))),k2_graph_5(A,B)) ) ) ) ) ) ) ) ).
fof(t22_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ( r1_tarski(k2_graph_5(B,C),A)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> r1_tarski(k1_graph_5(B,k4_finseq_4(k5_numbers,u2_graph_1(B),C,D)),A) ) ) ) ) ) ).
fof(t23_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ~ ( ~ r1_tarski(k2_graph_5(B,C),A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_1(E,u2_graph_1(B))
=> ! [F] :
( m2_finseq_1(F,u2_graph_1(B))
=> ~ ( r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(C))
& ~ r1_tarski(k1_graph_5(B,k4_finseq_4(k5_numbers,u2_graph_1(B),C,k1_nat_1(D,np__1))),A)
& k3_finseq_1(E) = D
& C = k8_finseq_1(u2_graph_1(B),E,F)
& r1_tarski(k2_graph_5(B,E),A) ) ) ) ) ) ) ) ).
fof(t24_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(A))
=> ( r1_tarski(k2_relat_1(B),k2_relat_1(C))
=> r1_tarski(k2_graph_5(A,B),k2_graph_5(A,C)) ) ) ) ) ).
fof(t25_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m2_finseq_1(D,u2_graph_1(C))
=> ! [E] :
( m2_finseq_1(E,u2_graph_1(C))
=> ( ( r1_tarski(k2_relat_1(D),k2_relat_1(E))
& r1_tarski(k4_xboole_0(k2_graph_5(C,E),A),B) )
=> r1_tarski(k4_xboole_0(k2_graph_5(C,D),A),B) ) ) ) ) ).
fof(t26_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m2_finseq_1(D,u2_graph_1(C))
=> ! [E] :
( m2_finseq_1(E,u2_graph_1(C))
=> ( r1_tarski(k4_xboole_0(k2_graph_5(C,k8_finseq_1(u2_graph_1(C),D,E)),A),B)
=> ( r1_tarski(k4_xboole_0(k2_graph_5(C,D),A),B)
& r1_tarski(k4_xboole_0(k2_graph_5(C,E),A),B) ) ) ) ) ) ).
fof(t27_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u2_graph_1(A))
=> ( ( B = k1_funct_1(u3_graph_1(A),C)
| B = k1_funct_1(u4_graph_1(A),C) )
=> r2_hidden(B,k1_graph_5(A,C)) ) ) ) ) ).
fof(t28_graph_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ( r2_hidden(A,k4_finseq_1(C))
=> ( ( D != k1_funct_1(u3_graph_1(B),k1_funct_1(C,A))
& D != k1_funct_1(u4_graph_1(B),k1_funct_1(C,A)) )
| r2_hidden(D,k2_graph_5(B,C)) ) ) ) ) ) ) ).
fof(t29_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ( k3_finseq_1(B) = np__1
=> k2_graph_5(A,B) = k1_graph_5(A,k4_finseq_4(k5_numbers,u2_graph_1(A),B,np__1)) ) ) ) ).
fof(t30_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(A))
=> ( r1_tarski(k2_graph_5(A,B),k2_graph_5(A,k8_finseq_1(u2_graph_1(A),B,C)))
& r1_tarski(k2_graph_5(A,C),k2_graph_5(A,k8_finseq_1(u2_graph_1(A),B,C))) ) ) ) ) ).
fof(t31_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ( ( C = k8_finseq_1(u2_graph_1(A),D,B)
& r1_xreal_0(np__1,k3_finseq_1(D))
& k3_finseq_1(B) = np__1 )
=> k2_graph_5(A,C) = k2_xboole_0(k2_graph_5(A,D),k1_tarski(k1_funct_1(u4_graph_1(A),k1_funct_1(B,np__1)))) ) ) ) ) ) ).
fof(t32_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ~ ( B != k1_funct_1(u3_graph_1(A),k1_funct_1(C,np__1))
& r2_hidden(B,k2_graph_5(A,C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(C))
& B = k1_funct_1(u4_graph_1(A),k1_funct_1(C,D)) ) ) ) ) ) ) ).
fof(d3_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ( r1_graph_5(A,B,C,D)
<=> ( B != k1_xboole_0
& k1_funct_1(u3_graph_1(A),k1_funct_1(B,np__1)) = C
& k1_funct_1(u4_graph_1(A),k1_funct_1(B,k3_finseq_1(B))) = D ) ) ) ) ) ) ).
fof(d4_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ! [E] :
( r2_graph_5(A,B,C,D,E)
<=> ( r1_graph_5(A,D,B,C)
& r1_tarski(k4_xboole_0(k2_graph_5(A,D),k1_tarski(C)),E) ) ) ) ) ) ) ).
fof(t33_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ( r1_graph_5(A,D,B,C)
=> ( r2_hidden(B,k2_graph_5(A,D))
& r2_hidden(C,k2_graph_5(A,D)) ) ) ) ) ) ) ).
fof(t34_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ( r2_hidden(A,k3_graph_5(B,C,D))
<=> ? [E] :
( v8_graph_1(E,B)
& m2_graph_1(E,B)
& E = A
& r1_graph_5(B,E,C,D) ) ) ) ) ) ).
fof(t35_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( ( v8_graph_1(E,B)
& m2_graph_1(E,B) )
=> ( r2_graph_5(B,C,D,E,A)
=> ( C = D
| r2_hidden(C,A) ) ) ) ) ) ) ).
fof(t36_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_graph_1(C))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(C))
=> ! [F] :
( ( v8_graph_1(F,C)
& m2_graph_1(F,C) )
=> ( ( r2_graph_5(C,D,E,F,A)
& r1_tarski(A,B) )
=> r2_graph_5(C,D,E,F,B) ) ) ) ) ) ).
fof(t37_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(A))
=> ! [F] :
( ( v8_graph_1(F,A)
& m2_graph_1(F,A) )
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(F))
& r1_graph_5(A,F,C,D)
& r1_graph_4(A,D,E,k1_funct_1(B,np__1))
& k3_finseq_1(B) = np__1
& ! [G] :
( ( v8_graph_1(G,A)
& m2_graph_1(G,A) )
=> ~ ( G = k8_finseq_1(u2_graph_1(A),F,B)
& r1_graph_5(A,G,C,E) ) ) ) ) ) ) ) ) ) ).
fof(t38_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(B))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(B))
=> ! [G] :
( ( v8_graph_1(G,B)
& m2_graph_1(G,B) )
=> ! [H] :
( ( v8_graph_1(H,B)
& m2_graph_1(H,B) )
=> ( ( G = k8_finseq_1(u2_graph_1(B),H,C)
& r1_xreal_0(np__1,k3_finseq_1(H))
& k3_finseq_1(C) = np__1
& r2_graph_5(B,D,E,H,A)
& r1_graph_4(B,E,F,k1_funct_1(C,np__1)) )
=> r2_graph_5(B,D,F,G,k2_xboole_0(A,k1_tarski(E))) ) ) ) ) ) ) ) ) ).
fof(d6_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ( r3_graph_5(A,B,C,D)
<=> ( v1_graph_4(B,A)
& r1_graph_5(A,B,C,D) ) ) ) ) ) ) ).
fof(d7_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ! [E] :
( r4_graph_5(A,B,C,D,E)
<=> ( v1_graph_4(B,A)
& r2_graph_5(A,C,D,B,E) ) ) ) ) ) ) ).
fof(t39_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ~ ( D = k1_xboole_0
& r3_graph_5(A,D,B,C) ) ) ) ) ) ).
fof(t40_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ( r3_graph_5(A,D,B,C)
=> r1_graph_5(A,D,B,C) ) ) ) ) ) ).
fof(t41_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> r1_tarski(k4_graph_5(A,B,C),k3_graph_5(A,B,C)) ) ) ) ).
fof(t42_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> r1_tarski(k6_graph_5(A,B),k7_graph_5(A)) ) ) ).
fof(t43_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> r1_tarski(k4_graph_5(A,B,C),k7_graph_5(A)) ) ) ) ).
fof(t44_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ( r1_graph_5(A,D,B,C)
=> r1_tarski(k6_graph_5(A,D),k4_graph_5(A,B,C)) ) ) ) ) ) ).
fof(t45_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( ( v8_graph_1(E,B)
& m2_graph_1(E,B) )
=> ( r2_graph_5(B,C,D,E,A)
=> r1_tarski(k6_graph_5(B,E),k5_graph_5(B,C,D,A)) ) ) ) ) ) ).
fof(t46_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( r2_hidden(B,k6_graph_5(A,C))
=> r1_xreal_0(k3_finseq_1(B),k3_finseq_1(C)) ) ) ) ) ).
fof(t47_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ( r1_graph_5(A,D,B,C)
=> ( k6_graph_5(A,D) != k1_xboole_0
& k4_graph_5(A,B,C) != k1_xboole_0 ) ) ) ) ) ) ).
fof(t48_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( ( v8_graph_1(E,B)
& m2_graph_1(E,B) )
=> ( r2_graph_5(B,C,D,E,A)
=> ( k6_graph_5(B,E) != k1_xboole_0
& k5_graph_5(B,C,D,A) != k1_xboole_0 ) ) ) ) ) ) ).
fof(t49_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> r1_tarski(k5_graph_5(B,C,D,A),k7_graph_5(B)) ) ) ) ).
fof(d13_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r5_graph_5(A,B)
<=> ( v1_funct_1(B)
& v1_funct_2(B,u2_graph_1(A),k8_graph_5)
& m2_relset_1(B,u2_graph_1(A),k8_graph_5) ) ) ) ) ).
fof(d14_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r6_graph_5(A,B)
<=> ( v1_funct_1(B)
& v1_funct_2(B,u2_graph_1(A),k1_numbers)
& m2_relset_1(B,u2_graph_1(A),k1_numbers) ) ) ) ) ).
fof(d15_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r6_graph_5(A,C)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( D = k9_graph_5(A,B,C)
<=> ( k4_finseq_1(B) = k4_finseq_1(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(B))
=> k1_goboard1(D,E) = k1_funct_1(C,k1_funct_1(B,E)) ) ) ) ) ) ) ) ) ) ).
fof(d16_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k10_graph_5(A,B,C) = k15_rvsum_1(k9_graph_5(A,B,C)) ) ) ) ).
fof(t50_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r5_graph_5(B,A)
=> r6_graph_5(B,A) ) ) ) ).
fof(t51_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( r5_graph_5(B,A)
& D = k9_graph_5(B,C,A) )
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> r1_xreal_0(np__0,k1_goboard1(D,E)) ) ) ) ) ) ) ) ).
fof(t52_graph_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m2_finseq_1(D,u2_graph_1(C))
=> ! [E] :
( m2_finseq_1(E,u2_graph_1(C))
=> ~ ( r1_tarski(k2_relat_1(D),k2_relat_1(E))
& r6_graph_5(C,B)
& r2_hidden(A,k4_finseq_1(D))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k4_finseq_1(E))
& k1_goboard1(k9_graph_5(C,E,B),F) = k1_goboard1(k9_graph_5(C,D,B),A) ) ) ) ) ) ) ) ) ).
fof(t53_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( m2_finseq_1(D,u2_graph_1(B))
=> ( ( k3_finseq_1(C) = np__1
& r1_tarski(k2_relat_1(C),k2_relat_1(D))
& r5_graph_5(B,A) )
=> r1_xreal_0(k10_graph_5(B,C,A),k10_graph_5(B,D,A)) ) ) ) ) ) ).
fof(t54_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ( r5_graph_5(B,A)
=> r1_xreal_0(np__0,k10_graph_5(B,C,A)) ) ) ) ) ).
fof(t55_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ( ( C = k1_xboole_0
& r6_graph_5(B,A) )
=> k10_graph_5(B,C,A) = np__0 ) ) ) ) ).
fof(t56_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(u2_graph_1(B)))) )
=> ~ ( E = k4_graph_5(B,C,D)
& ! [F] :
( m2_finseq_1(F,u2_graph_1(B))
=> ~ ( r2_hidden(F,E)
& ! [G] :
( m2_finseq_1(G,u2_graph_1(B))
=> ( r2_hidden(G,E)
=> r1_xreal_0(k10_graph_5(B,F,A),k10_graph_5(B,G,A)) ) ) ) ) ) ) ) ) ) ) ).
fof(t57_graph_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_graph_1(C))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(C))
=> ! [F] :
( ( ~ v1_xboole_0(F)
& v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k3_finseq_2(u2_graph_1(C)))) )
=> ~ ( F = k5_graph_5(C,D,E,A)
& ! [G] :
( m2_finseq_1(G,u2_graph_1(C))
=> ~ ( r2_hidden(G,F)
& ! [H] :
( m2_finseq_1(H,u2_graph_1(C))
=> ( r2_hidden(H,F)
=> r1_xreal_0(k10_graph_5(C,G,B),k10_graph_5(C,H,B)) ) ) ) ) ) ) ) ) ) ) ).
fof(t58_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( m2_finseq_1(D,u2_graph_1(B))
=> ( r6_graph_5(B,A)
=> k10_graph_5(B,k8_finseq_1(u2_graph_1(B),C,D),A) = k3_real_1(k10_graph_5(B,C,A),k10_graph_5(B,D,A)) ) ) ) ) ) ).
fof(t59_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( m2_finseq_1(D,u2_graph_1(B))
=> ( ( v2_funct_1(C)
& r1_tarski(k2_relat_1(C),k2_relat_1(D))
& r5_graph_5(B,A) )
=> r1_xreal_0(k10_graph_5(B,C,A),k10_graph_5(B,D,A)) ) ) ) ) ) ).
fof(t60_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u2_graph_1(B))
=> ! [D] :
( ( v8_graph_1(D,B)
& m2_graph_1(D,B) )
=> ( ( r2_hidden(C,k6_graph_5(B,D))
& r5_graph_5(B,A) )
=> r1_xreal_0(k10_graph_5(B,C,A),k10_graph_5(B,D,A)) ) ) ) ) ) ).
fof(d17_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( r7_graph_5(A,B,C,D,E)
<=> ( r1_graph_5(A,D,B,C)
& ! [F] :
( ( v8_graph_1(F,A)
& m2_graph_1(F,A) )
=> ( r1_graph_5(A,F,B,C)
=> r1_xreal_0(k10_graph_5(A,D,E),k10_graph_5(A,F,E)) ) ) ) ) ) ) ) ) ) ).
fof(d18_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ! [E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( r8_graph_5(A,B,C,D,E,F)
<=> ( r2_graph_5(A,B,C,D,E)
& ! [G] :
( ( v8_graph_1(G,A)
& m2_graph_1(G,A) )
=> ( r2_graph_5(A,B,C,G,E)
=> r1_xreal_0(k10_graph_5(A,D,F),k10_graph_5(A,G,F)) ) ) ) ) ) ) ) ) ) ).
fof(t61_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) )
=> r1_xreal_0(k3_finseq_1(B),k2_graph_1(A)) ) ) ).
fof(t62_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) )
=> r1_xreal_0(k3_finseq_1(B),k3_graph_1(A)) ) ) ).
fof(t63_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ~ ( k4_graph_5(B,C,D) != k1_xboole_0
& ! [E] :
( m2_finseq_1(E,u2_graph_1(B))
=> ~ ( r2_hidden(E,k4_graph_5(B,C,D))
& ! [F] :
( m2_finseq_1(F,u2_graph_1(B))
=> ( r2_hidden(F,k4_graph_5(B,C,D))
=> r1_xreal_0(k10_graph_5(B,E,A),k10_graph_5(B,F,A)) ) ) ) ) ) ) ) ) ) ).
fof(t64_graph_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& v7_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_graph_1(C))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(C))
=> ~ ( k5_graph_5(C,D,E,A) != k1_xboole_0
& ! [F] :
( m2_finseq_1(F,u2_graph_1(C))
=> ~ ( r2_hidden(F,k5_graph_5(C,D,E,A))
& ! [G] :
( m2_finseq_1(G,u2_graph_1(C))
=> ( r2_hidden(G,k5_graph_5(C,D,E,A))
=> r1_xreal_0(k10_graph_5(C,F,B),k10_graph_5(C,G,B)) ) ) ) ) ) ) ) ) ) ).
fof(t65_graph_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v8_graph_1(C,B)
& m2_graph_1(C,B) )
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(B))
=> ~ ( r1_graph_5(B,C,D,E)
& r5_graph_5(B,A)
& ! [F] :
( ( v8_graph_1(F,B)
& v1_graph_4(F,B)
& m2_graph_1(F,B) )
=> ~ r7_graph_5(B,D,E,F,A) ) ) ) ) ) ) ) ).
fof(t66_graph_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& v7_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( ( v8_graph_1(D,C)
& m2_graph_1(D,C) )
=> ! [E] :
( m1_subset_1(E,u1_graph_1(C))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(C))
=> ~ ( r2_graph_5(C,E,F,D,A)
& r5_graph_5(C,B)
& ! [G] :
( ( v8_graph_1(G,C)
& v1_graph_4(G,C)
& m2_graph_1(G,C) )
=> ~ r8_graph_5(C,E,F,G,A,B) ) ) ) ) ) ) ) ).
fof(t67_graph_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& v7_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( ( v8_graph_1(D,C)
& m2_graph_1(D,C) )
=> ! [E] :
( m1_subset_1(E,u1_graph_1(C))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(C))
=> ( ( r5_graph_5(C,B)
& r8_graph_5(C,E,F,D,A,B)
& ! [G] :
( ( v8_graph_1(G,C)
& m2_graph_1(G,C) )
=> ! [H] :
( m1_subset_1(H,u1_graph_1(C))
=> ( r8_graph_5(C,E,H,G,A,B)
=> ( r2_hidden(H,A)
| r1_xreal_0(k10_graph_5(C,D,B),k10_graph_5(C,G,B)) ) ) ) ) )
=> ( E = F
| r7_graph_5(C,E,F,D,B) ) ) ) ) ) ) ) ).
fof(t68_graph_5,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v2_graph_1(D)
& v7_graph_1(D)
& l1_graph_1(D) )
=> ! [E] :
( ( v8_graph_1(E,D)
& m2_graph_1(E,D) )
=> ! [F] :
( m1_subset_1(F,u1_graph_1(D))
=> ! [G] :
( m1_subset_1(G,u1_graph_1(D))
=> ( ( r5_graph_5(D,C)
& r8_graph_5(D,F,G,E,A,C)
& r1_tarski(A,B)
& ! [H] :
( ( v8_graph_1(H,D)
& m2_graph_1(H,D) )
=> ! [I] :
( m1_subset_1(I,u1_graph_1(D))
=> ( r8_graph_5(D,F,I,H,A,C)
=> ( r2_hidden(I,A)
| r1_xreal_0(k10_graph_5(D,E,C),k10_graph_5(D,H,C)) ) ) ) ) )
=> ( F = G
| r8_graph_5(D,F,G,E,B,C) ) ) ) ) ) ) ) ).
fof(d19_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> ! [C,D] :
( m1_subset_1(D,u1_graph_1(A))
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( r9_graph_5(A,B,C,D,E)
<=> ! [F] :
( m1_subset_1(F,u1_graph_1(A))
=> ~ ( r2_hidden(F,C)
& F != D
& ! [G] :
( ( v8_graph_1(G,A)
& m2_graph_1(G,A) )
=> ~ ( r8_graph_5(A,D,F,G,C,E)
& r1_xreal_0(k10_graph_5(A,G,E),k10_graph_5(A,B,E)) ) ) ) ) ) ) ) ) ) ).
fof(t69_graph_5,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v2_graph_1(D)
& v3_graph_1(D)
& v7_graph_1(D)
& l1_graph_1(D) )
=> ! [E] :
( ( v8_graph_1(E,D)
& m2_graph_1(E,D) )
=> ! [F] :
( ( v8_graph_1(F,D)
& m2_graph_1(F,D) )
=> ! [G] :
( ( v8_graph_1(G,D)
& m2_graph_1(G,D) )
=> ! [H] :
( m1_subset_1(H,u1_graph_1(D))
=> ! [I] :
( m1_subset_1(I,u1_graph_1(D))
=> ! [J] :
( m1_subset_1(J,u1_graph_1(D))
=> ( ( r2_hidden(A,u2_graph_1(D))
& r5_graph_5(D,C)
& r1_xreal_0(np__1,k3_finseq_1(E))
& r8_graph_5(D,H,I,E,B,C)
& G = k7_finseq_1(E,k9_finseq_1(A))
& r8_graph_5(D,H,J,F,B,C)
& r1_graph_4(D,I,J,A)
& r9_graph_5(D,E,B,H,C) )
=> ( H = I
| H = J
| ( ( r1_xreal_0(k10_graph_5(D,F,C),k10_graph_5(D,G,C))
=> r8_graph_5(D,H,J,F,k2_xboole_0(B,k1_tarski(I)),C) )
& ( r1_xreal_0(k10_graph_5(D,G,C),k10_graph_5(D,F,C))
=> r8_graph_5(D,H,J,G,k2_xboole_0(B,k1_tarski(I)),C) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(s1_graph_5,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_graph_5
& ! [B] :
( m1_subset_1(B,f2_s1_graph_5)
=> ( r2_hidden(B,f1_s1_graph_5)
=> k1_funct_1(A,B) = f3_s1_graph_5(B) ) ) ) ).
fof(s2_graph_5,axiom,
? [A] :
( m1_subset_1(A,f1_s2_graph_5)
& ! [B] :
( m1_subset_1(B,f1_s2_graph_5)
=> r1_xreal_0(f2_s2_graph_5(A),f2_s2_graph_5(B)) ) ) ).
fof(dt_m1_graph_5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_finseq_2(A))) )
=> ! [C] :
( m1_graph_5(C,A,B)
=> m2_finseq_1(C,A) ) ) ).
fof(existence_m1_graph_5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_finseq_2(A))) )
=> ? [C] : m1_graph_5(C,A,B) ) ).
fof(redefinition_m1_graph_5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_finseq_2(A))) )
=> ! [C] :
( m1_graph_5(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_k1_graph_5,axiom,
$true ).
fof(dt_k2_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_finseq_1(B,u2_graph_1(A)) )
=> m1_subset_1(k2_graph_5(A,B),k1_zfmisc_1(u1_graph_1(A))) ) ).
fof(dt_k3_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,u1_graph_1(A)) )
=> m1_subset_1(k3_graph_5(A,B,C),k1_zfmisc_1(k3_finseq_2(u2_graph_1(A)))) ) ).
fof(dt_k4_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,u1_graph_1(A)) )
=> m1_subset_1(k4_graph_5(A,B,C),k1_zfmisc_1(k3_finseq_2(u2_graph_1(A)))) ) ).
fof(dt_k5_graph_5,axiom,
! [A,B,C,D] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,u1_graph_1(A)) )
=> m1_subset_1(k5_graph_5(A,B,C,D),k1_zfmisc_1(k3_finseq_2(u2_graph_1(A)))) ) ).
fof(dt_k6_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> m1_subset_1(k6_graph_5(A,B),k1_zfmisc_1(k3_finseq_2(u2_graph_1(A)))) ) ).
fof(dt_k7_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> m1_subset_1(k7_graph_5(A),k1_zfmisc_1(k3_finseq_2(u2_graph_1(A)))) ) ).
fof(dt_k8_graph_5,axiom,
m1_subset_1(k8_graph_5,k1_zfmisc_1(k1_numbers)) ).
fof(dt_k9_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_finseq_1(B,u2_graph_1(A))
& v1_relat_1(C)
& v1_funct_1(C) )
=> m2_finseq_1(k9_graph_5(A,B,C),k1_numbers) ) ).
fof(dt_k10_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_finseq_1(B,u2_graph_1(A))
& v1_relat_1(C)
& v1_funct_1(C) )
=> m1_subset_1(k10_graph_5(A,B,C),k1_numbers) ) ).
fof(t3_graph_5,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( C = a_2_0_graph_5(A,B)
=> v1_finset_1(C) ) ) ) ).
fof(t4_graph_5,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( C = a_2_1_graph_5(A,B)
=> v1_finset_1(C) ) ) ) ).
fof(d2_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u2_graph_1(A))
=> k2_graph_5(A,B) = a_2_2_graph_5(A,B) ) ) ).
fof(d5_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> k3_graph_5(A,B,C) = a_3_0_graph_5(A,B,C) ) ) ) ).
fof(d8_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> k4_graph_5(A,B,C) = a_3_1_graph_5(A,B,C) ) ) ) ).
fof(d9_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] : k5_graph_5(A,B,C,D) = a_4_0_graph_5(A,B,C,D) ) ) ) ).
fof(d10_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> k6_graph_5(A,B) = a_2_3_graph_5(A,B) ) ) ).
fof(d11_graph_5,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> k7_graph_5(A) = a_1_0_graph_5(A) ) ).
fof(d12_graph_5,axiom,
k8_graph_5 = a_0_0_graph_5 ).
fof(fraenkel_a_2_0_graph_5,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_0_graph_5(B,C))
<=> ? [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
& A = D
& r1_xreal_0(np__1,k3_finseq_1(D))
& r1_xreal_0(k3_finseq_1(D),C) ) ) ) ).
fof(fraenkel_a_2_1_graph_5,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_1_graph_5(B,C))
<=> ? [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
& A = D
& r1_xreal_0(k3_finseq_1(D),C) ) ) ) ).
fof(fraenkel_a_2_2_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(B)
& l1_graph_1(B)
& m2_finseq_1(C,u2_graph_1(B)) )
=> ( r2_hidden(A,a_2_2_graph_5(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_graph_1(B))
& A = D
& ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r2_hidden(E,k4_finseq_1(C))
& r2_hidden(D,k1_graph_5(B,k4_finseq_4(k5_numbers,u2_graph_1(B),C,E))) ) ) ) ) ).
fof(fraenkel_a_3_0_graph_5,axiom,
! [A,B,C,D] :
( ( v2_graph_1(B)
& l1_graph_1(B)
& m1_subset_1(C,u1_graph_1(B))
& m1_subset_1(D,u1_graph_1(B)) )
=> ( r2_hidden(A,a_3_0_graph_5(B,C,D))
<=> ? [E] :
( v8_graph_1(E,B)
& m2_graph_1(E,B)
& A = E
& r1_graph_5(B,E,C,D) ) ) ) ).
fof(fraenkel_a_3_1_graph_5,axiom,
! [A,B,C,D] :
( ( v2_graph_1(B)
& l1_graph_1(B)
& m1_subset_1(C,u1_graph_1(B))
& m1_subset_1(D,u1_graph_1(B)) )
=> ( r2_hidden(A,a_3_1_graph_5(B,C,D))
<=> ? [E] :
( v8_graph_1(E,B)
& v1_graph_4(E,B)
& m2_graph_1(E,B)
& A = E
& r3_graph_5(B,E,C,D) ) ) ) ).
fof(fraenkel_a_4_0_graph_5,axiom,
! [A,B,C,D,E] :
( ( v2_graph_1(B)
& l1_graph_1(B)
& m1_subset_1(C,u1_graph_1(B))
& m1_subset_1(D,u1_graph_1(B)) )
=> ( r2_hidden(A,a_4_0_graph_5(B,C,D,E))
<=> ? [F] :
( v8_graph_1(F,B)
& v1_graph_4(F,B)
& m2_graph_1(F,B)
& A = F
& r4_graph_5(B,F,C,D,E) ) ) ) ).
fof(fraenkel_a_2_3_graph_5,axiom,
! [A,B,C] :
( ( v2_graph_1(B)
& l1_graph_1(B)
& v8_graph_1(C,B)
& m2_graph_1(C,B) )
=> ( r2_hidden(A,a_2_3_graph_5(B,C))
<=> ? [D] :
( v8_graph_1(D,B)
& v1_graph_4(D,B)
& m2_graph_1(D,B)
& A = D
& D != k1_xboole_0
& k1_funct_1(u3_graph_1(B),k1_funct_1(D,np__1)) = k1_funct_1(u3_graph_1(B),k1_funct_1(C,np__1))
& k1_funct_1(u4_graph_1(B),k1_funct_1(D,k3_finseq_1(D))) = k1_funct_1(u4_graph_1(B),k1_funct_1(C,k3_finseq_1(C)))
& r1_tarski(k2_relat_1(D),k2_relat_1(C)) ) ) ) ).
fof(fraenkel_a_1_0_graph_5,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r2_hidden(A,a_1_0_graph_5(B))
<=> ? [C] :
( v8_graph_1(C,B)
& v1_graph_4(C,B)
& m2_graph_1(C,B)
& A = C ) ) ) ).
fof(fraenkel_a_0_0_graph_5,axiom,
! [A] :
( r2_hidden(A,a_0_0_graph_5)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& A = B
& r1_xreal_0(np__0,B) ) ) ).
%------------------------------------------------------------------------------