SET007 Axioms: SET007+519.ax
%------------------------------------------------------------------------------
% File : SET007+519 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Euler Circuits and Paths
% 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 113 ( 5 unt; 0 def)
% Number of atoms : 855 ( 135 equ)
% Maximal formula atoms : 26 ( 7 avg)
% Number of connectives : 814 ( 72 ~; 23 |; 396 &)
% ( 23 <=>; 300 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 38 ( 36 usr; 1 prp; 0-4 aty)
% Number of functors : 57 ( 57 usr; 5 con; 0-4 aty)
% Number of variables : 384 ( 366 !; 18 ?)
% 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_3,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B)
& v1_abian(B) )
=> ( v1_xreal_0(k6_xcmplx_0(A,B))
& v1_int_1(k6_xcmplx_0(A,B))
& v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_abian(k6_xcmplx_0(A,B)) ) ) ).
fof(rc1_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v1_msscyc_1(B,A) ) ) ).
fof(fc2_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> v1_finset_1(k2_graph_3(A,B,C)) ) ).
fof(fc3_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> v1_finset_1(k3_graph_3(A,B,C)) ) ).
fof(fc4_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> v1_finset_1(k4_graph_3(A,B,C)) ) ).
fof(fc5_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& v1_xboole_0(C) )
=> ( v1_xboole_0(k2_graph_3(A,B,C))
& v1_finset_1(k2_graph_3(A,B,C))
& v1_membered(k2_graph_3(A,B,C))
& v2_membered(k2_graph_3(A,B,C))
& v3_membered(k2_graph_3(A,B,C))
& v4_membered(k2_graph_3(A,B,C))
& v5_membered(k2_graph_3(A,B,C)) ) ) ).
fof(fc6_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& v1_xboole_0(C) )
=> ( v1_xboole_0(k3_graph_3(A,B,C))
& v1_finset_1(k3_graph_3(A,B,C))
& v1_membered(k3_graph_3(A,B,C))
& v2_membered(k3_graph_3(A,B,C))
& v3_membered(k3_graph_3(A,B,C))
& v4_membered(k3_graph_3(A,B,C))
& v5_membered(k3_graph_3(A,B,C)) ) ) ).
fof(fc7_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& v1_xboole_0(C) )
=> ( v1_xboole_0(k4_graph_3(A,B,C))
& v1_finset_1(k4_graph_3(A,B,C))
& v1_membered(k4_graph_3(A,B,C))
& v2_membered(k4_graph_3(A,B,C))
& v3_membered(k4_graph_3(A,B,C))
& v4_membered(k4_graph_3(A,B,C))
& v5_membered(k4_graph_3(A,B,C)) ) ) ).
fof(fc8_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> v1_finset_1(k5_graph_3(A,B)) ) ).
fof(fc9_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> v1_finset_1(k6_graph_3(A,B)) ) ).
fof(fc10_graph_3,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_graph_1(k8_graph_3(A,B,C))
& v2_graph_1(k8_graph_3(A,B,C))
& v7_graph_1(k8_graph_3(A,B,C)) ) ) ).
fof(fc11_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,u1_graph_1(A)) )
=> ( v1_graph_1(k8_graph_3(A,B,C))
& v2_graph_1(k8_graph_3(A,B,C))
& v6_graph_1(k8_graph_3(A,B,C)) ) ) ).
fof(t1_graph_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( v1_abian(A)
<=> v1_abian(B) )
<=> v1_abian(k6_xcmplx_0(A,B)) ) ) ) ).
fof(t2_graph_3,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(k1_graph_2(A,B,C)))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(E,k4_finseq_1(A))
& k1_funct_1(A,E) = k1_funct_1(k1_graph_2(A,B,C),D)
& k1_nat_1(E,np__1) = k1_nat_1(B,D)
& r1_xreal_0(B,E)
& r1_xreal_0(E,C) ) ) ) ) ) ) ) ).
fof(t3_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ~ ( r1_graph_2(A,C,B)
& v1_xboole_0(C) ) ) ) ) ).
fof(t4_graph_3,axiom,
$true ).
fof(t5_graph_3,axiom,
$true ).
fof(t6_graph_3,axiom,
$true ).
fof(t7_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( r2_hidden(B,u2_graph_1(A))
=> ( v2_funct_1(k9_finseq_1(B))
& m2_graph_1(k9_finseq_1(B),A) ) ) ) ).
fof(t8_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( v2_funct_1(k2_graph_2(u2_graph_1(A),B,C,D))
& m2_graph_1(k2_graph_2(u2_graph_1(A),B,C,D),A) ) ) ) ) ) ).
fof(t9_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ! [C] :
( ( v2_funct_1(C)
& m2_graph_1(C,A) )
=> ! [D] :
( m2_finseq_1(D,u1_graph_1(A))
=> ! [E] :
( m2_finseq_1(E,u1_graph_1(A))
=> ( ( r1_xboole_0(k2_relat_1(B),k2_relat_1(C))
& r1_graph_2(A,D,B)
& r1_graph_2(A,E,C)
& k1_funct_1(D,k3_finseq_1(D)) = k1_funct_1(E,np__1) )
=> ( v2_funct_1(k8_finseq_1(u2_graph_1(A),B,C))
& m2_graph_1(k8_finseq_1(u2_graph_1(A),B,C),A) ) ) ) ) ) ) ) ).
fof(t10_graph_3,axiom,
$true ).
fof(t11_graph_3,axiom,
$true ).
fof(t12_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ( B = k1_xboole_0
=> v1_msscyc_1(B,A) ) ) ) ).
fof(t13_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_funct_1(C)
& v1_msscyc_1(C,A)
& m2_graph_1(C,A) )
=> ( v2_funct_1(k8_finseq_1(u2_graph_1(A),k2_graph_2(u2_graph_1(A),C,k1_nat_1(B,np__1),k3_finseq_1(C)),k2_graph_2(u2_graph_1(A),C,np__1,B)))
& v1_msscyc_1(k8_finseq_1(u2_graph_1(A),k2_graph_2(u2_graph_1(A),C,k1_nat_1(B,np__1),k3_finseq_1(C)),k2_graph_2(u2_graph_1(A),C,np__1,B)),A)
& m2_graph_1(k8_finseq_1(u2_graph_1(A),k2_graph_2(u2_graph_1(A),C,k1_nat_1(B,np__1),k3_finseq_1(C)),k2_graph_2(u2_graph_1(A),C,np__1,B)),A) ) ) ) ) ).
fof(t14_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(B))
=> ( k3_finseq_1(k8_finseq_1(u2_graph_1(A),k2_graph_2(u2_graph_1(A),B,k1_nat_1(C,np__1),k3_finseq_1(B)),k2_graph_2(u2_graph_1(A),B,np__1,C))) = k3_finseq_1(B)
& k2_relat_1(k8_finseq_1(u2_graph_1(A),k2_graph_2(u2_graph_1(A),B,k1_nat_1(C,np__1),k3_finseq_1(B)),k2_graph_2(u2_graph_1(A),B,np__1,C))) = k2_relat_1(B)
& k1_funct_1(k8_finseq_1(u2_graph_1(A),k2_graph_2(u2_graph_1(A),B,k1_nat_1(C,np__1),k3_finseq_1(B)),k2_graph_2(u2_graph_1(A),B,np__1,C)),np__1) = k1_funct_1(B,k1_nat_1(C,np__1)) ) ) ) ) ) ).
fof(t15_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_funct_1(C)
& v1_msscyc_1(C,A)
& m2_graph_1(C,A) )
=> ~ ( r2_hidden(B,k4_finseq_1(C))
& ! [D] :
( ( v2_funct_1(D)
& v1_msscyc_1(D,A)
& m2_graph_1(D,A) )
=> ~ ( k1_funct_1(D,np__1) = k1_funct_1(C,B)
& k3_finseq_1(D) = k3_finseq_1(C)
& k2_relat_1(D) = k2_relat_1(C) ) ) ) ) ) ) ).
fof(t16_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B,C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ( ( C = k1_funct_1(u3_graph_1(A),B)
& D = k1_funct_1(u4_graph_1(A),B) )
=> r1_graph_2(A,k2_finseq_4(u1_graph_1(A),D,C),k9_finseq_1(B)) ) ) ) ) ).
fof(t17_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( ( r2_hidden(D,u2_graph_1(A))
& r1_graph_2(A,C,B)
& k1_funct_1(C,k3_finseq_1(C)) = k1_funct_1(u3_graph_1(A),D) )
=> ( m1_graph_1(k7_finseq_1(B,k9_finseq_1(D)),A)
& ? [E] :
( m2_finseq_1(E,u1_graph_1(A))
& E = k3_graph_2(C,k10_finseq_1(k1_funct_1(u3_graph_1(A),D),k1_funct_1(u4_graph_1(A),D)))
& r1_graph_2(A,E,k7_finseq_1(B,k9_finseq_1(D)))
& k1_funct_1(E,np__1) = k1_funct_1(C,np__1)
& k1_funct_1(E,k3_finseq_1(E)) = k1_funct_1(u4_graph_1(A),D) ) ) ) ) ) ) ).
fof(t18_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( ( r2_hidden(D,u2_graph_1(A))
& r1_graph_2(A,C,B)
& k1_funct_1(C,k3_finseq_1(C)) = k1_funct_1(u4_graph_1(A),D) )
=> ( m1_graph_1(k7_finseq_1(B,k9_finseq_1(D)),A)
& ? [E] :
( m2_finseq_1(E,u1_graph_1(A))
& E = k3_graph_2(C,k10_finseq_1(k1_funct_1(u4_graph_1(A),D),k1_funct_1(u3_graph_1(A),D)))
& r1_graph_2(A,E,k7_finseq_1(B,k9_finseq_1(D)))
& k1_funct_1(E,np__1) = k1_funct_1(C,np__1)
& k1_funct_1(E,k3_finseq_1(E)) = k1_funct_1(u3_graph_1(A),D) ) ) ) ) ) ) ).
fof(t19_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ( r1_graph_2(A,C,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(B))
& ~ ( k1_funct_1(C,D) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,D))
& k1_funct_1(C,k1_nat_1(D,np__1)) = k1_funct_1(u3_graph_1(A),k1_funct_1(B,D)) )
& ~ ( k1_funct_1(C,D) = k1_funct_1(u3_graph_1(A),k1_funct_1(B,D))
& k1_funct_1(C,k1_nat_1(D,np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,D)) ) ) ) ) ) ) ) ).
fof(t20_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( ( r1_graph_2(A,C,B)
& r2_hidden(D,k2_relat_1(B)) )
=> ( r2_hidden(k1_funct_1(u4_graph_1(A),D),k2_relat_1(C))
& r2_hidden(k1_funct_1(u3_graph_1(A),D),k2_relat_1(C)) ) ) ) ) ) ).
fof(t21_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> k1_graph_3(A,k1_xboole_0) = k1_xboole_0 ) ).
fof(t22_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B,C] :
~ ( r2_hidden(B,u2_graph_1(A))
& r2_hidden(B,C)
& v1_xboole_0(k1_graph_3(A,C)) ) ) ).
fof(t23_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v6_graph_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ~ ( B != C
& ! [D] :
( m1_graph_1(D,A)
=> ! [E] :
( m2_finseq_1(E,u1_graph_1(A))
=> ~ ( ~ v1_xboole_0(D)
& r1_graph_2(A,E,D)
& k1_funct_1(E,np__1) = B
& k1_funct_1(E,k3_finseq_1(E)) = C ) ) ) ) ) ) ) ) ).
fof(t24_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& l1_graph_1(A) )
=> ! [B,C] :
( m1_subset_1(C,u1_graph_1(A))
=> ~ ( ~ r1_xboole_0(B,u2_graph_1(A))
& ~ r2_hidden(C,k1_graph_3(A,B))
& ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ! [E] :
( m1_subset_1(E,u2_graph_1(A))
=> ~ ( r2_hidden(D,k1_graph_3(A,B))
& ~ r2_hidden(E,B)
& ( D = k1_funct_1(u4_graph_1(A),E)
| D = k1_funct_1(u3_graph_1(A),E) ) ) ) ) ) ) ) ).
fof(d1_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C,D] :
( m1_subset_1(D,k1_zfmisc_1(u2_graph_1(A)))
=> ( D = k2_graph_3(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(E,u2_graph_1(A))
& r2_hidden(E,C)
& k1_funct_1(u4_graph_1(A),E) = B ) ) ) ) ) ) ).
fof(d2_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C,D] :
( m1_subset_1(D,k1_zfmisc_1(u2_graph_1(A)))
=> ( D = k3_graph_3(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(E,u2_graph_1(A))
& r2_hidden(E,C)
& k1_funct_1(u3_graph_1(A),E) = B ) ) ) ) ) ) ).
fof(d3_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] : k4_graph_3(A,B,C) = k4_subset_1(u2_graph_1(A),k2_graph_3(A,B,C),k3_graph_3(A,B,C)) ) ) ).
fof(d4_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k5_graph_3(A,B) = k2_graph_3(A,B,u2_graph_1(A)) ) ) ).
fof(d5_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k6_graph_3(A,B) = k3_graph_3(A,B,u2_graph_1(A)) ) ) ).
fof(t25_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] : r1_tarski(k2_graph_3(A,B,C),k5_graph_3(A,B)) ) ) ).
fof(t26_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] : r1_tarski(k3_graph_3(A,B,C),k6_graph_3(A,B)) ) ) ).
fof(t27_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k4_card_1(k5_graph_3(A,B)) = k4_graph_1(A,B) ) ) ).
fof(t28_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k4_card_1(k6_graph_3(A,B)) = k5_graph_1(A,B) ) ) ).
fof(d6_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] : k7_graph_3(A,B,C) = k1_nat_1(k4_card_1(k2_graph_3(A,B,C)),k4_card_1(k3_graph_3(A,B,C))) ) ) ).
fof(t29_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k6_graph_1(A,B) = k7_graph_3(A,B,u2_graph_1(A)) ) ) ).
fof(t30_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ~ ( k7_graph_3(B,C,A) != np__0
& v1_xboole_0(k4_graph_3(B,C,A)) ) ) ) ).
fof(t31_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(C)
& v7_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_graph_1(C))
=> ~ ( r2_hidden(A,u2_graph_1(C))
& ~ r2_hidden(A,B)
& ( D = k1_funct_1(u4_graph_1(C),A)
| D = k1_funct_1(u3_graph_1(C),A) )
& k6_graph_1(C,D) = k7_graph_3(C,D,B) ) ) ) ).
fof(t32_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C,D] :
( r1_tarski(C,D)
=> k4_card_1(k2_graph_3(A,B,k4_xboole_0(D,C))) = k6_xcmplx_0(k4_card_1(k2_graph_3(A,B,D)),k4_card_1(k2_graph_3(A,B,C))) ) ) ) ).
fof(t33_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C,D] :
( r1_tarski(C,D)
=> k4_card_1(k3_graph_3(A,B,k4_xboole_0(D,C))) = k6_xcmplx_0(k4_card_1(k3_graph_3(A,B,D)),k4_card_1(k3_graph_3(A,B,C))) ) ) ) ).
fof(t34_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C,D] :
( r1_tarski(C,D)
=> k7_graph_3(A,B,k4_xboole_0(D,C)) = k6_xcmplx_0(k7_graph_3(A,B,D),k7_graph_3(A,B,C)) ) ) ) ).
fof(t35_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ( k2_graph_3(B,C,A) = k2_graph_3(B,C,k3_xboole_0(A,u2_graph_1(B)))
& k3_graph_3(B,C,A) = k3_graph_3(B,C,k3_xboole_0(A,u2_graph_1(B))) ) ) ) ).
fof(t36_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> k7_graph_3(B,C,A) = k7_graph_3(B,C,k3_xboole_0(A,u2_graph_1(B))) ) ) ).
fof(t37_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_graph_1(C,A)
=> ! [D] :
( m2_finseq_1(D,u1_graph_1(A))
=> ( r1_graph_2(A,D,C)
=> ( v1_xboole_0(C)
| ( r2_hidden(B,k2_relat_1(D))
<=> k7_graph_3(A,B,k2_relat_1(C)) != np__0 ) ) ) ) ) ) ) ).
fof(t38_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k6_graph_1(A,B) != np__0 ) ) ).
fof(d7_graph_3,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] :
( ( v1_graph_1(D)
& v2_graph_1(D)
& l1_graph_1(D) )
=> ( D = k8_graph_3(A,B,C)
<=> ( u1_graph_1(D) = u1_graph_1(A)
& u2_graph_1(D) = k2_xboole_0(u2_graph_1(A),k1_tarski(u2_graph_1(A)))
& u3_graph_1(D) = k1_funct_4(u3_graph_1(A),k3_cqc_lang(u2_graph_1(A),B))
& u4_graph_1(D) = k1_funct_4(u4_graph_1(A),k3_cqc_lang(u2_graph_1(A),C)) ) ) ) ) ) ) ).
fof(t39_graph_3,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))
=> ( r2_hidden(u2_graph_1(A),u2_graph_1(k8_graph_3(A,B,C)))
& u2_graph_1(A) = k4_xboole_0(u2_graph_1(k8_graph_3(A,B,C)),k1_tarski(u2_graph_1(A)))
& k1_funct_1(u3_graph_1(k8_graph_3(A,B,C)),u2_graph_1(A)) = B
& k1_funct_1(u4_graph_1(k8_graph_3(A,B,C)),u2_graph_1(A)) = C ) ) ) ) ).
fof(t40_graph_3,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,u2_graph_1(B))
=> ( k1_funct_1(u3_graph_1(k8_graph_3(B,C,D)),A) = k1_funct_1(u3_graph_1(B),A)
& k1_funct_1(u4_graph_1(k8_graph_3(B,C,D)),A) = k1_funct_1(u4_graph_1(B),A) ) ) ) ) ) ).
fof(t41_graph_3,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] :
( m1_graph_1(D,A)
=> ! [E] :
( m2_finseq_1(E,u1_graph_1(A))
=> ! [F] :
( m2_finseq_1(F,u1_graph_1(k8_graph_3(A,B,C)))
=> ( ( F = E
& r1_graph_2(A,E,D) )
=> r1_graph_2(k8_graph_3(A,B,C),F,D) ) ) ) ) ) ) ) ).
fof(t42_graph_3,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] :
( m1_graph_1(D,A)
=> m1_graph_1(D,k8_graph_3(A,B,C)) ) ) ) ) ).
fof(t43_graph_3,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] :
( ( v2_funct_1(D)
& m2_graph_1(D,A) )
=> ( v2_funct_1(D)
& m2_graph_1(D,k8_graph_3(A,B,C)) ) ) ) ) ) ).
fof(t44_graph_3,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] :
( m1_subset_1(E,u1_graph_1(k8_graph_3(B,C,D)))
=> ( E = C
=> ( C = D
| k2_graph_3(k8_graph_3(B,C,D),E,A) = k2_graph_3(B,C,A) ) ) ) ) ) ) ).
fof(t45_graph_3,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] :
( m1_subset_1(E,u1_graph_1(k8_graph_3(B,D,C)))
=> ( E = C
=> ( D = C
| k3_graph_3(k8_graph_3(B,D,C),E,A) = k3_graph_3(B,C,A) ) ) ) ) ) ) ).
fof(t46_graph_3,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] :
( m1_subset_1(E,u1_graph_1(k8_graph_3(B,C,D)))
=> ( ( E = C
& r2_hidden(u2_graph_1(B),A) )
=> ( C = D
| ( k3_graph_3(k8_graph_3(B,C,D),E,A) = k2_xboole_0(k3_graph_3(B,C,A),k1_tarski(u2_graph_1(B)))
& r1_xboole_0(k3_graph_3(B,C,A),k1_tarski(u2_graph_1(B))) ) ) ) ) ) ) ) ).
fof(t47_graph_3,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] :
( m1_subset_1(E,u1_graph_1(k8_graph_3(B,D,C)))
=> ( ( E = C
& r2_hidden(u2_graph_1(B),A) )
=> ( D = C
| ( k2_graph_3(k8_graph_3(B,D,C),E,A) = k2_xboole_0(k2_graph_3(B,C,A),k1_tarski(u2_graph_1(B)))
& r1_xboole_0(k2_graph_3(B,C,A),k1_tarski(u2_graph_1(B))) ) ) ) ) ) ) ) ).
fof(t48_graph_3,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] :
( m1_subset_1(E,u1_graph_1(B))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(k8_graph_3(B,D,E)))
=> ( F = C
=> ( C = D
| C = E
| k2_graph_3(k8_graph_3(B,D,E),F,A) = k2_graph_3(B,C,A) ) ) ) ) ) ) ) ).
fof(t49_graph_3,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] :
( m1_subset_1(E,u1_graph_1(B))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(k8_graph_3(B,D,E)))
=> ( F = C
=> ( C = D
| C = E
| k3_graph_3(k8_graph_3(B,D,E),F,A) = k3_graph_3(B,C,A) ) ) ) ) ) ) ) ).
fof(t50_graph_3,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] :
( ( v2_funct_1(D)
& m2_graph_1(D,k8_graph_3(A,B,C)) )
=> ( ~ r2_hidden(u2_graph_1(A),k2_relat_1(D))
=> ( v2_funct_1(D)
& m2_graph_1(D,A) ) ) ) ) ) ) ).
fof(t51_graph_3,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] :
( m2_finseq_1(D,u1_graph_1(A))
=> ! [E] :
( ( v2_funct_1(E)
& m2_graph_1(E,k8_graph_3(A,B,C)) )
=> ! [F] :
( m2_finseq_1(F,u1_graph_1(k8_graph_3(A,B,C)))
=> ( ( D = F
& r1_graph_2(k8_graph_3(A,B,C),F,E) )
=> ( r2_hidden(u2_graph_1(A),k2_relat_1(E))
| r1_graph_2(A,D,E) ) ) ) ) ) ) ) ) ).
fof(t52_graph_3,axiom,
! [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))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(B))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(k8_graph_3(B,D,E)))
=> ( ( F = C
& r2_hidden(u2_graph_1(B),A) )
=> ( D = E
| ( C != D
& C != E )
| k7_graph_3(k8_graph_3(B,D,E),F,A) = k1_nat_1(k7_graph_3(B,C,A),np__1) ) ) ) ) ) ) ) ).
fof(t53_graph_3,axiom,
! [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))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(B))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(k8_graph_3(B,D,E)))
=> ( F = C
=> ( C = D
| C = E
| k7_graph_3(k8_graph_3(B,D,E),F,A) = k7_graph_3(B,C,A) ) ) ) ) ) ) ) ).
fof(t54_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( ( v2_funct_1(C)
& v1_msscyc_1(C,A)
& m2_graph_1(C,A) )
=> v1_abian(k7_graph_3(A,B,k2_relat_1(C))) ) ) ) ).
fof(t55_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( ( v2_funct_1(D)
& m2_graph_1(D,A) )
=> ( r1_graph_2(A,C,D)
=> ( v1_msscyc_1(D,A)
| ( v1_abian(k7_graph_3(A,B,k2_relat_1(D)))
<=> ( B != k1_funct_1(C,np__1)
& B != k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) ) ) ) ) ).
fof(d8_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,u2_graph_1(A)) )
=> ( B = k9_graph_3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v2_funct_1(C)
& v1_msscyc_1(C,A)
& m2_graph_1(C,A) ) ) ) ) ) ).
fof(t56_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> m2_finseq_2(k1_xboole_0,u2_graph_1(A),k9_graph_3(A)) ) ).
fof(d10_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_2(B,u2_graph_1(A),k9_graph_3(A))
=> ! [C] :
( m2_finseq_2(C,u2_graph_1(A),k9_graph_3(A))
=> ( r1_xboole_0(k2_relat_1(B),k2_relat_1(C))
=> ( r1_xboole_0(k1_graph_3(A,k2_relat_1(B)),k1_graph_3(A,k2_relat_1(C)))
| ! [D] :
( m2_finseq_2(D,u2_graph_1(A),k9_graph_3(A))
=> ( D = k11_graph_3(A,B,C)
<=> ? [E] :
( m1_subset_1(E,u1_graph_1(A))
& E = k8_subset_1(k5_subset_1(u1_graph_1(A),k1_graph_3(A,k2_relat_1(B)),k1_graph_3(A,k2_relat_1(C))))
& D = k8_finseq_1(u2_graph_1(A),k10_graph_3(A,E,B),k10_graph_3(A,E,C)) ) ) ) ) ) ) ) ) ).
fof(t58_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_2(B,u2_graph_1(A),k9_graph_3(A))
=> ! [C] :
( m2_finseq_2(C,u2_graph_1(A),k9_graph_3(A))
=> ~ ( ~ r1_xboole_0(k1_graph_3(A,k2_relat_1(B)),k1_graph_3(A,k2_relat_1(C)))
& r1_xboole_0(k2_relat_1(B),k2_relat_1(C))
& ~ ( B = k1_xboole_0
& C = k1_xboole_0 )
& v1_xboole_0(k11_graph_3(A,B,C)) ) ) ) ) ).
fof(t59_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m2_finseq_2(D,u2_graph_1(B),k12_graph_3(B,C,A))
=> ! [E] :
( v1_finset_1(E)
=> ( E = u2_graph_1(B)
=> ( k7_graph_3(B,C,A) = np__0
| r1_xreal_0(k3_finseq_1(D),k4_card_1(E)) ) ) ) ) ) ) ).
fof(t60_graph_3,axiom,
! [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))
=> v1_abian(k7_graph_3(B,D,A)) )
=> ( k7_graph_3(B,C,A) = np__0
| ! [D] :
( m1_graph_3(D,u2_graph_1(B),k9_graph_3(B),k13_graph_3(B,C,A))
=> ( ~ v1_xboole_0(D)
& r1_tarski(k2_relat_1(D),A)
& r2_hidden(C,k1_graph_3(B,k2_relat_1(D))) ) ) ) ) ) ) ).
fof(t62_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_2(B,u2_graph_1(A),k9_graph_3(A))
=> ( ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> v1_abian(k6_graph_1(A,C)) )
=> ( k2_relat_1(B) = u2_graph_1(A)
| v1_xboole_0(B)
| ( ~ v1_xboole_0(k14_graph_3(A,B))
& ~ r1_xreal_0(k4_card_1(k2_relat_1(k14_graph_3(A,B))),k4_card_1(k2_relat_1(B))) ) ) ) ) ) ).
fof(d14_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ( v1_graph_3(B,A)
<=> k2_relat_1(B) = u2_graph_1(A) ) ) ) ).
fof(t63_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ( ( v1_graph_3(B,A)
& r1_graph_2(A,C,B) )
=> k2_relat_1(C) = u1_graph_1(A) ) ) ) ) ).
fof(t64_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ( ? [B] :
( v2_funct_1(B)
& v1_msscyc_1(B,A)
& m2_graph_1(B,A)
& v1_graph_3(B,A) )
<=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> v1_abian(k6_graph_1(A,B)) ) ) ) ).
fof(t65_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ( ~ ( ? [B] :
( v2_funct_1(B)
& m2_graph_1(B,A)
& ~ v1_msscyc_1(B,A)
& v1_graph_3(B,A) )
& ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ( v1_abian(k6_graph_1(A,D))
<=> ( D != B
& D != C ) ) ) ) ) ) )
& ~ ( ? [B] :
( m1_subset_1(B,u1_graph_1(A))
& ? [C] :
( m1_subset_1(C,u1_graph_1(A))
& B != C
& ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ( v1_abian(k6_graph_1(A,D))
<=> ( D != B
& D != C ) ) ) ) )
& ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ~ ( ~ v1_msscyc_1(B,A)
& v1_graph_3(B,A) ) ) ) ) ) ).
fof(dt_m1_graph_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ! [D] :
( m1_graph_3(D,A,B,C)
=> m2_finseq_1(D,A) ) ) ).
fof(existence_m1_graph_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ? [D] : m1_graph_3(D,A,B,C) ) ).
fof(redefinition_m1_graph_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ! [D] :
( m1_graph_3(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_k1_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> m1_subset_1(k1_graph_3(A,B),k1_zfmisc_1(u1_graph_1(A))) ) ).
fof(redefinition_k1_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> k1_graph_3(A,B) = k5_graph_2(A,B) ) ).
fof(dt_k2_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m1_subset_1(k2_graph_3(A,B,C),k1_zfmisc_1(u2_graph_1(A))) ) ).
fof(dt_k3_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m1_subset_1(k3_graph_3(A,B,C),k1_zfmisc_1(u2_graph_1(A))) ) ).
fof(dt_k4_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m1_subset_1(k4_graph_3(A,B,C),k1_zfmisc_1(u2_graph_1(A))) ) ).
fof(dt_k5_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m1_subset_1(k5_graph_3(A,B),k1_zfmisc_1(u2_graph_1(A))) ) ).
fof(dt_k6_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m1_subset_1(k6_graph_3(A,B),k1_zfmisc_1(u2_graph_1(A))) ) ).
fof(dt_k7_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m2_subset_1(k7_graph_3(A,B,C),k1_numbers,k5_numbers) ) ).
fof(dt_k8_graph_3,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)) )
=> ( v1_graph_1(k8_graph_3(A,B,C))
& v2_graph_1(k8_graph_3(A,B,C))
& l1_graph_1(k8_graph_3(A,B,C)) ) ) ).
fof(dt_k9_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( ~ v1_xboole_0(k9_graph_3(A))
& m1_finseq_2(k9_graph_3(A),u2_graph_1(A)) ) ) ).
fof(dt_k10_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A))
& m1_subset_1(C,k9_graph_3(A)) )
=> m2_finseq_2(k10_graph_3(A,B,C),u2_graph_1(A),k9_graph_3(A)) ) ).
fof(dt_k11_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,k9_graph_3(A))
& m1_subset_1(C,k9_graph_3(A)) )
=> m2_finseq_2(k11_graph_3(A,B,C),u2_graph_1(A),k9_graph_3(A)) ) ).
fof(dt_k12_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> ( ~ v1_xboole_0(k12_graph_3(A,B,C))
& m1_finseq_2(k12_graph_3(A,B,C),u2_graph_1(A)) ) ) ).
fof(dt_k13_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> ( ~ v1_xboole_0(k13_graph_3(A,B,C))
& m1_subset_1(k13_graph_3(A,B,C),k1_zfmisc_1(k9_graph_3(A))) ) ) ).
fof(dt_k14_graph_3,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,k9_graph_3(A)) )
=> m2_finseq_2(k14_graph_3(A,B),u2_graph_1(A),k9_graph_3(A)) ) ).
fof(t57_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m2_finseq_2(C,u2_graph_1(A),k9_graph_3(A))
=> ( r2_hidden(B,k1_graph_3(A,k2_relat_1(C)))
=> ( ~ v1_xboole_0(a_3_0_graph_3(A,B,C))
& m1_subset_1(a_3_0_graph_3(A,B,C),k1_zfmisc_1(k9_graph_3(A))) ) ) ) ) ) ).
fof(d9_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m2_finseq_2(C,u2_graph_1(A),k9_graph_3(A))
=> ( r2_hidden(B,k1_graph_3(A,k2_relat_1(C)))
=> k10_graph_3(A,B,C) = k8_subset_1(a_3_0_graph_3(A,B,C)) ) ) ) ) ).
fof(d11_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( k7_graph_3(A,B,C) != np__0
=> k12_graph_3(A,B,C) = a_3_1_graph_3(A,B,C) ) ) ) ).
fof(d12_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> v1_abian(k7_graph_3(A,D,C)) )
=> ( k7_graph_3(A,B,C) = np__0
| k13_graph_3(A,B,C) = a_3_2_graph_3(A,B,C) ) ) ) ) ).
fof(t61_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_2(B,u2_graph_1(A),k9_graph_3(A))
=> ~ ( k2_relat_1(B) != u2_graph_1(A)
& ~ v1_xboole_0(B)
& ~ ( ~ v1_xboole_0(a_2_0_graph_3(A,B))
& m1_subset_1(a_2_0_graph_3(A,B),k1_zfmisc_1(u1_graph_1(A))) ) ) ) ) ).
fof(d13_graph_3,axiom,
! [A] :
( ( v2_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_2(B,u2_graph_1(A),k9_graph_3(A))
=> ~ ( k2_relat_1(B) != u2_graph_1(A)
& ~ v1_xboole_0(B)
& ~ ! [C] :
( m2_finseq_2(C,u2_graph_1(A),k9_graph_3(A))
=> ( C = k14_graph_3(A,B)
<=> ? [D] :
( m2_finseq_2(D,u2_graph_1(A),k9_graph_3(A))
& ? [E] :
( m1_subset_1(E,u1_graph_1(A))
& E = k8_subset_1(a_2_0_graph_3(A,B))
& D = k8_subset_1(k13_graph_3(A,E,k4_xboole_0(u2_graph_1(A),k2_relat_1(B))))
& C = k11_graph_3(A,B,D) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_graph_3,axiom,
! [A,B,C,D] :
( ( v2_graph_1(B)
& l1_graph_1(B)
& m1_subset_1(C,u1_graph_1(B))
& m2_finseq_2(D,u2_graph_1(B),k9_graph_3(B)) )
=> ( r2_hidden(A,a_3_0_graph_3(B,C,D))
<=> ? [E] :
( m2_finseq_2(E,u2_graph_1(B),k9_graph_3(B))
& A = E
& k2_relat_1(E) = k2_relat_1(D)
& ? [F] :
( m2_finseq_1(F,u1_graph_1(B))
& r1_graph_2(B,F,E)
& k1_funct_1(F,np__1) = C ) ) ) ) ).
fof(fraenkel_a_3_1_graph_3,axiom,
! [A,B,C,D] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B)
& m1_subset_1(C,u1_graph_1(B)) )
=> ( r2_hidden(A,a_3_1_graph_3(B,C,D))
<=> ? [E] :
( m2_finseq_2(E,D,k3_finseq_2(D))
& A = E
& v2_funct_1(E)
& m2_graph_1(E,B)
& ~ v1_xboole_0(E)
& ? [F] :
( m2_finseq_1(F,u1_graph_1(B))
& r1_graph_2(B,F,E)
& k1_funct_1(F,np__1) = C ) ) ) ) ).
fof(fraenkel_a_3_2_graph_3,axiom,
! [A,B,C,D] :
( ( v2_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B)
& m1_subset_1(C,u1_graph_1(B)) )
=> ( r2_hidden(A,a_3_2_graph_3(B,C,D))
<=> ? [E] :
( m2_finseq_2(E,u2_graph_1(B),k9_graph_3(B))
& A = E
& r1_tarski(k2_relat_1(E),D)
& ~ v1_xboole_0(E)
& ? [F] :
( m2_finseq_1(F,u1_graph_1(B))
& r1_graph_2(B,F,E)
& k1_funct_1(F,np__1) = C ) ) ) ) ).
fof(fraenkel_a_2_0_graph_3,axiom,
! [A,B,C] :
( ( v2_graph_1(B)
& v6_graph_1(B)
& v7_graph_1(B)
& l1_graph_1(B)
& m2_finseq_2(C,u2_graph_1(B),k9_graph_3(B)) )
=> ( r2_hidden(A,a_2_0_graph_3(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_graph_1(B))
& A = D
& r2_hidden(D,k1_graph_3(B,k2_relat_1(C)))
& k6_graph_1(B,D) != k7_graph_3(B,D,k2_relat_1(C)) ) ) ) ).
%------------------------------------------------------------------------------