## SET007 Axioms: SET007+91.ax

```%------------------------------------------------------------------------------
% File     : SET007+91 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Three-Argument Operations and Four-Argument Operations
% 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    : multop_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   24 (   4 unit)
%            Number of atoms       :  229 (  20 equality)
%            Maximal formula depth :   24 (  14 average)
%            Number of connectives :  243 (  38 ~  ;   0  |;  99  &)
%                                         (   0 <=>; 106 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   1 propositional; 0-5 arity)
%            Number of functors    :   41 (  25 constant; 0-10 arity)
%            Number of variables   :  159 (   0 singleton; 147 !;  12 ?)
%            Maximal term depth    :    3 (   1 average)
% 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(d1_multop_1,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] : k1_multop_1(A,B,C,D) = k1_funct_1(A,k3_mcart_1(B,C,D)) ) )).

fof(t1_multop_1,axiom,(
\$true )).

fof(t2_multop_1,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_zfmisc_1(B,C,D),A)
& m2_relset_1(E,k3_zfmisc_1(B,C,D),A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k3_zfmisc_1(B,C,D),A)
& m2_relset_1(F,k3_zfmisc_1(B,C,D),A) )
=> ( ! [G,H,I] :
( ( r2_hidden(G,B)
& r2_hidden(H,C)
& r2_hidden(I,D) )
=> k1_funct_1(E,k3_mcart_1(G,H,I)) = k1_funct_1(F,k3_mcart_1(G,H,I)) )
=> E = F ) ) ) ) )).

fof(t3_multop_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,k3_zfmisc_1(A,B,C),D)
& m2_relset_1(E,k3_zfmisc_1(A,B,C),D) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k3_zfmisc_1(A,B,C),D)
& m2_relset_1(F,k3_zfmisc_1(A,B,C),D) )
=> ( ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,C)
=> k8_funct_2(k3_zfmisc_1(A,B,C),D,E,k4_domain_1(A,B,C,G,H,I)) = k8_funct_2(k3_zfmisc_1(A,B,C),D,F,k4_domain_1(A,B,C,G,H,I)) ) ) )
=> E = F ) ) ) ) ) ) ) )).

fof(t4_multop_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,k3_zfmisc_1(A,B,C),D)
& m2_relset_1(E,k3_zfmisc_1(A,B,C),D) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k3_zfmisc_1(A,B,C),D)
& m2_relset_1(F,k3_zfmisc_1(A,B,C),D) )
=> ( ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,C)
=> k2_multop_1(A,B,C,D,E,G,H,I) = k2_multop_1(A,B,C,D,F,G,H,I) ) ) )
=> E = F ) ) ) ) ) ) ) )).

fof(d2_multop_1,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D,E] : k3_multop_1(A,B,C,D,E) = k1_funct_1(A,k4_mcart_1(B,C,D,E)) ) )).

fof(t5_multop_1,axiom,(
\$true )).

fof(t6_multop_1,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D,E,F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k4_zfmisc_1(B,C,D,E),A)
& m2_relset_1(F,k4_zfmisc_1(B,C,D,E),A) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k4_zfmisc_1(B,C,D,E),A)
& m2_relset_1(G,k4_zfmisc_1(B,C,D,E),A) )
=> ( ! [H,I,J,K] :
( ( r2_hidden(H,B)
& r2_hidden(I,C)
& r2_hidden(J,D)
& r2_hidden(K,E) )
=> k1_funct_1(F,k4_mcart_1(H,I,J,K)) = k1_funct_1(G,k4_mcart_1(H,I,J,K)) )
=> F = G ) ) ) ) )).

fof(t7_multop_1,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
& m2_relset_1(F,k4_zfmisc_1(A,B,C,D),E) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k4_zfmisc_1(A,B,C,D),E)
& m2_relset_1(G,k4_zfmisc_1(A,B,C,D),E) )
=> ( ! [H] :
( m1_subset_1(H,A)
=> ! [I] :
( m1_subset_1(I,B)
=> ! [J] :
( m1_subset_1(J,C)
=> ! [K] :
( m1_subset_1(K,D)
=> k8_funct_2(k4_zfmisc_1(A,B,C,D),E,F,k5_domain_1(A,B,C,D,H,I,J,K)) = k8_funct_2(k4_zfmisc_1(A,B,C,D),E,G,k5_domain_1(A,B,C,D,H,I,J,K)) ) ) ) )
=> F = G ) ) ) ) ) ) ) ) )).

fof(t8_multop_1,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
& m2_relset_1(F,k4_zfmisc_1(A,B,C,D),E) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k4_zfmisc_1(A,B,C,D),E)
& m2_relset_1(G,k4_zfmisc_1(A,B,C,D),E) )
=> ( ! [H] :
( m1_subset_1(H,A)
=> ! [I] :
( m1_subset_1(I,B)
=> ! [J] :
( m1_subset_1(J,C)
=> ! [K] :
( m1_subset_1(K,D)
=> k4_multop_1(A,B,C,D,E,F,H,I,J,K) = k4_multop_1(A,B,C,D,E,G,H,I,J,K) ) ) ) )
=> F = G ) ) ) ) ) ) ) ) )).

fof(s1_multop_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_multop_1)
=> ! [B] :
( m1_subset_1(B,f2_s1_multop_1)
=> ! [C] :
( m1_subset_1(C,f3_s1_multop_1)
=> ? [D] :
( m1_subset_1(D,f4_s1_multop_1)
& p1_s1_multop_1(A,B,C,D) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k3_zfmisc_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1),f4_s1_multop_1)
& m2_relset_1(A,k3_zfmisc_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1),f4_s1_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s1_multop_1)
=> ! [C] :
( m1_subset_1(C,f2_s1_multop_1)
=> ! [D] :
( m1_subset_1(D,f3_s1_multop_1)
=> p1_s1_multop_1(B,C,D,k8_funct_2(k3_zfmisc_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1),f4_s1_multop_1,A,k4_domain_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1,B,C,D))) ) ) ) ) )).

fof(s2_multop_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s2_multop_1)
=> ! [B] :
( m1_subset_1(B,f1_s2_multop_1)
=> ! [C] :
( m1_subset_1(C,f1_s2_multop_1)
=> ? [D] :
( m1_subset_1(D,f1_s2_multop_1)
& p1_s2_multop_1(A,B,C,D) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k3_zfmisc_1(f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1),f1_s2_multop_1)
& m2_relset_1(A,k3_zfmisc_1(f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1),f1_s2_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s2_multop_1)
=> ! [C] :
( m1_subset_1(C,f1_s2_multop_1)
=> ! [D] :
( m1_subset_1(D,f1_s2_multop_1)
=> p1_s2_multop_1(B,C,D,k2_multop_1(f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1,A,B,C,D)) ) ) ) ) )).

fof(s3_multop_1,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k3_zfmisc_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1),f4_s3_multop_1)
& m2_relset_1(A,k3_zfmisc_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1),f4_s3_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s3_multop_1)
=> ! [C] :
( m1_subset_1(C,f2_s3_multop_1)
=> ! [D] :
( m1_subset_1(D,f3_s3_multop_1)
=> k8_funct_2(k3_zfmisc_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1),f4_s3_multop_1,A,k4_domain_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1,B,C,D)) = f5_s3_multop_1(B,C,D) ) ) ) ) )).

fof(s4_multop_1,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k3_zfmisc_1(f1_s4_multop_1,f2_s4_multop_1,f3_s4_multop_1),f4_s4_multop_1)
& m2_relset_1(A,k3_zfmisc_1(f1_s4_multop_1,f2_s4_multop_1,f3_s4_multop_1),f4_s4_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s4_multop_1)
=> ! [C] :
( m1_subset_1(C,f2_s4_multop_1)
=> ! [D] :
( m1_subset_1(D,f3_s4_multop_1)
=> k2_multop_1(f1_s4_multop_1,f2_s4_multop_1,f3_s4_multop_1,f4_s4_multop_1,A,B,C,D) = f5_s4_multop_1(B,C,D) ) ) ) ) )).

fof(s5_multop_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s5_multop_1)
=> ! [B] :
( m1_subset_1(B,f2_s5_multop_1)
=> ! [C] :
( m1_subset_1(C,f3_s5_multop_1)
=> ! [D] :
( m1_subset_1(D,f4_s5_multop_1)
=> ? [E] :
( m1_subset_1(E,f5_s5_multop_1)
& p1_s5_multop_1(A,B,C,D,E) ) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k4_zfmisc_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1),f5_s5_multop_1)
& m2_relset_1(A,k4_zfmisc_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1),f5_s5_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s5_multop_1)
=> ! [C] :
( m1_subset_1(C,f2_s5_multop_1)
=> ! [D] :
( m1_subset_1(D,f3_s5_multop_1)
=> ! [E] :
( m1_subset_1(E,f4_s5_multop_1)
=> p1_s5_multop_1(B,C,D,E,k8_funct_2(k4_zfmisc_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1),f5_s5_multop_1,A,k5_domain_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1,B,C,D,E))) ) ) ) ) ) )).

fof(s6_multop_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s6_multop_1)
=> ! [B] :
( m1_subset_1(B,f1_s6_multop_1)
=> ! [C] :
( m1_subset_1(C,f1_s6_multop_1)
=> ! [D] :
( m1_subset_1(D,f1_s6_multop_1)
=> ? [E] :
( m1_subset_1(E,f1_s6_multop_1)
& p1_s6_multop_1(A,B,C,D,E) ) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k4_zfmisc_1(f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1),f1_s6_multop_1)
& m2_relset_1(A,k4_zfmisc_1(f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1),f1_s6_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s6_multop_1)
=> ! [C] :
( m1_subset_1(C,f1_s6_multop_1)
=> ! [D] :
( m1_subset_1(D,f1_s6_multop_1)
=> ! [E] :
( m1_subset_1(E,f1_s6_multop_1)
=> p1_s6_multop_1(B,C,D,E,k4_multop_1(f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,A,B,C,D,E)) ) ) ) ) ) )).

fof(s7_multop_1,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k4_zfmisc_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1),f5_s7_multop_1)
& m2_relset_1(A,k4_zfmisc_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1),f5_s7_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s7_multop_1)
=> ! [C] :
( m1_subset_1(C,f2_s7_multop_1)
=> ! [D] :
( m1_subset_1(D,f3_s7_multop_1)
=> ! [E] :
( m1_subset_1(E,f4_s7_multop_1)
=> k8_funct_2(k4_zfmisc_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1),f5_s7_multop_1,A,k5_domain_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1,B,C,D,E)) = f6_s7_multop_1(B,C,D,E) ) ) ) ) ) )).

fof(s8_multop_1,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k4_zfmisc_1(f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1),f1_s8_multop_1)
& m2_relset_1(A,k4_zfmisc_1(f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1),f1_s8_multop_1)
& ! [B] :
( m1_subset_1(B,f1_s8_multop_1)
=> ! [C] :
( m1_subset_1(C,f1_s8_multop_1)
=> ! [D] :
( m1_subset_1(D,f1_s8_multop_1)
=> ! [E] :
( m1_subset_1(E,f1_s8_multop_1)
=> k4_multop_1(f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,A,B,C,D,E) = f2_s8_multop_1(B,C,D,E) ) ) ) ) ) )).

fof(dt_k1_multop_1,axiom,(
\$true )).

fof(dt_k2_multop_1,axiom,(
! [A,B,C,D,E,F,G,H] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k3_zfmisc_1(A,B,C),D)
& m1_relset_1(E,k3_zfmisc_1(A,B,C),D)
& m1_subset_1(F,A)
& m1_subset_1(G,B)
& m1_subset_1(H,C) )
=> m1_subset_1(k2_multop_1(A,B,C,D,E,F,G,H),D) ) )).

fof(redefinition_k2_multop_1,axiom,(
! [A,B,C,D,E,F,G,H] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k3_zfmisc_1(A,B,C),D)
& m1_relset_1(E,k3_zfmisc_1(A,B,C),D)
& m1_subset_1(F,A)
& m1_subset_1(G,B)
& m1_subset_1(H,C) )
=> k2_multop_1(A,B,C,D,E,F,G,H) = k1_multop_1(E,F,G,H) ) )).

fof(dt_k3_multop_1,axiom,(
\$true )).

fof(dt_k4_multop_1,axiom,(
! [A,B,C,D,E,F,G,H,I,J] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& ~ v1_xboole_0(E)
& v1_funct_1(F)
& v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
& m1_relset_1(F,k4_zfmisc_1(A,B,C,D),E)
& m1_subset_1(G,A)
& m1_subset_1(H,B)
& m1_subset_1(I,C)
& m1_subset_1(J,D) )
=> m1_subset_1(k4_multop_1(A,B,C,D,E,F,G,H,I,J),E) ) )).

fof(redefinition_k4_multop_1,axiom,(
! [A,B,C,D,E,F,G,H,I,J] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& ~ v1_xboole_0(E)
& v1_funct_1(F)
& v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
& m1_relset_1(F,k4_zfmisc_1(A,B,C,D),E)
& m1_subset_1(G,A)
& m1_subset_1(H,B)
& m1_subset_1(I,C)
& m1_subset_1(J,D) )
=> k4_multop_1(A,B,C,D,E,F,G,H,I,J) = k3_multop_1(F,G,H,I,J) ) )).
%------------------------------------------------------------------------------
```