SET007 Axioms: SET007+579.ax

```%------------------------------------------------------------------------------
% File     : SET007+579 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Public-Key Cryptography and Pepin's Test
% Version  : [Urb08] axioms.
% English  : Public-Key Cryptography and Pepin's Test for the Primality of
%            Fermat Numbers

% 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    : pepin [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   77 (  14 unit)
%            Number of atoms       :  315 (  66 equality)
%            Maximal formula depth :   18 (   6 average)
%            Number of connectives :  285 (  47 ~  ;  20  |;  47  &)
%                                         (   6 <=>; 165 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-3 arity)
%            Number of functors    :   32 (  11 constant; 0-3 arity)
%            Number of variables   :  133 (   0 singleton; 133 !;   0 ?)
%            Maximal term depth    :    5 (   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(t1_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_int_2(A,k1_nat_1(A,np__1)) ) )).

fof(t2_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(A)
& ~ r2_int_2(B,A)
& k6_nat_1(B,A) != A ) ) ) )).

fof(t3_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,k2_nat_1(B,C))
& r2_int_2(B,A) )
=> r1_nat_1(A,C) ) ) ) ) )).

fof(t4_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(C,B)
& r2_int_2(A,C) )
=> r1_nat_1(k2_nat_1(A,C),B) ) ) ) ) )).

fof(t5_pepin,axiom,(
! [A] :
( v1_int_1(A)
=> ( ~ r1_xreal_0(A,np__1)
=> k6_int_1(np__1,A) = np__1 ) ) )).

fof(t6_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B != np__0
=> ( r1_nat_1(B,A)
<=> k4_nat_1(A,B) = np__0 ) ) ) ) )).

fof(t7_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,k4_nat_1(B,A))
=> ( A = np__0
| r1_nat_1(A,B) ) ) ) ) )).

fof(t8_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k4_nat_1(B,A) = C
=> ( r1_xreal_0(A,np__0)
| r2_int_1(A,k6_xcmplx_0(B,C)) ) ) ) ) ) )).

fof(t9_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( v1_int_2(B)
=> ( k2_nat_1(A,B) = np__0
| r1_xreal_0(B,k4_nat_1(C,k2_nat_1(A,B)))
| k4_nat_1(C,k2_nat_1(A,B)) = k4_nat_1(C,B) ) ) ) ) ) )).

fof(t10_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> k6_int_1(k2_xcmplx_0(k3_xcmplx_0(B,A),np__1),A) = k4_nat_1(np__1,A) ) ) )).

fof(t11_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k4_nat_1(k2_nat_1(B,C),A) = k4_nat_1(C,A)
& r2_int_2(C,A) )
=> ( r1_xreal_0(A,np__1)
| k4_nat_1(B,A) = np__1 ) ) ) ) ) )).

fof(t12_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k4_nat_1(k3_euler_2(A,B),C) = k4_nat_1(k3_euler_2(k4_nat_1(A,C),B),C) ) ) ) )).

fof(t13_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( A != np__0
=> k4_nat_1(k1_pepin(A),k1_nat_1(A,np__1)) = np__1 ) ) )).

fof(t14_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k4_nat_1(C,B) = A
=> ( r1_xreal_0(B,k1_pepin(A))
| k4_nat_1(k1_pepin(C),B) = k1_pepin(A) ) ) ) ) ) )).

fof(t15_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( v1_int_2(A)
& k4_nat_1(B,A) = k4_xcmplx_0(np__1) )
=> k4_nat_1(k1_pepin(B),A) = np__1 ) ) ) )).

fof(t16_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( v1_abian(A)
& v1_abian(k1_nat_1(A,np__1)) ) ) )).

fof(t17_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__2)
& v1_int_2(A)
& v1_abian(A) ) ) )).

fof(t18_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> v1_abian(k3_series_1(np__2,A)) ) ) )).

fof(t19_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ v1_abian(A)
& ~ v1_abian(B)
& v1_abian(k2_nat_1(A,B)) ) ) ) )).

fof(t20_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ v1_abian(A)
& v1_abian(k3_euler_2(A,B)) ) ) ) )).

fof(t21_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( v1_abian(B)
=> ( r1_xreal_0(A,np__0)
| v1_abian(k3_euler_2(B,A)) ) ) ) ) )).

fof(t22_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_nat_1(np__2,A)
<=> v1_abian(A) ) ) )).

fof(t23_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( v1_abian(k2_nat_1(A,B))
& ~ v1_abian(A)
& ~ v1_abian(B) ) ) ) )).

fof(t24_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_euler_2(A,np__2) = k1_pepin(A) ) )).

fof(t25_pepin,axiom,(
\$true )).

fof(t26_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(k3_euler_2(A,B),np__1) ) ) ) )).

fof(t27_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& B != np__0
& k3_euler_2(A,B) != k2_nat_1(A,k3_euler_2(A,k5_binarith(B,np__1))) ) ) ) )).

fof(t28_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k4_nat_1(B,np__2) = np__0
=> k1_pepin(k3_euler_2(A,k3_nat_1(B,np__2))) = k3_euler_2(A,B) ) ) ) )).

fof(t29_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( A = np__0
| k3_nat_1(k3_euler_2(A,B),A) = k3_euler_2(A,k5_binarith(B,np__1)) ) ) ) ) )).

fof(t30_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_euler_2(np__2,k1_nat_1(A,np__1)) = k1_nat_1(k3_euler_2(np__2,A),k3_euler_2(np__2,A)) ) )).

fof(t31_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k3_euler_2(A,B) = k3_euler_2(A,C)
=> ( r1_xreal_0(A,np__1)
| B = C ) ) ) ) ) )).

fof(t32_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
<=> r1_nat_1(k3_euler_2(np__2,A),k3_euler_2(np__2,B)) ) ) ) )).

fof(t33_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(A)
& r1_nat_1(B,k3_euler_2(A,C))
& B != np__1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> B != k2_nat_1(A,D) ) ) ) ) ) )).

fof(t34_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( v1_int_2(A)
=> ( C = np__0
| r1_xreal_0(k3_euler_2(A,k1_nat_1(B,np__1)),C)
| ( r1_nat_1(C,k3_euler_2(A,k1_nat_1(B,np__1)))
<=> r1_nat_1(C,k3_euler_2(A,B)) ) ) ) ) ) ) )).

fof(t35_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(A)
& r1_nat_1(B,k3_euler_2(A,C))
& B != np__0
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( B = k3_euler_2(A,D)
& r1_xreal_0(D,C) ) ) ) ) ) ) )).

fof(t36_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k4_nat_1(B,A) = np__1
=> ( r1_xreal_0(A,np__1)
| k4_nat_1(k3_euler_2(B,C),A) = np__1 ) ) ) ) ) )).

fof(t37_pepin,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k4_nat_1(k2_newton(B,A),B) = np__0 ) ) ) )).

fof(t38_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( v1_int_2(A)
& r2_int_2(B,A) )
=> k4_nat_1(k3_euler_2(B,k5_binarith(A,np__1)),A) = np__1 ) ) ) )).

fof(t39_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( v1_int_2(A)
& r1_nat_1(B,k3_euler_2(A,C)) )
=> ( r1_xreal_0(B,np__1)
| r1_nat_1(B,k3_nat_1(k3_euler_2(A,C),A))
| B = k3_euler_2(A,C) ) ) ) ) ) )).

fof(t40_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__1)
=> ( k4_nat_1(A,B) = np__1
<=> r1_int_1(A,np__1,B) ) ) ) ) )).

fof(t41_pepin,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(A,B,C)
=> r1_int_1(k2_pepin(A),k2_pepin(B),C) ) ) ) ) )).

fof(t42_pepin,axiom,(
\$true )).

fof(t43_pepin,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ( ( r1_int_1(A,B,C)
& r1_int_1(A,D,C) )
=> r1_int_1(B,D,C) ) ) ) ) ) )).

fof(t44_pepin,axiom,(
v1_int_2(np__3) )).

fof(t45_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& k1_euler_1(A) = np__0 ) ) )).

fof(t46_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& r1_xreal_0(A,k4_xcmplx_0(A)) ) ) )).

fof(t47_pepin,axiom,(
\$true )).

fof(t48_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( A != np__0
=> k3_nat_1(A,A) = np__1 ) ) )).

fof(d1_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_pepin(A,B,C) = k4_nat_1(k3_euler_2(B,A),C) ) ) ) )).

fof(t49_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( v1_int_2(A)
& v1_int_2(B)
& C = k2_nat_1(A,B)
& r2_int_2(D,k1_euler_1(C))
& k4_nat_1(k2_nat_1(D,E),k1_euler_1(C)) = np__1 )
=> ( A = B
| ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,F)
=> k3_pepin(E,k3_pepin(D,F,C),C) = F ) ) ) ) ) ) ) ) ) )).

fof(d2_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_int_2(A,B)
=> ( r1_xreal_0(B,np__1)
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k4_pepin(A,B)
<=> ( ~ r1_xreal_0(C,np__0)
& k4_nat_1(k3_euler_2(A,C),B) = np__1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k4_nat_1(k3_euler_2(A,D),B) = np__1
=> ( r1_xreal_0(D,np__0)
| ( ~ r1_xreal_0(C,np__0)
& r1_xreal_0(C,D) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t50_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__1)
=> k4_pepin(np__1,A) = np__1 ) ) )).

fof(t51_pepin,axiom,(
\$true )).

fof(t52_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k4_nat_1(k3_euler_2(C,B),A) = np__1
& r2_int_2(C,A) )
=> ( r1_xreal_0(A,np__1)
| r1_xreal_0(B,np__0)
| r1_nat_1(k4_pepin(C,A),B) ) ) ) ) ) )).

fof(t53_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_int_2(B,A)
& r1_nat_1(k4_pepin(B,A),C) )
=> ( r1_xreal_0(A,np__1)
| k4_nat_1(k3_euler_2(B,C),A) = np__1 ) ) ) ) ) )).

fof(t54_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( v1_int_2(A)
& r2_int_2(B,A) )
=> r1_nat_1(k4_pepin(B,A),k5_binarith(A,np__1)) ) ) ) )).

fof(d3_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k5_pepin(A) = k1_nat_1(k3_euler_2(np__2,k3_euler_2(np__2,A)),np__1) ) )).

fof(t55_pepin,axiom,(
k5_pepin(np__0) = np__3 )).

fof(t56_pepin,axiom,(
k5_pepin(np__1) = np__5 )).

fof(t57_pepin,axiom,(
k5_pepin(np__2) = np__17 )).

fof(t58_pepin,axiom,(
k5_pepin(np__3) = np__257 )).

fof(t59_pepin,axiom,(
k5_pepin(np__4) = k1_nat_1(k2_nat_1(np__256,np__256),np__1) )).

fof(t60_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k5_pepin(A),np__2) ) )).

fof(t61_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(A)
& ~ r1_xreal_0(A,np__2)
& r1_nat_1(A,k5_pepin(B))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> A != k1_nat_1(k2_nat_1(C,k3_euler_2(np__2,k1_nat_1(B,np__1))),np__1) ) ) ) ) )).

fof(t62_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( A != np__0
=> r2_int_2(np__3,k5_pepin(A)) ) ) )).

fof(t63_pepin,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_int_1(k3_euler_2(np__3,k3_nat_1(k5_binarith(k5_pepin(A),np__1),np__2)),k4_xcmplx_0(np__1),k5_pepin(A))
=> v1_int_2(k5_pepin(A)) ) ) )).

fof(t64_pepin,axiom,(
v1_int_2(np__5) )).

fof(t65_pepin,axiom,(
v1_int_2(np__17) )).

fof(t66_pepin,axiom,(
v1_int_2(np__257) )).

fof(t67_pepin,axiom,(
v1_int_2(k1_nat_1(k2_nat_1(np__256,np__256),np__1)) )).

fof(dt_k1_pepin,axiom,(
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k1_pepin(A),k1_numbers,k5_numbers) ) )).

fof(redefinition_k1_pepin,axiom,(
! [A] :
( m1_subset_1(A,k5_numbers)
=> k1_pepin(A) = k5_square_1(A) ) )).

fof(dt_k2_pepin,axiom,(
! [A] :
( v1_int_1(A)
=> m2_subset_1(k2_pepin(A),k1_numbers,k5_numbers) ) )).

fof(redefinition_k2_pepin,axiom,(
! [A] :
( v1_int_1(A)
=> k2_pepin(A) = k5_square_1(A) ) )).

fof(dt_k3_pepin,axiom,(
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )
=> m2_subset_1(k3_pepin(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k4_pepin,axiom,(
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k4_pepin(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k5_pepin,axiom,(
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k5_pepin(A),k1_numbers,k5_numbers) ) )).
%------------------------------------------------------------------------------
```