SET007 Axioms: SET007+263.ax
%------------------------------------------------------------------------------
% File : SET007+263 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : From Double Loops to Fields
% 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 : algstr_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 46 ( 28 unt; 0 def)
% Number of atoms : 461 ( 96 equ)
% Maximal formula atoms : 64 ( 10 avg)
% Number of connectives : 474 ( 59 ~; 6 |; 245 &)
% ( 6 <=>; 158 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 1 con; 0-3 aty)
% Number of variables : 161 ( 155 !; 6 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_algstr_2,axiom,
( ~ v3_struct_0(k3_vectsp_1)
& v3_rlvect_1(k3_vectsp_1)
& v4_rlvect_1(k3_vectsp_1)
& v5_rlvect_1(k3_vectsp_1)
& v6_rlvect_1(k3_vectsp_1)
& v2_group_1(k3_vectsp_1)
& v4_group_1(k3_vectsp_1)
& v7_group_1(k3_vectsp_1)
& v3_vectsp_1(k3_vectsp_1)
& v4_vectsp_1(k3_vectsp_1)
& v5_vectsp_1(k3_vectsp_1)
& v6_vectsp_1(k3_vectsp_1)
& v7_vectsp_1(k3_vectsp_1)
& v8_vectsp_1(k3_vectsp_1)
& v9_vectsp_1(k3_vectsp_1)
& ~ v10_vectsp_1(k3_vectsp_1)
& v1_algstr_1(k3_vectsp_1)
& v2_algstr_1(k3_vectsp_1)
& v3_algstr_1(k3_vectsp_1)
& v4_algstr_1(k3_vectsp_1)
& v5_algstr_1(k3_vectsp_1)
& v6_algstr_1(k3_vectsp_1)
& v9_algstr_1(k3_vectsp_1)
& v10_algstr_1(k3_vectsp_1)
& v11_algstr_1(k3_vectsp_1) ) ).
fof(rc1_algstr_2,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A)
& v6_algstr_1(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& v11_algstr_1(A) ) ).
fof(d1_algstr_2,axiom,
$true ).
fof(d2_algstr_2,axiom,
$true ).
fof(d3_algstr_2,axiom,
$true ).
fof(d4_algstr_2,axiom,
$true ).
fof(d5_algstr_2,axiom,
$true ).
fof(d6_algstr_2,axiom,
$true ).
fof(d7_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v5_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k1_algstr_2(A,B)
<=> k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) ) ) ) ).
fof(d8_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v5_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_algstr_2(A,B,C) = k2_rlvect_1(A,B,k1_algstr_2(A,C)) ) ) ) ).
fof(t1_algstr_2,axiom,
$true ).
fof(t2_algstr_2,axiom,
$true ).
fof(t3_algstr_2,axiom,
$true ).
fof(t4_algstr_2,axiom,
$true ).
fof(t5_algstr_2,axiom,
$true ).
fof(t6_algstr_2,axiom,
$true ).
fof(t7_algstr_2,axiom,
$true ).
fof(t8_algstr_2,axiom,
$true ).
fof(t9_algstr_2,axiom,
$true ).
fof(t10_algstr_2,axiom,
$true ).
fof(t11_algstr_2,axiom,
$true ).
fof(t12_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_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))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
& k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k2_group_1(A),B) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,B,D) != C ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,D,B) != C ) ) ) )
& ! [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)
=> ( B = k1_rlvect_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,C,B) = k1_group_1(A,D,B)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_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,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ) ) ).
fof(t13_algstr_2,axiom,
$true ).
fof(t14_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_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,k1_algstr_2(A,C)) = k1_algstr_2(A,k1_group_1(A,B,C)) ) ) ) ).
fof(t15_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v2_algstr_1(A)
& v5_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_algstr_2(A,k1_algstr_2(A,B)) = B ) ) ).
fof(t16_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
=> k1_group_1(A,k1_algstr_2(A,k2_group_1(A)),k1_algstr_2(A,k2_group_1(A))) = k2_group_1(A) ) ).
fof(t17_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_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,k2_algstr_2(A,C,D)) = k2_algstr_2(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ).
fof(t18_algstr_2,axiom,
$true ).
fof(t19_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v5_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_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))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
& k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k2_group_1(A),B) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,B,D) != C ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,D,B) != C ) ) ) )
& ! [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)
=> ( B = k1_rlvect_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,C,B) = k1_group_1(A,D,B)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_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,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) ).
fof(t20_algstr_2,axiom,
$true ).
fof(t21_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v5_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,k1_algstr_2(A,B),C) = k1_algstr_2(A,k1_group_1(A,B,C)) ) ) ) ).
fof(t22_algstr_2,axiom,
$true ).
fof(t23_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v5_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
=> k1_group_1(A,k1_algstr_2(A,k2_group_1(A)),k1_algstr_2(A,k2_group_1(A))) = k2_group_1(A) ) ).
fof(t24_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v5_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_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,k2_algstr_2(A,B,C),D) = k2_algstr_2(A,k1_group_1(A,B,D),k1_group_1(A,C,D)) ) ) ) ) ).
fof(t25_algstr_2,axiom,
$true ).
fof(t26_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_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))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
& k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k2_group_1(A),B) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,B,D) != C ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,D,B) != C ) ) ) )
& ! [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)
=> ( B = k1_rlvect_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,C,B) = k1_group_1(A,D,B)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_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,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,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,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) ).
fof(t27_algstr_2,axiom,
$true ).
fof(t28_algstr_2,axiom,
$true ).
fof(t29_algstr_2,axiom,
$true ).
fof(t30_algstr_2,axiom,
$true ).
fof(t31_algstr_2,axiom,
$true ).
fof(t32_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_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))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
& k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) != k2_group_1(A) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_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,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,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,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) ).
fof(t33_algstr_2,axiom,
$true ).
fof(t34_algstr_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& v11_algstr_1(A)
& l3_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_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))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
& k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) != k2_group_1(A) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_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,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) )
& ! [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(dt_k1_algstr_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v5_algstr_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_algstr_2(A,B),u1_struct_0(A)) ) ).
fof(dt_k2_algstr_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v5_algstr_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k2_algstr_2(A,B,C),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------