SET007 Axioms: SET007+575.ax


%------------------------------------------------------------------------------
% File     : SET007+575 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Basic Properties of SCM  over Ring
% 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    : scmring2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (   7 unit)
%            Number of atoms       :  877 (  67 equality)
%            Maximal formula depth :   18 (  13 average)
%            Number of connectives :  903 (  91 ~  ;   0  |; 655  &)
%                                         (   5 <=>; 152 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   1 propositional; 0-3 arity)
%            Number of functors    :   58 (  13 constant; 0-4 arity)
%            Number of variables   :  179 (   0 singleton; 176 !;   3 ?)
%            Maximal term depth    :    7 (   2 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(fc1_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_scmring2(A))
        & v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & ~ v2_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) )).

fof(fc2_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_scmring2(A))
        & v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & ~ v2_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v5_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) )).

fof(fc3_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => ~ v3_ami_1(k4_scmring2(A,B,C),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc4_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => ~ v3_ami_1(k5_scmring2(A,B,C),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc5_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => ~ v3_ami_1(k6_scmring2(A,B,C),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc6_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => ~ v3_ami_1(k7_scmring2(A,B,C),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc7_scmring2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) )
     => ~ v3_ami_1(k9_scmring2(A,B),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc8_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_subset_1(C,u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) )
     => ~ v3_ami_1(k10_scmring2(A,C,B),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc9_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_subset_1(C,u1_struct_0(A)) )
     => ~ v3_ami_1(k8_scmring2(A,B,C),k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) )).

fof(fc10_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_scmring2(A))
        & v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & ~ v2_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v4_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v5_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v6_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v7_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v8_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & v10_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) )).

fof(d1_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_ami_1(B,k1_tarski(u1_struct_0(A)))
            & l1_ami_1(B,k1_tarski(u1_struct_0(A))) )
         => ( B = k1_scmring2(A)
          <=> ( u1_struct_0(B) = k5_numbers
              & u1_ami_1(k1_tarski(u1_struct_0(A)),B) = np__0
              & u2_ami_1(k1_tarski(u1_struct_0(A)),B) = k3_ami_2
              & u3_ami_1(k1_tarski(u1_struct_0(A)),B) = k1_gr_cy_1(np__8)
              & u4_ami_1(k1_tarski(u1_struct_0(A)),B) = k1_scmring1(A)
              & u5_ami_1(k1_tarski(u1_struct_0(A)),B) = k2_scmring1(A)
              & u6_ami_1(k1_tarski(u1_struct_0(A)),B) = k16_scmring1(A) ) ) ) ) )).

fof(d2_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_scmring2(A)))
         => ( m1_scmring2(B,A)
          <=> r2_hidden(B,k4_xboole_0(u1_struct_0(k1_scmring2(A)),k2_xboole_0(k3_ami_2,k1_tarski(np__0)))) ) ) ) )).

fof(t1_scmring2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v1_scmring1(B)
        & l3_vectsp_1(B) )
     => ( m1_scmring2(A,B)
      <=> r2_hidden(A,k2_ami_2) ) ) )).

fof(t2_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => r2_hidden(k4_tarski(np__0,k1_xboole_0),k1_scmring1(A)) ) )).

fof(t3_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => m2_subset_1(k4_tarski(np__0,k1_xboole_0),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(t4_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B,C] :
          ( ( ~ v3_struct_0(C)
            & v3_rlvect_1(C)
            & v4_rlvect_1(C)
            & v5_rlvect_1(C)
            & v6_rlvect_1(C)
            & v4_group_1(C)
            & v6_vectsp_1(C)
            & v7_vectsp_1(C)
            & v8_vectsp_1(C)
            & v1_scmring1(C)
            & l3_vectsp_1(C) )
         => ! [D] :
              ( m1_scmring2(D,C)
             => ! [E] :
                  ( m1_scmring2(E,C)
                 => ( r2_hidden(B,k2_enumset1(np__1,np__2,np__3,np__4))
                   => r2_hidden(k4_tarski(B,k10_finseq_1(D,E)),k1_scmring1(A)) ) ) ) ) ) )).

fof(t5_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v4_group_1(C)
                & v6_vectsp_1(C)
                & v7_vectsp_1(C)
                & v8_vectsp_1(C)
                & v1_scmring1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_scmring2(D,C)
                 => r2_hidden(k4_tarski(np__5,k10_finseq_1(D,B)),k1_scmring1(A)) ) ) ) ) )).

fof(t6_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v1_scmring1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_struct_0(C,k1_scmring2(B),u2_ami_1(k1_tarski(u1_struct_0(B)),k1_scmring2(B)))
             => r2_hidden(k4_tarski(np__6,k12_finseq_1(u2_ami_1(k1_tarski(u1_struct_0(B)),k1_scmring2(B)),C)),k1_scmring1(A)) ) ) ) )).

fof(t7_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v1_scmring1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_scmring2(C,B)
             => ! [D] :
                  ( m1_struct_0(D,k1_scmring2(B),u2_ami_1(k1_tarski(u1_struct_0(B)),k1_scmring2(B)))
                 => r2_hidden(k4_tarski(np__7,k10_finseq_1(D,C)),k1_scmring1(A)) ) ) ) ) )).

fof(d3_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => k4_scmring2(A,B,C) = k4_tarski(np__1,k10_finseq_1(B,C)) ) ) ) )).

fof(d4_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => k5_scmring2(A,B,C) = k4_tarski(np__2,k10_finseq_1(B,C)) ) ) ) )).

fof(d5_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => k6_scmring2(A,B,C) = k4_tarski(np__3,k10_finseq_1(B,C)) ) ) ) )).

fof(d6_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => k7_scmring2(A,B,C) = k4_tarski(np__4,k10_finseq_1(B,C)) ) ) ) )).

fof(d7_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k8_scmring2(A,B,C) = k4_tarski(np__5,k10_finseq_1(B,C)) ) ) ) )).

fof(d8_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_struct_0(B,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => k9_scmring2(A,B) = k4_tarski(np__6,k12_finseq_1(u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),B)) ) ) )).

fof(d9_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_struct_0(B,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => ! [C] :
              ( m1_scmring2(C,A)
             => k10_scmring2(A,B,C) = k4_tarski(np__7,k10_finseq_1(B,C)) ) ) ) )).

fof(t8_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
        <=> ~ ( B != k4_tarski(np__0,k1_xboole_0)
              & ! [C] :
                  ( m1_scmring2(C,A)
                 => ! [D] :
                      ( m1_scmring2(D,A)
                     => B != k4_scmring2(A,C,D) ) )
              & ! [C] :
                  ( m1_scmring2(C,A)
                 => ! [D] :
                      ( m1_scmring2(D,A)
                     => B != k5_scmring2(A,C,D) ) )
              & ! [C] :
                  ( m1_scmring2(C,A)
                 => ! [D] :
                      ( m1_scmring2(D,A)
                     => B != k6_scmring2(A,C,D) ) )
              & ! [C] :
                  ( m1_scmring2(C,A)
                 => ! [D] :
                      ( m1_scmring2(D,A)
                     => B != k7_scmring2(A,C,D) ) )
              & ! [C] :
                  ( m1_struct_0(C,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
                 => B != k9_scmring2(A,C) )
              & ! [C] :
                  ( m1_scmring2(C,A)
                 => ! [D] :
                      ( m1_struct_0(D,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
                     => B != k10_scmring2(A,D,C) ) )
              & ! [C] :
                  ( m1_scmring2(C,A)
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => B != k8_scmring2(A,C,D) ) ) ) ) ) )).

fof(t9_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)) = np__0 ) )).

fof(t10_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
         => ! [C] :
              ( m1_subset_1(C,k4_card_3(k2_scmring1(A)))
             => ( C = B
               => k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),B) = k3_scmring1(A,C) ) ) ) ) )).

fof(d10_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_struct_0(B,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => ! [C] :
              ( m1_struct_0(C,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
             => ( C = k11_scmring2(A,B)
              <=> ? [D] :
                    ( m2_subset_1(D,k5_numbers,k3_ami_2)
                    & D = B
                    & C = k15_ami_2(D) ) ) ) ) ) )).

fof(t11_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_struct_0(B,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k3_ami_2)
             => ( C = B
               => k15_ami_2(C) = k11_scmring2(A,B) ) ) ) ) )).

fof(t12_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
         => ! [C] :
              ( m2_subset_1(C,k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_scmring1(A))
                 => ( D = C
                   => ! [E] :
                        ( m1_subset_1(E,k4_card_3(k2_scmring1(A)))
                       => ( E = B
                         => k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),C,B) = k15_scmring1(A,D,E) ) ) ) ) ) ) ) )).

fof(t13_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                 => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k4_scmring2(A,B,C),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),D))
                    & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k4_scmring2(A,B,C),D),B) = k3_scmring2(A,D,C)
                    & ! [E] :
                        ( m1_scmring2(E,A)
                       => ( E != B
                         => k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k4_scmring2(A,B,C),D),E) = k3_scmring2(A,D,E) ) ) ) ) ) ) ) )).

fof(t14_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                 => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k5_scmring2(A,B,C),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),D))
                    & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k5_scmring2(A,B,C),D),B) = k4_rlvect_1(A,k3_scmring2(A,D,B),k3_scmring2(A,D,C))
                    & ! [E] :
                        ( m1_scmring2(E,A)
                       => ( E != B
                         => k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k5_scmring2(A,B,C),D),E) = k3_scmring2(A,D,E) ) ) ) ) ) ) ) )).

fof(t15_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                 => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k6_scmring2(A,B,C),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),D))
                    & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k6_scmring2(A,B,C),D),B) = k6_rlvect_1(A,k3_scmring2(A,D,B),k3_scmring2(A,D,C))
                    & ! [E] :
                        ( m1_scmring2(E,A)
                       => ( E != B
                         => k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k6_scmring2(A,B,C),D),E) = k3_scmring2(A,D,E) ) ) ) ) ) ) ) )).

fof(t16_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                 => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k7_scmring2(A,B,C),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),D))
                    & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k7_scmring2(A,B,C),D),B) = k1_group_1(A,k3_scmring2(A,D,B),k3_scmring2(A,D,C))
                    & ! [E] :
                        ( m1_scmring2(E,A)
                       => ( E != B
                         => k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k7_scmring2(A,B,C),D),E) = k3_scmring2(A,D,E) ) ) ) ) ) ) ) )).

fof(t17_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_struct_0(C,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
             => ! [D] :
                  ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                 => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k9_scmring2(A,C),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = C
                    & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k9_scmring2(A,C),D),B) = k3_scmring2(A,D,B) ) ) ) ) ) )).

fof(t18_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => ! [C] :
              ( m1_scmring2(C,A)
             => ! [D] :
                  ( m1_struct_0(D,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                     => ( ( k3_scmring2(A,E,B) = k1_rlvect_1(A)
                         => k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k10_scmring2(A,D,B),E),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = D )
                        & ( k3_scmring2(A,E,B) != k1_rlvect_1(A)
                         => k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k10_scmring2(A,D,B),E),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),E)) )
                        & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k10_scmring2(A,D,B),E),C) = k3_scmring2(A,E,C) ) ) ) ) ) ) )).

fof(t19_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_scmring2(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                 => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k8_scmring2(A,C,B),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),D))
                    & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k8_scmring2(A,C,B),D),C) = B
                    & ! [E] :
                        ( m1_scmring2(E,A)
                       => ( E != C
                         => k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k8_scmring2(A,C,B),D),E) = k3_scmring2(A,D,E) ) ) ) ) ) ) ) )).

fof(t20_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => ~ ( ? [C] :
                  ( m1_subset_1(C,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
                  & k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),B,C),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),C)) )
              & v3_ami_1(B,k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) ) ) )).

fof(t21_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => ( B = k4_tarski(np__0,k1_xboole_0)
           => v3_ami_1(B,k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) ) ) )).

fof(t22_scmring2,axiom,(
    $true )).

fof(t23_scmring2,axiom,(
    $true )).

fof(t24_scmring2,axiom,(
    $true )).

fof(t25_scmring2,axiom,(
    $true )).

fof(t26_scmring2,axiom,(
    $true )).

fof(t27_scmring2,axiom,(
    $true )).

fof(t28_scmring2,axiom,(
    $true )).

fof(t29_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
         => ( v3_ami_1(B,k1_tarski(u1_struct_0(A)),k1_scmring2(A))
           => B = k5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)) ) ) ) )).

fof(t30_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => k5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)) = k4_tarski(np__0,k1_xboole_0) ) )).

fof(dt_m1_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_scmring2(B,A)
         => m1_subset_1(B,u1_struct_0(k1_scmring2(A))) ) ) )).

fof(existence_m1_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ? [B] : m1_scmring2(B,A) ) )).

fof(dt_k1_scmring2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ( v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A)))
        & l1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) )).

fof(dt_k2_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
        & m1_subset_1(C,k2_ami_2) )
     => m1_subset_1(k2_scmring2(A,B,C),u1_struct_0(A)) ) )).

fof(redefinition_k2_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
        & m1_subset_1(C,k2_ami_2) )
     => k2_scmring2(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k3_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
        & m1_scmring2(C,A) )
     => m1_subset_1(k3_scmring2(A,B,C),u1_struct_0(A)) ) )).

fof(redefinition_k3_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))))
        & m1_scmring2(C,A) )
     => k3_scmring2(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k4_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => m2_subset_1(k4_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k5_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => m2_subset_1(k5_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k6_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => m2_subset_1(k6_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k7_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_scmring2(C,A) )
     => m2_subset_1(k7_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k8_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_scmring2(B,A)
        & m1_subset_1(C,u1_struct_0(A)) )
     => m2_subset_1(k8_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k9_scmring2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) )
     => m2_subset_1(k9_scmring2(A,B),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k10_scmring2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))
        & m1_scmring2(C,A) )
     => m2_subset_1(k10_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).

fof(dt_k11_scmring2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) )
     => m1_struct_0(k11_scmring2(A,B),k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) )).
%------------------------------------------------------------------------------