SET007 Axioms: SET007+299.ax
%------------------------------------------------------------------------------
% File : SET007+299 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Natural transformations. Discrete categories
% 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 : nattra_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 109 ( 10 unt; 0 def)
% Number of atoms : 946 ( 105 equ)
% Maximal formula atoms : 32 ( 8 avg)
% Number of connectives : 898 ( 61 ~; 0 |; 386 &)
% ( 22 <=>; 429 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-6 aty)
% Number of functors : 56 ( 56 usr; 4 con; 0-7 aty)
% Number of variables : 504 ( 489 !; 15 ?)
% 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(rc1_nattra_1,axiom,
? [A] :
( l1_cat_1(A)
& v2_cat_1(A)
& v2_nattra_1(A) ) ).
fof(rc2_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ? [B] :
( m3_cat_2(B,A)
& v1_cat_1(B)
& v2_cat_1(B)
& v2_nattra_1(B) ) ) ).
fof(t1_nattra_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,A)
& m2_relset_1(E,C,A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,D,B)
& m2_relset_1(F,D,B) )
=> ! [G] :
( ( ~ v1_xboole_0(G)
& m1_subset_1(G,k1_zfmisc_1(C)) )
=> ! [H] :
( ( ~ v1_xboole_0(H)
& m1_subset_1(H,k1_zfmisc_1(D)) )
=> k1_nattra_1(k2_zfmisc_1(C,D),k2_zfmisc_1(A,B),k9_cat_2(C,D,A,B,E,F),k12_mcart_1(C,D,G,H)) = k9_cat_2(G,H,A,B,k1_nattra_1(C,A,E,G),k1_nattra_1(D,B,F,H)) ) ) ) ) ) ) ) ) ).
fof(t2_nattra_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ! [G] :
( ( v1_funct_1(G)
& m2_relset_1(G,k12_mcart_1(A,A,C,C),C) )
=> ( G = k1_realset1(E,C)
=> ! [H] :
( ( v1_funct_1(H)
& m2_relset_1(H,k12_mcart_1(B,B,D,D),D) )
=> ( H = k1_realset1(F,D)
=> k2_nattra_1(A,B,C,D,G,H) = k1_realset1(k10_cat_2(A,B,E,F),k12_mcart_1(A,B,C,D)) ) ) ) ) ) ) ) ) ) ) ).
fof(t3_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_cat_1(C,A,B,B)
=> r2_hidden(C,k6_cat_1(A,B,B)) ) ) ) ).
fof(t4_nattra_1,axiom,
! [A,B,C] :
( m1_subset_1(C,u2_cat_1(k8_cat_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(k8_cat_1(A,B)))
=> C = D ) ) ).
fof(t5_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> r2_hidden(k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,k10_cat_1(A,B),k10_cat_1(A,B)),k10_cat_1(A,B)),u5_cat_1(A)) ) ) ).
fof(t6_nattra_1,axiom,
! [A,B] : u5_cat_1(k8_cat_1(A,B)) = k1_tarski(k4_tarski(k4_tarski(B,B),B)) ).
fof(t7_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> m3_cat_2(k8_cat_1(B,k10_cat_1(A,B)),A) ) ) ).
fof(t8_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ( u3_cat_1(B) = k7_relat_1(u3_cat_1(A),u2_cat_1(B))
& u4_cat_1(B) = k7_relat_1(u4_cat_1(A),u2_cat_1(B))
& u5_cat_1(B) = k1_realset1(u5_cat_1(A),u2_cat_1(B))
& u6_cat_1(B) = k7_relat_1(u6_cat_1(A),u1_cat_1(B)) ) ) ) ).
fof(t9_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_cat_1(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u2_cat_1(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,C,B)
& m2_relset_1(D,C,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,B)
& m2_relset_1(E,C,B) )
=> ( ( D = k1_nattra_1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),C)
& E = k1_nattra_1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),C) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,k2_zfmisc_1(C,C),C) )
=> ( F = k1_realset1(u5_cat_1(A),C)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,B,C)
& m2_relset_1(G,B,C) )
=> ( G = k1_nattra_1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),B)
=> m3_cat_2(g1_cat_1(B,C,D,E,F,G),A) ) ) ) ) ) ) ) ) ) ) ).
fof(t10_nattra_1,axiom,
! [A] :
( ( v1_cat_1(A)
& v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v1_cat_1(B)
& m3_cat_2(B,A) )
=> ( ( u1_cat_1(B) = u1_cat_1(A)
& u2_cat_1(B) = u2_cat_1(A) )
=> B = A ) ) ) ).
fof(d1_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ( k6_cat_1(A,D,E) != k1_xboole_0
=> ! [F] :
( m1_cat_1(F,A,D,E)
=> k3_nattra_1(A,B,C,D,E,F) = k8_funct_2(u2_cat_1(A),u2_cat_1(B),C,F) ) ) ) ) ) ) ) ).
fof(t11_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,C,A)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m1_subset_1(F,u1_cat_1(C))
=> ! [G] :
( m1_subset_1(G,u1_cat_1(C))
=> ( k6_cat_1(C,F,G) != k1_xboole_0
=> ! [H] :
( m1_cat_1(H,C,F,G)
=> k3_nattra_1(C,B,k14_cat_1(C,A,B,D,E),F,G,H) = k3_nattra_1(A,B,E,k13_cat_1(C,A,D,F),k13_cat_1(C,A,D,G),k3_nattra_1(C,A,D,F,G,H)) ) ) ) ) ) ) ) ) ) ).
fof(t12_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ( k6_cat_1(A,E,F) != k1_xboole_0
=> ! [G] :
( m1_cat_1(G,A,E,F)
=> k3_nattra_1(A,B,C,E,F,G) = k3_nattra_1(A,B,D,E,F,G) ) ) ) )
=> C = D ) ) ) ) ) ).
fof(t13_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(B))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(B))
=> ~ ( k6_cat_1(B,D,E) != k1_xboole_0
& k6_cat_1(B,E,F) != k1_xboole_0
& ~ ! [G] :
( m1_cat_1(G,B,D,E)
=> ! [H] :
( m1_cat_1(H,B,E,F)
=> k3_nattra_1(B,A,C,D,F,k9_cat_1(B,D,E,F,G,H)) = k9_cat_1(A,k13_cat_1(B,A,C,D),k13_cat_1(B,A,C,E),k13_cat_1(B,A,C,F),k3_nattra_1(B,A,C,D,E,G),k3_nattra_1(B,A,C,E,F,H)) ) ) ) ) ) ) ) ) ) ).
fof(t14_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(B))
=> ( k3_nattra_1(A,B,C,D,D,k10_cat_1(A,D)) = k10_cat_1(B,E)
=> k13_cat_1(A,B,C,D) = E ) ) ) ) ) ) ).
fof(t15_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k3_nattra_1(B,A,C,D,D,k10_cat_1(B,D)) = k10_cat_1(A,k13_cat_1(B,A,C,D)) ) ) ) ) ).
fof(t16_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ( k6_cat_1(A,B,C) != k1_xboole_0
=> ! [D] :
( m1_cat_1(D,A,B,C)
=> k3_nattra_1(A,A,k15_cat_1(A),B,C,D) = D ) ) ) ) ) ).
fof(t17_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ( ~ r1_xboole_0(k6_cat_1(A,B,C),k6_cat_1(A,D,E))
=> ( B = D
& C = E ) ) ) ) ) ) ) ).
fof(d2_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_nattra_1(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> k6_cat_1(B,k13_cat_1(A,B,C,E),k13_cat_1(A,B,D,E)) != k1_xboole_0 ) ) ) ) ) ) ).
fof(t18_nattra_1,axiom,
$true ).
fof(t19_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( ( r1_nattra_1(A,B,C,D)
& r1_nattra_1(A,B,D,E) )
=> r1_nattra_1(A,B,C,E) ) ) ) ) ) ) ).
fof(d3_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_nattra_1(A,B,C,D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_cat_1(A),u2_cat_1(B))
& m2_relset_1(E,u1_cat_1(A),u2_cat_1(B)) )
=> ( m1_nattra_1(E,A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> m1_cat_1(k8_funct_2(u1_cat_1(A),u2_cat_1(B),E,F),B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,D,F)) ) ) ) ) ) ) ) ) ).
fof(d4_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m1_nattra_1(D,A,B,C,C)
=> ( D = k4_nattra_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> k8_funct_2(u1_cat_1(A),u2_cat_1(B),D,E) = k10_cat_1(B,k13_cat_1(A,B,C,E)) ) ) ) ) ) ) ).
fof(d5_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_nattra_1(A,B,C,D)
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> k5_nattra_1(A,B,C,D,E,F) = k8_funct_2(u1_cat_1(A),u2_cat_1(B),E,F) ) ) ) ) ) ) ) ).
fof(d6_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( ( r1_nattra_1(A,B,C,D)
& r1_nattra_1(A,B,D,E) )
=> ! [F] :
( m1_nattra_1(F,A,B,C,D)
=> ! [G] :
( m1_nattra_1(G,A,B,D,E)
=> ! [H] :
( m1_nattra_1(H,A,B,C,E)
=> ( H = k6_nattra_1(A,B,C,D,E,F,G)
<=> ! [I] :
( m1_subset_1(I,u1_cat_1(A))
=> k5_nattra_1(A,B,C,E,H,I) = k9_cat_1(B,k13_cat_1(A,B,C,I),k13_cat_1(A,B,D,I),k13_cat_1(A,B,E,I),k5_nattra_1(A,B,C,D,F,I),k5_nattra_1(A,B,D,E,G,I)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m2_cat_1(D,B,A)
=> ( r1_nattra_1(B,A,C,D)
=> ! [E] :
( m1_nattra_1(E,B,A,C,D)
=> ! [F] :
( m1_nattra_1(F,B,A,C,D)
=> ( ! [G] :
( m1_subset_1(G,u1_cat_1(B))
=> k5_nattra_1(B,A,C,D,E,G) = k5_nattra_1(B,A,C,D,F,G) )
=> E = F ) ) ) ) ) ) ) ) ).
fof(t21_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k5_nattra_1(B,A,C,C,k4_nattra_1(B,A,C),D) = k10_cat_1(A,k13_cat_1(B,A,C,D)) ) ) ) ) ).
fof(t22_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_nattra_1(A,B,C,D)
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ( k6_nattra_1(A,B,C,D,D,E,k4_nattra_1(A,B,D)) = E
& k6_nattra_1(A,B,C,C,D,k4_nattra_1(A,B,C),E) = E ) ) ) ) ) ) ) ).
fof(t23_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,A,B)
=> ( ( r1_nattra_1(A,B,C,D)
& r1_nattra_1(A,B,D,E)
& r1_nattra_1(A,B,E,F) )
=> ! [G] :
( m1_nattra_1(G,A,B,C,D)
=> ! [H] :
( m1_nattra_1(H,A,B,D,E)
=> ! [I] :
( m1_nattra_1(I,A,B,E,F)
=> k6_nattra_1(A,B,C,D,F,G,k6_nattra_1(A,B,D,E,F,H,I)) = k6_nattra_1(A,B,C,E,F,k6_nattra_1(A,B,C,D,E,G,H),I) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r2_nattra_1(A,B,C,D)
<=> ( r1_nattra_1(A,B,C,D)
& ? [E] :
( m1_nattra_1(E,A,B,C,D)
& ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ! [G] :
( m1_subset_1(G,u1_cat_1(A))
=> ( k6_cat_1(A,F,G) != k1_xboole_0
=> ! [H] :
( m1_cat_1(H,A,F,G)
=> k9_cat_1(B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,C,G),k13_cat_1(A,B,D,G),k3_nattra_1(A,B,C,F,G,H),k5_nattra_1(A,B,C,D,E,G)) = k9_cat_1(B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,D,F),k13_cat_1(A,B,D,G),k5_nattra_1(A,B,C,D,E,F),k3_nattra_1(A,B,D,F,G,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_nattra_1,axiom,
$true ).
fof(t25_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( ( r2_nattra_1(A,B,C,D)
& r2_nattra_1(A,B,D,E) )
=> r2_nattra_1(A,B,C,E) ) ) ) ) ) ) ).
fof(d8_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r2_nattra_1(A,B,C,D)
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ( m2_nattra_1(E,A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ! [G] :
( m1_subset_1(G,u1_cat_1(A))
=> ( k6_cat_1(A,F,G) != k1_xboole_0
=> ! [H] :
( m1_cat_1(H,A,F,G)
=> k9_cat_1(B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,C,G),k13_cat_1(A,B,D,G),k3_nattra_1(A,B,C,F,G,H),k5_nattra_1(A,B,C,D,E,G)) = k9_cat_1(B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,D,F),k13_cat_1(A,B,D,G),k5_nattra_1(A,B,C,D,E,F),k3_nattra_1(A,B,D,F,G,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( ( r2_nattra_1(A,B,C,D)
& r2_nattra_1(A,B,D,E) )
=> ! [F] :
( m2_nattra_1(F,A,B,C,D)
=> ! [G] :
( m2_nattra_1(G,A,B,D,E)
=> k8_nattra_1(A,B,C,D,E,F,G) = k6_nattra_1(A,B,C,D,E,F,G) ) ) ) ) ) ) ) ) ).
fof(t26_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r2_nattra_1(A,B,C,D)
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> ( k8_nattra_1(A,B,C,D,D,E,k7_nattra_1(A,B,D)) = E
& k8_nattra_1(A,B,C,C,D,k7_nattra_1(A,B,C),E) = E ) ) ) ) ) ) ) ).
fof(t27_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m2_cat_1(D,B,A)
=> ! [E] :
( m2_cat_1(E,B,A)
=> ( ( r2_nattra_1(B,A,C,D)
& r2_nattra_1(B,A,D,E) )
=> ! [F] :
( m2_nattra_1(F,B,A,C,D)
=> ! [G] :
( m2_nattra_1(G,B,A,D,E)
=> ! [H] :
( m1_subset_1(H,u1_cat_1(B))
=> k5_nattra_1(B,A,C,E,k8_nattra_1(B,A,C,D,E,F,G),H) = k9_cat_1(A,k13_cat_1(B,A,C,H),k13_cat_1(B,A,D,H),k13_cat_1(B,A,E,H),k5_nattra_1(B,A,C,D,F,H),k5_nattra_1(B,A,D,E,G,H)) ) ) ) ) ) ) ) ) ) ).
fof(t28_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_nattra_1(G,A,B,C,D)
=> ! [H] :
( m2_nattra_1(H,A,B,D,E)
=> ( ( r2_nattra_1(A,B,C,D)
& r2_nattra_1(A,B,D,E)
& r2_nattra_1(A,B,E,F) )
=> ! [I] :
( m2_nattra_1(I,A,B,E,F)
=> k8_nattra_1(A,B,C,D,F,G,k8_nattra_1(A,B,D,E,F,H,I)) = k8_nattra_1(A,B,C,E,F,k8_nattra_1(A,B,C,D,E,G,H),I) ) ) ) ) ) ) ) ) ) ) ).
fof(d10_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ( v1_nattra_1(E,A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> v5_cat_1(k5_nattra_1(A,B,C,D,E,F),B) ) ) ) ) ) ) ) ).
fof(d11_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r3_nattra_1(A,B,C,D)
<=> ( r2_nattra_1(A,B,C,D)
& ? [E] :
( m2_nattra_1(E,A,B,C,D)
& v1_nattra_1(E,A,B,C,D) ) ) ) ) ) ) ) ).
fof(d12_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_nattra_1(A,B,C,D)
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ( v1_nattra_1(E,A,B,C,D)
=> ! [F] :
( m1_nattra_1(F,A,B,D,C)
=> ( F = k9_nattra_1(A,B,C,D,E)
<=> ! [G] :
( m1_subset_1(G,u1_cat_1(A))
=> k5_nattra_1(A,B,D,C,F,G) = k11_cat_1(B,k13_cat_1(A,B,C,G),k13_cat_1(A,B,D,G),k5_nattra_1(A,B,C,D,E,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(d13_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> ( ( r2_nattra_1(A,B,C,D)
& v1_nattra_1(E,A,B,C,D) )
=> k10_nattra_1(A,B,C,D,E) = k9_nattra_1(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t29_nattra_1,axiom,
$true ).
fof(t30_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> ( ( r2_nattra_1(A,B,C,D)
& v1_nattra_1(E,A,B,C,D) )
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> k5_nattra_1(A,B,D,C,k10_nattra_1(A,B,C,D,E),F) = k11_cat_1(B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,D,F),k5_nattra_1(A,B,C,D,E,F)) ) ) ) ) ) ) ) ).
fof(t31_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r3_nattra_1(A,B,C,D)
=> r3_nattra_1(A,B,D,C) ) ) ) ) ) ).
fof(t32_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( ( r3_nattra_1(A,B,C,D)
& r3_nattra_1(A,B,D,E) )
=> r3_nattra_1(A,B,C,E) ) ) ) ) ) ) ).
fof(d14_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r3_nattra_1(A,B,C,D)
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> ( m3_nattra_1(E,A,B,C,D)
<=> v1_nattra_1(E,A,B,C,D) ) ) ) ) ) ) ) ).
fof(t33_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> m3_nattra_1(k7_nattra_1(A,B,C),A,B,C,C) ) ) ) ).
fof(t34_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( ( r3_nattra_1(A,B,C,D)
& r3_nattra_1(A,B,D,E) )
=> ! [F] :
( m3_nattra_1(F,A,B,C,D)
=> ! [G] :
( m3_nattra_1(G,A,B,D,E)
=> m3_nattra_1(k8_nattra_1(A,B,C,D,E,F,G),A,B,C,E) ) ) ) ) ) ) ) ) ).
fof(d15_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( m4_nattra_1(C,A,B)
<=> ! [D] :
~ ( r2_hidden(D,C)
& ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_nattra_1(G,A,B,E,F)
=> ~ ( D = k4_tarski(k4_tarski(E,F),G)
& r2_nattra_1(A,B,E,F) ) ) ) ) ) ) ) ) ) ).
fof(d16_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m4_nattra_1(C,A,B)
=> ( C = k11_nattra_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( m2_cat_1(E,A,B)
& ? [F] :
( m2_cat_1(F,A,B)
& ? [G] :
( m2_nattra_1(G,A,B,E,F)
& D = k4_tarski(k4_tarski(E,F),G)
& r2_nattra_1(A,B,E,F) ) ) ) ) ) ) ) ) ).
fof(d17_nattra_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,C,D)
& m2_relset_1(F,C,D) )
=> ( r4_nattra_1(A,B,C,D,E,F)
<=> ( A = C
& ! [G] :
( m1_subset_1(G,A)
=> k8_funct_2(A,B,E,G) = k1_funct_1(F,G) ) ) ) ) ) ) ) ) ) ).
fof(t35_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> ( r2_nattra_1(A,B,C,D)
<=> r2_hidden(k4_tarski(k4_tarski(C,D),E),k11_nattra_1(A,B)) ) ) ) ) ) ) ).
fof(d18_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& l1_cat_1(C) )
=> ( C = k12_nattra_1(A,B)
<=> ( u1_cat_1(C) = k7_cat_2(A,B)
& u2_cat_1(C) = k11_nattra_1(A,B)
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ( k2_cat_1(C,D) = k1_mcart_1(k1_mcart_1(D))
& k3_cat_1(C,D) = k2_mcart_1(k1_mcart_1(D)) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(C))
=> ( k2_cat_1(C,E) = k3_cat_1(C,D)
=> r2_hidden(k13_cat_2(C,C,E,D),k1_relat_1(u5_cat_1(C))) ) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(C))
=> ~ ( r2_hidden(k13_cat_2(C,C,E,D),k1_relat_1(u5_cat_1(C)))
& ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_cat_1(G,A,B)
=> ! [H] :
( m2_cat_1(H,A,B)
=> ! [I] :
( m2_nattra_1(I,A,B,F,G)
=> ! [J] :
( m2_nattra_1(J,A,B,G,H)
=> ~ ( D = k4_tarski(k4_tarski(F,G),I)
& E = k4_tarski(k4_tarski(G,H),J)
& k1_funct_1(u5_cat_1(C),k13_cat_2(C,C,E,D)) = k4_tarski(k4_tarski(F,H),k8_nattra_1(A,B,F,G,H,I,J)) ) ) ) ) ) ) ) ) )
& ! [D] :
( m1_subset_1(D,u1_cat_1(C))
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( E = D
=> k10_cat_1(C,D) = k4_tarski(k4_tarski(E,E),k7_nattra_1(A,B,E)) ) ) ) ) ) ) ) ) ).
fof(t36_nattra_1,axiom,
$true ).
fof(t37_nattra_1,axiom,
$true ).
fof(t38_nattra_1,axiom,
$true ).
fof(t39_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> ! [F] :
( m1_subset_1(F,u2_cat_1(k12_nattra_1(A,B)))
=> ( F = k4_tarski(k4_tarski(C,D),E)
=> ( k2_cat_1(k12_nattra_1(A,B),F) = C
& k3_cat_1(k12_nattra_1(A,B),F) = D ) ) ) ) ) ) ) ) ).
fof(t40_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(k12_nattra_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(k12_nattra_1(A,B)))
=> ! [E] :
( m1_cat_1(E,k12_nattra_1(A,B),C,D)
=> ~ ( k6_cat_1(k12_nattra_1(A,B),C,D) != k1_xboole_0
& ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_cat_1(G,A,B)
=> ! [H] :
( m2_nattra_1(H,A,B,F,G)
=> ~ ( C = F
& D = G
& E = k4_tarski(k4_tarski(F,G),H) ) ) ) ) ) ) ) ) ) ) ).
fof(t41_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_nattra_1(G,A,B,E,F)
=> ! [H] :
( m2_nattra_1(H,A,B,C,D)
=> ! [I] :
( m1_subset_1(I,u2_cat_1(k12_nattra_1(A,B)))
=> ! [J] :
( m1_subset_1(J,u2_cat_1(k12_nattra_1(A,B)))
=> ( ( I = k4_tarski(k4_tarski(E,F),G)
& J = k4_tarski(k4_tarski(C,D),H) )
=> ( r2_hidden(k13_cat_2(k12_nattra_1(A,B),k12_nattra_1(A,B),J,I),k1_relat_1(u5_cat_1(k12_nattra_1(A,B))))
<=> r4_nattra_1(u2_cat_1(A),u2_cat_1(B),u2_cat_1(A),u2_cat_1(B),F,C) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t42_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_nattra_1(F,A,B,C,D)
=> ! [G] :
( m2_nattra_1(G,A,B,D,E)
=> ! [H] :
( m1_subset_1(H,u2_cat_1(k12_nattra_1(A,B)))
=> ! [I] :
( m1_subset_1(I,u2_cat_1(k12_nattra_1(A,B)))
=> ( ( H = k4_tarski(k4_tarski(C,D),F)
& I = k4_tarski(k4_tarski(D,E),G) )
=> k4_cat_1(k12_nattra_1(A,B),H,I) = k4_tarski(k4_tarski(C,E),k8_nattra_1(A,B,C,D,E,F,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(d19_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v2_nattra_1(A)
<=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ? [C] :
( m1_subset_1(C,u1_cat_1(A))
& B = k10_cat_1(A,C) ) ) ) ) ).
fof(t43_nattra_1,axiom,
$true ).
fof(t44_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& v2_nattra_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k6_cat_1(A,B,B) = k1_tarski(k10_cat_1(A,B)) ) ) ).
fof(t45_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v2_nattra_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ? [C] :
( v1_finset_1(C)
& C = k6_cat_1(A,B,B)
& k4_card_1(C) = np__1 ) )
& ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ( B != C
=> k6_cat_1(A,B,C) = k1_xboole_0 ) ) ) ) ) ) ).
fof(t46_nattra_1,axiom,
! [A,B] : v2_nattra_1(k8_cat_1(A,B)) ).
fof(t47_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& v2_nattra_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> v2_nattra_1(B) ) ) ).
fof(t48_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( ( v2_nattra_1(A)
& v2_nattra_1(B) )
=> v2_nattra_1(k11_cat_2(A,B)) ) ) ) ).
fof(t49_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& v2_nattra_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m2_cat_1(D,B,A)
=> ( r1_nattra_1(B,A,C,D)
=> r4_nattra_1(u2_cat_1(B),u2_cat_1(A),u2_cat_1(B),u2_cat_1(A),C,D) ) ) ) ) ) ).
fof(t50_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& v2_nattra_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m1_nattra_1(D,B,A,C,C)
=> r4_nattra_1(u1_cat_1(B),u2_cat_1(A),u1_cat_1(B),u2_cat_1(A),D,k7_nattra_1(B,A,C)) ) ) ) ) ).
fof(t51_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( v2_nattra_1(A)
=> v2_nattra_1(k12_nattra_1(B,A)) ) ) ) ).
fof(t52_nattra_1,axiom,
! [A] :
( ( v1_cat_1(A)
& v2_cat_1(A)
& l1_cat_1(A) )
=> ( v2_nattra_1(A)
=> k13_nattra_1(A) = A ) ) ).
fof(t53_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k13_nattra_1(k13_nattra_1(A)) = k13_nattra_1(A) ) ).
fof(t54_nattra_1,axiom,
! [A,B] : k13_nattra_1(k8_cat_1(A,B)) = k8_cat_1(A,B) ).
fof(t55_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> k13_nattra_1(k11_cat_2(A,B)) = k11_cat_2(k13_nattra_1(A),k13_nattra_1(B)) ) ) ).
fof(s1_nattra_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_nattra_1)
=> ~ r1_xboole_0(f2_s1_nattra_1,f3_s1_nattra_1(A)) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s1_nattra_1,f2_s1_nattra_1)
& m2_relset_1(A,f1_s1_nattra_1,f2_s1_nattra_1)
& ! [B] :
( m1_subset_1(B,f1_s1_nattra_1)
=> r2_hidden(k8_funct_2(f1_s1_nattra_1,f2_s1_nattra_1,A,B),f3_s1_nattra_1(B)) ) ) ) ).
fof(dt_m1_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ( v1_funct_1(E)
& v1_funct_2(E,u1_cat_1(A),u2_cat_1(B))
& m2_relset_1(E,u1_cat_1(A),u2_cat_1(B)) ) ) ) ).
fof(existence_m1_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ? [E] : m1_nattra_1(E,A,B,C,D) ) ).
fof(dt_m2_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> m1_nattra_1(E,A,B,C,D) ) ) ).
fof(existence_m2_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ? [E] : m2_nattra_1(E,A,B,C,D) ) ).
fof(dt_m3_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ! [E] :
( m3_nattra_1(E,A,B,C,D)
=> m2_nattra_1(E,A,B,C,D) ) ) ).
fof(existence_m3_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ? [E] : m3_nattra_1(E,A,B,C,D) ) ).
fof(dt_m4_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m4_nattra_1(C,A,B)
=> ~ v1_xboole_0(C) ) ) ).
fof(existence_m4_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m4_nattra_1(C,A,B) ) ).
fof(reflexivity_r1_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> r1_nattra_1(A,B,C,C) ) ).
fof(reflexivity_r2_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> r2_nattra_1(A,B,C,C) ) ).
fof(reflexivity_r3_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> r3_nattra_1(A,B,C,C) ) ).
fof(symmetry_r4_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& v1_funct_2(F,C,D)
& m1_relset_1(F,C,D) )
=> ( r4_nattra_1(A,B,C,D,E,F)
=> r4_nattra_1(A,B,C,D,F,E) ) ) ).
fof(reflexivity_r4_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& v1_funct_2(F,C,D)
& m1_relset_1(F,C,D) )
=> r4_nattra_1(A,B,C,D,E,E) ) ).
fof(redefinition_r4_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& v1_funct_2(F,C,D)
& m1_relset_1(F,C,D) )
=> ( r4_nattra_1(A,B,C,D,E,F)
<=> E = F ) ) ).
fof(dt_k1_nattra_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k1_nattra_1(A,B,C,D))
& v1_funct_2(k1_nattra_1(A,B,C,D),D,B)
& m2_relset_1(k1_nattra_1(A,B,C,D),D,B) ) ) ).
fof(redefinition_k1_nattra_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> k1_nattra_1(A,B,C,D) = k7_relat_1(C,D) ) ).
fof(dt_k2_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& v1_funct_1(E)
& m1_relset_1(E,k12_mcart_1(A,A,C,C),C)
& v1_funct_1(F)
& m1_relset_1(F,k12_mcart_1(B,B,D,D),D) )
=> ( v1_funct_1(k2_nattra_1(A,B,C,D,E,F))
& m2_relset_1(k2_nattra_1(A,B,C,D,E,F),k12_mcart_1(k2_zfmisc_1(A,B),k2_zfmisc_1(A,B),k12_mcart_1(A,B,C,D),k12_mcart_1(A,B,C,D)),k12_mcart_1(A,B,C,D)) ) ) ).
fof(redefinition_k2_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& v1_funct_1(E)
& m1_relset_1(E,k12_mcart_1(A,A,C,C),C)
& v1_funct_1(F)
& m1_relset_1(F,k12_mcart_1(B,B,D,D),D) )
=> k2_nattra_1(A,B,C,D,E,F) = k3_funct_4(E,F) ) ).
fof(dt_k3_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m1_subset_1(D,u1_cat_1(A))
& m1_subset_1(E,u1_cat_1(A))
& m1_cat_1(F,A,D,E) )
=> m1_cat_1(k3_nattra_1(A,B,C,D,E,F),B,k13_cat_1(A,B,C,D),k13_cat_1(A,B,C,E)) ) ).
fof(dt_k4_nattra_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> m1_nattra_1(k4_nattra_1(A,B,C),A,B,C,C) ) ).
fof(dt_k5_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m1_nattra_1(E,A,B,C,D)
& m1_subset_1(F,u1_cat_1(A)) )
=> m1_cat_1(k5_nattra_1(A,B,C,D,E,F),B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,D,F)) ) ).
fof(dt_k6_nattra_1,axiom,
! [A,B,C,D,E,F,G] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m2_cat_1(E,A,B)
& m1_nattra_1(F,A,B,C,D)
& m1_nattra_1(G,A,B,D,E) )
=> m1_nattra_1(k6_nattra_1(A,B,C,D,E,F,G),A,B,C,E) ) ).
fof(dt_k7_nattra_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> m2_nattra_1(k7_nattra_1(A,B,C),A,B,C,C) ) ).
fof(redefinition_k7_nattra_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> k7_nattra_1(A,B,C) = k4_nattra_1(A,B,C) ) ).
fof(dt_k8_nattra_1,axiom,
! [A,B,C,D,E,F,G] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m2_cat_1(E,A,B)
& m2_nattra_1(F,A,B,C,D)
& m2_nattra_1(G,A,B,D,E) )
=> m2_nattra_1(k8_nattra_1(A,B,C,D,E,F,G),A,B,C,E) ) ).
fof(dt_k9_nattra_1,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m1_nattra_1(E,A,B,C,D) )
=> m1_nattra_1(k9_nattra_1(A,B,C,D,E),A,B,D,C) ) ).
fof(dt_k10_nattra_1,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m2_nattra_1(E,A,B,C,D) )
=> m2_nattra_1(k10_nattra_1(A,B,C,D,E),A,B,D,C) ) ).
fof(dt_k11_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m4_nattra_1(k11_nattra_1(A,B),A,B) ) ).
fof(dt_k12_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v1_cat_1(k12_nattra_1(A,B))
& v2_cat_1(k12_nattra_1(A,B))
& l1_cat_1(k12_nattra_1(A,B)) ) ) ).
fof(dt_k13_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v1_cat_1(k13_nattra_1(A))
& v2_nattra_1(k13_nattra_1(A))
& m3_cat_2(k13_nattra_1(A),A) ) ) ).
fof(d20_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v1_cat_1(B)
& v2_nattra_1(B)
& m3_cat_2(B,A) )
=> ( B = k13_nattra_1(A)
<=> ( u1_cat_1(B) = u1_cat_1(A)
& u2_cat_1(B) = a_1_0_nattra_1(A) ) ) ) ) ).
fof(fraenkel_a_1_0_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( r2_hidden(A,a_1_0_nattra_1(B))
<=> ? [C] :
( m1_subset_1(C,u1_cat_1(B))
& A = k10_cat_1(B,C) ) ) ) ).
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