## SET007 Axioms: SET007+212.ax

```%------------------------------------------------------------------------------
% File     : SET007+212 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Groups
% 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    : group_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  138 (  27 unit)
%            Number of atoms       :  816 ( 139 equality)
%            Maximal formula depth :   16 (   7 average)
%            Number of connectives :  791 ( 113 ~  ;   5  |; 356  &)
%                                         (  19 <=>; 298 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   34 (   1 propositional; 0-3 arity)
%            Number of functors    :   35 (   7 constant; 0-6 arity)
%            Number of variables   :  297 (   0 singleton; 283 !;  14 ?)
%            Maximal term depth    :    4 (   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(rc1_group_1,axiom,(
? [A] :
( l1_group_1(A)
& v1_group_1(A) ) )).

fof(rc2_group_1,axiom,(
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A) ) )).

fof(fc1_group_1,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( ~ v3_struct_0(g1_group_1(A,B))
& v1_group_1(g1_group_1(A,B)) ) ) )).

fof(cc1_group_1,axiom,(
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A) ) ) ) )).

fof(rc3_group_1,axiom,(
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A) ) )).

fof(rc4_group_1,axiom,(
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A) ) )).

fof(fc2_group_1,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,A) )
=> ( ~ v3_struct_0(g1_rlvect_1(A,B,C))
& v1_rlvect_1(g1_rlvect_1(A,B,C)) ) ) )).

fof(d1_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,C) ) ) ) )).

fof(d2_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v2_group_1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = C
& k1_group_1(A,B,C) = C ) ) ) ) ) )).

fof(d3_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v3_group_1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = C
& k1_group_1(A,B,C) = C
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,C,D) = B
& k1_group_1(A,D,C) = B ) ) ) ) ) ) )).

fof(d4_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v4_group_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ) )).

fof(t1_group_1,axiom,(
\$true )).

fof(t2_group_1,axiom,(
\$true )).

fof(t3_group_1,axiom,(
\$true )).

fof(t4_group_1,axiom,(
\$true )).

fof(t5_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = C
& k1_group_1(A,B,C) = C
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,C,D) = B
& k1_group_1(A,D,C) = B ) ) ) )
| ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) ) ) ) ) )).

fof(t6_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,B,D) = C )
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,D,B) = C ) ) ) ) )
=> ( v4_group_1(A)
& v3_group_1(A) ) ) ) )).

fof(t7_group_1,axiom,
( v4_group_1(g1_group_1(k1_numbers,k33_binop_2))
& v3_group_1(g1_group_1(k1_numbers,k33_binop_2)) )).

fof(d5_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v2_group_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B = k2_group_1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = C
& k1_group_1(A,B,C) = C ) ) ) ) ) ) )).

fof(t8_group_1,axiom,(
\$true )).

fof(t9_group_1,axiom,(
\$true )).

fof(t10_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = C
& k1_group_1(A,B,C) = C ) )
=> B = k2_group_1(A) ) ) ) )).

fof(d6_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k3_group_1(A,B)
<=> ( k1_group_1(A,B,C) = k2_group_1(A)
& k1_group_1(A,C,B) = k2_group_1(A) ) ) ) ) ) )).

fof(t11_group_1,axiom,(
\$true )).

fof(t12_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( k1_group_1(A,B,C) = k2_group_1(A)
& k1_group_1(A,C,B) = k2_group_1(A) )
=> C = k3_group_1(A,B) ) ) ) ) )).

fof(t13_group_1,axiom,(
\$true )).

fof(t14_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
| k1_group_1(A,C,B) = k1_group_1(A,D,B) )
=> C = D ) ) ) ) ) )).

fof(t15_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( k1_group_1(A,B,C) = B
| k1_group_1(A,C,B) = B )
=> C = k2_group_1(A) ) ) ) ) )).

fof(t16_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k3_group_1(A,k2_group_1(A)) = k2_group_1(A) ) )).

fof(t17_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k3_group_1(A,B) = k3_group_1(A,C)
=> B = C ) ) ) ) )).

fof(t18_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k3_group_1(A,B) = k2_group_1(A)
=> B = k2_group_1(A) ) ) ) )).

fof(t19_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_group_1(A,k3_group_1(A,B)) = B ) ) )).

fof(t20_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( k1_group_1(A,B,C) = k2_group_1(A)
| k1_group_1(A,C,B) = k2_group_1(A) )
=> ( B = k3_group_1(A,C)
& C = k3_group_1(A,B) ) ) ) ) ) )).

fof(t21_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = D
<=> C = k1_group_1(A,k3_group_1(A,B),D) ) ) ) ) ) )).

fof(t22_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = D
<=> B = k1_group_1(A,D,k3_group_1(A,C)) ) ) ) ) ) )).

fof(t23_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,B,D) = C ) ) ) ) )).

fof(t24_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,D,B) = C ) ) ) ) )).

fof(t25_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_group_1(A,k1_group_1(A,B,C)) = k1_group_1(A,k3_group_1(A,C),k3_group_1(A,B)) ) ) ) )).

fof(t26_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k1_group_1(A,C,B)
<=> k3_group_1(A,k1_group_1(A,B,C)) = k1_group_1(A,k3_group_1(A,B),k3_group_1(A,C)) ) ) ) ) )).

fof(t27_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k1_group_1(A,C,B)
<=> k1_group_1(A,k3_group_1(A,B),k3_group_1(A,C)) = k1_group_1(A,k3_group_1(A,C),k3_group_1(A,B)) ) ) ) ) )).

fof(t28_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k1_group_1(A,C,B)
<=> k1_group_1(A,B,k3_group_1(A,C)) = k1_group_1(A,k3_group_1(A,C),B) ) ) ) ) )).

fof(d7_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( B = k4_group_1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = k3_group_1(A,C) ) ) ) ) )).

fof(t29_group_1,axiom,(
\$true )).

fof(t30_group_1,axiom,(
\$true )).

fof(t31_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_group_1(A) )
=> v2_binop_1(u1_group_1(A),u1_struct_0(A)) ) )).

fof(t32_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> r3_binop_1(u1_struct_0(A),k2_group_1(A),u1_group_1(A)) ) )).

fof(t33_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k2_group_1(A) ) )).

fof(t34_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> v1_setwiseo(u1_group_1(A),u1_struct_0(A)) ) )).

fof(t35_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> r1_finseqop(u1_struct_0(A),k4_group_1(A),u1_group_1(A)) ) )).

fof(t36_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> v1_finseqop(u1_group_1(A),u1_struct_0(A)) ) )).

fof(t37_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k6_finseqop(u1_struct_0(A),u1_group_1(A)) = k4_group_1(A) ) )).

fof(d8_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) )
=> ( B = k5_group_1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,np__0) = k2_group_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,k1_nat_1(D,np__1)) = k1_group_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,D),C) ) ) ) ) ) ) )).

fof(d9_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( r1_xreal_0(np__0,B)
=> k6_group_1(A,B,C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,k1_int_2(B)) )
& ( ~ r1_xreal_0(np__0,B)
=> k6_group_1(A,B,C) = k3_group_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,k1_int_2(B))) ) ) ) ) ) )).

fof(d10_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k6_group_1(A,B,C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,B) ) ) ) )).

fof(t38_group_1,axiom,(
\$true )).

fof(t39_group_1,axiom,(
\$true )).

fof(t40_group_1,axiom,(
\$true )).

fof(t41_group_1,axiom,(
\$true )).

fof(t42_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> k6_group_1(B,A,k2_group_1(B)) = k2_group_1(B) ) ) )).

fof(t43_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_group_1(A,np__0,B) = k2_group_1(A) ) ) )).

fof(t44_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_group_1(A,np__1,B) = B ) ) )).

fof(t45_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_group_1(A,np__2,B) = k1_group_1(A,B,B) ) ) )).

fof(t46_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_group_1(A,np__3,B) = k1_group_1(A,k1_group_1(A,B,B),B) ) ) )).

fof(t47_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k6_group_1(A,np__2,B) = k2_group_1(A)
<=> k3_group_1(A,B) = B ) ) ) )).

fof(t48_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k1_nat_1(A,B),D) = k1_group_1(C,k6_group_1(C,A,D),k6_group_1(C,B,D)) ) ) ) ) )).

fof(t49_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k6_group_1(B,k1_nat_1(A,np__1),C) = k1_group_1(B,k6_group_1(B,A,C),C)
& k6_group_1(B,k1_nat_1(A,np__1),C) = k1_group_1(B,C,k6_group_1(B,A,C)) ) ) ) ) )).

fof(t50_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k2_nat_1(A,B),D) = k6_group_1(C,B,k6_group_1(C,A,D)) ) ) ) ) )).

fof(t51_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k6_group_1(B,A,k3_group_1(B,C)) = k3_group_1(B,k6_group_1(B,A,C)) ) ) ) )).

fof(t52_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( k1_group_1(B,C,D) = k1_group_1(B,D,C)
=> k1_group_1(B,C,k6_group_1(B,A,D)) = k1_group_1(B,k6_group_1(B,A,D),C) ) ) ) ) ) )).

fof(t53_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ( k1_group_1(C,D,E) = k1_group_1(C,E,D)
=> k1_group_1(C,k6_group_1(C,A,D),k6_group_1(C,B,E)) = k1_group_1(C,k6_group_1(C,B,E),k6_group_1(C,A,D)) ) ) ) ) ) ) )).

fof(t54_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( k1_group_1(B,C,D) = k1_group_1(B,D,C)
=> k6_group_1(B,A,k1_group_1(B,C,D)) = k1_group_1(B,k6_group_1(B,A,C),k6_group_1(B,A,D)) ) ) ) ) ) )).

fof(t55_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( r1_xreal_0(np__0,A)
=> k6_group_1(B,A,C) = k6_group_1(B,k1_int_2(A),C) ) ) ) ) )).

fof(t56_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( ~ r1_xreal_0(np__0,A)
=> k6_group_1(B,A,C) = k3_group_1(B,k6_group_1(B,k1_int_2(A),C)) ) ) ) ) )).

fof(t57_group_1,axiom,(
\$true )).

fof(t58_group_1,axiom,(
\$true )).

fof(t59_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( A = np__0
=> k6_group_1(B,A,C) = k2_group_1(B) ) ) ) ) )).

fof(t60_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( r1_xreal_0(A,np__0)
=> k6_group_1(B,A,C) = k3_group_1(B,k6_group_1(B,k1_int_2(A),C)) ) ) ) ) )).

fof(t61_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> k6_group_1(B,A,k2_group_1(B)) = k2_group_1(B) ) ) )).

fof(t62_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_group_1(A,k7_binop_2(np__1),B) = k3_group_1(A,B) ) ) )).

fof(t63_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k2_xcmplx_0(A,B),D) = k1_group_1(C,k6_group_1(C,A,D),k6_group_1(C,B,D)) ) ) ) ) )).

fof(t64_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k2_xcmplx_0(A,B),D) = k1_group_1(C,k6_group_1(C,A,D),k6_group_1(C,B,D)) ) ) ) ) )).

fof(t65_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k2_xcmplx_0(B,A),D) = k1_group_1(C,k6_group_1(C,B,D),k6_group_1(C,A,D)) ) ) ) ) )).

fof(t66_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k6_group_1(B,k2_xcmplx_0(A,np__1),C) = k1_group_1(B,k6_group_1(B,A,C),C)
& k6_group_1(B,k2_xcmplx_0(A,np__1),C) = k1_group_1(B,C,k6_group_1(B,A,C)) ) ) ) ) )).

fof(t67_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k3_xcmplx_0(A,B),D) = k6_group_1(C,B,k6_group_1(C,A,D)) ) ) ) ) )).

fof(t68_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k3_xcmplx_0(A,B),D) = k6_group_1(C,B,k6_group_1(C,A,D)) ) ) ) ) )).

fof(t69_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k6_group_1(C,k3_xcmplx_0(B,A),D) = k6_group_1(C,A,k6_group_1(C,B,D)) ) ) ) ) )).

fof(t70_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k6_group_1(B,k4_xcmplx_0(A),C) = k3_group_1(B,k6_group_1(B,A,C)) ) ) ) )).

fof(t71_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k6_group_1(B,k7_binop_2(A),C) = k3_group_1(B,k6_group_1(B,A,C)) ) ) ) )).

fof(t72_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k6_group_1(B,A,k3_group_1(B,C)) = k3_group_1(B,k6_group_1(B,A,C)) ) ) ) )).

fof(t73_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( k1_group_1(B,C,D) = k1_group_1(B,D,C)
=> k6_group_1(B,A,k1_group_1(B,C,D)) = k1_group_1(B,k6_group_1(B,A,C),k6_group_1(B,A,D)) ) ) ) ) ) )).

fof(t74_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ( k1_group_1(C,D,E) = k1_group_1(C,E,D)
=> k1_group_1(C,k6_group_1(C,A,D),k6_group_1(C,B,E)) = k1_group_1(C,k6_group_1(C,B,E),k6_group_1(C,A,D)) ) ) ) ) ) ) )).

fof(t75_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ( k1_group_1(C,D,E) = k1_group_1(C,E,D)
=> k1_group_1(C,k6_group_1(C,A,D),k6_group_1(C,B,E)) = k1_group_1(C,k6_group_1(C,B,E),k6_group_1(C,A,D)) ) ) ) ) ) ) )).

fof(t76_group_1,axiom,(
\$true )).

fof(t77_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( k1_group_1(B,C,D) = k1_group_1(B,D,C)
=> k1_group_1(B,C,k6_group_1(B,A,D)) = k1_group_1(B,k6_group_1(B,A,D),C) ) ) ) ) ) )).

fof(d11_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v5_group_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k6_group_1(A,C,B) = k2_group_1(A)
=> C = np__0 ) ) ) ) ) )).

fof(t78_group_1,axiom,(
\$true )).

fof(t79_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ~ v5_group_1(k2_group_1(A),A) ) )).

fof(d12_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( v5_group_1(B,A)
=> ( C = k7_group_1(A,B)
<=> C = np__0 ) )
& ( ~ v5_group_1(B,A)
=> ( C = k7_group_1(A,B)
<=> ( k6_group_1(A,C,B) = k2_group_1(A)
& C != np__0
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k6_group_1(A,D,B) = k2_group_1(A)
=> ( D = np__0
| r1_xreal_0(C,D) ) ) ) ) ) ) ) ) ) ) )).

fof(t80_group_1,axiom,(
\$true )).

fof(t81_group_1,axiom,(
\$true )).

fof(t82_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_group_1(A,k7_group_1(A,B),B) = k2_group_1(A) ) ) )).

fof(t83_group_1,axiom,(
\$true )).

fof(t84_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k7_group_1(A,k2_group_1(A)) = np__1 ) )).

fof(t85_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k7_group_1(A,B) = np__1
=> B = k2_group_1(A) ) ) ) )).

fof(t86_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k6_group_1(B,A,C) = k2_group_1(B)
=> r1_nat_1(k7_group_1(B,C),A) ) ) ) ) )).

fof(d13_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k8_group_1(A) = k1_card_1(u1_struct_0(A)) ) )).

fof(d14_group_1,axiom,(
! [A] :
( l1_struct_0(A)
=> ( v6_group_1(A)
<=> v1_finset_1(u1_struct_0(A)) ) ) )).

fof(d15_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v6_group_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k9_group_1(A)
<=> ? [C] :
( v1_finset_1(C)
& C = u1_struct_0(A)
& B = k4_card_1(C) ) ) ) ) ) )).

fof(t87_group_1,axiom,(
\$true )).

fof(t88_group_1,axiom,(
\$true )).

fof(t89_group_1,axiom,(
\$true )).

fof(t90_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v6_group_1(A)
=> r1_xreal_0(np__1,k9_group_1(A)) ) ) )).

fof(d16_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v7_group_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) )).

fof(t91_group_1,axiom,(
\$true )).

fof(t92_group_1,axiom,
( ~ v3_struct_0(g1_group_1(k1_numbers,k33_binop_2))
& v3_group_1(g1_group_1(k1_numbers,k33_binop_2))
& v4_group_1(g1_group_1(k1_numbers,k33_binop_2))
& v7_group_1(g1_group_1(k1_numbers,k33_binop_2))
& l1_group_1(g1_group_1(k1_numbers,k33_binop_2)) )).

fof(t93_group_1,axiom,(
\$true )).

fof(t94_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_group_1(A,k10_group_1(A,B,C)) = k10_group_1(A,k3_group_1(A,B),k3_group_1(A,C)) ) ) ) )).

fof(t95_group_1,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k6_group_1(B,A,k10_group_1(B,C,D)) = k10_group_1(B,k6_group_1(B,A,C),k6_group_1(B,A,D)) ) ) ) ) )).

fof(t96_group_1,axiom,(
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k6_group_1(B,A,k10_group_1(B,C,D)) = k10_group_1(B,k6_group_1(B,A,C),k6_group_1(B,A,D)) ) ) ) ) )).

fof(t97_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ( v3_rlvect_1(g1_rlvect_1(u1_struct_0(A),u1_group_1(A),k2_group_1(A)))
& v4_rlvect_1(g1_rlvect_1(u1_struct_0(A),u1_group_1(A),k2_group_1(A)))
& v5_rlvect_1(g1_rlvect_1(u1_struct_0(A),u1_group_1(A),k2_group_1(A)))
& v6_rlvect_1(g1_rlvect_1(u1_struct_0(A),u1_group_1(A),k2_group_1(A))) ) ) )).

fof(dt_l1_group_1,axiom,(
! [A] :
( l1_group_1(A)
=> l1_struct_0(A) ) )).

fof(existence_l1_group_1,axiom,(
? [A] : l1_group_1(A) )).

fof(abstractness_v1_group_1,axiom,(
! [A] :
( l1_group_1(A)
=> ( v1_group_1(A)
=> A = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) )).

fof(dt_k1_group_1,axiom,(
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k2_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) )).

fof(dt_k3_group_1,axiom,(
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k3_group_1(A,B),u1_struct_0(A)) ) )).

fof(dt_k4_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_funct_1(k4_group_1(A))
& v1_funct_2(k4_group_1(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k4_group_1(A),u1_struct_0(A),u1_struct_0(A)) ) ) )).

fof(dt_k5_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v1_funct_1(k5_group_1(A))
& v1_funct_2(k5_group_1(A),k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
& m2_relset_1(k5_group_1(A),k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) ) ) )).

fof(dt_k6_group_1,axiom,(
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_int_1(B)
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k6_group_1(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k7_group_1,axiom,(
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m2_subset_1(k7_group_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k8_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> v1_card_1(k8_group_1(A)) ) )).

fof(dt_k9_group_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m2_subset_1(k9_group_1(A),k1_numbers,k5_numbers) ) )).

fof(dt_k10_group_1,axiom,(
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k10_group_1(A,B,C),u1_struct_0(A)) ) )).

fof(commutativity_k10_group_1,axiom,(
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k10_group_1(A,B,C) = k10_group_1(A,C,B) ) )).

fof(redefinition_k10_group_1,axiom,(
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k10_group_1(A,B,C) = k1_group_1(A,B,C) ) )).

fof(dt_u1_group_1,axiom,(
! [A] :
( l1_group_1(A)
=> ( v1_funct_1(u1_group_1(A))
& v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) )).

fof(dt_g1_group_1,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_group_1(g1_group_1(A,B))
& l1_group_1(g1_group_1(A,B)) ) ) )).

fof(free_g1_group_1,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C,D] :
( g1_group_1(A,B) = g1_group_1(C,D)
=> ( A = C
& B = D ) ) ) )).
%------------------------------------------------------------------------------
```