SET007 Axioms: SET007+574.ax


%------------------------------------------------------------------------------
% File     : SET007+574 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Construction 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    : scmring1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   70 (   0 unit)
%            Number of atoms       :  551 (  99 equality)
%            Maximal formula depth :   26 (   9 average)
%            Number of connectives :  571 (  90 ~  ;   0  |; 294  &)
%                                         (  17 <=>; 170 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   44 (   0 propositional; 1-3 arity)
%            Number of functors    :   67 (  18 constant; 0-6 arity)
%            Number of variables   :  230 (   0 singleton; 175 !;  55 ?)
%            Maximal term depth    :    8 (   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(cc1_scmring1,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ( ~ v1_xboole_0(A)
        & ~ v1_realset1(A) ) ) )).

fof(cc2_scmring1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( ~ v6_group_1(A)
       => ( ~ v3_struct_0(A)
          & ~ v3_realset2(A) ) ) ) )).

fof(cc3_scmring1,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v3_realset2(A) )
       => ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A) ) ) ) )).

fof(cc4_scmring1,axiom,(
    ! [A] :
      ( l3_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v3_realset2(A) )
       => ( ~ v3_struct_0(A)
          & v4_vectsp_1(A)
          & v6_vectsp_1(A) ) ) ) )).

fof(cc5_scmring1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_ami_2)
     => ( v4_ordinal2(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xcmplx_0(A) ) ) )).

fof(fc1_scmring1,axiom,
    ( v1_relat_1(k4_ami_2)
    & ~ v1_xboole_0(k4_ami_2)
    & ~ v1_realset1(k4_ami_2) )).

fof(fc2_scmring1,axiom,
    ( ~ v1_xboole_0(k3_ami_2)
    & ~ v1_finset_1(k3_ami_2)
    & ~ v1_realset1(k3_ami_2)
    & v1_membered(k3_ami_2)
    & v2_membered(k3_ami_2)
    & v3_membered(k3_ami_2)
    & v4_membered(k3_ami_2)
    & v5_membered(k3_ami_2) )).

fof(fc3_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ( v1_relat_1(k1_scmring1(A))
        & ~ v1_xboole_0(k1_scmring1(A))
        & ~ v1_realset1(k1_scmring1(A)) ) ) )).

fof(cc6_scmring1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( ( ~ v3_struct_0(A)
          & v3_realset2(A) )
       => ( ~ v3_struct_0(A)
          & v1_scmring1(A) ) ) ) )).

fof(rc1_scmring1,axiom,(
    ? [A] :
      ( l1_struct_0(A)
      & v1_struct_0(A)
      & ~ v3_struct_0(A)
      & v6_group_1(A)
      & v3_realset2(A)
      & v1_scmring1(A) ) )).

fof(rc2_scmring1,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v3_group_1(A)
      & v4_group_1(A)
      & v6_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_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)
      & v3_realset2(A)
      & v1_scmring1(A) ) )).

fof(rc3_scmring1,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v3_group_1(A)
      & v4_group_1(A)
      & v6_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_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)
      & v3_realset2(A)
      & v1_scmring1(A) ) )).

fof(fc4_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
        & m1_relset_1(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
        & m1_subset_1(C,k1_scmring1(A)) )
     => ( v1_relat_1(k1_funct_1(B,C))
        & v1_funct_1(k1_funct_1(B,C)) ) ) )).

fof(d2_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ( v1_scmring1(A)
      <=> ( u1_struct_0(A) != k3_ami_2
          & u1_struct_0(A) != k1_scmring1(A) ) ) ) )).

fof(d3_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)))
            & m2_relset_1(B,k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2))) )
         => ( B = k2_scmring1(A)
          <=> ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)),B,np__0) = k3_ami_2
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)),B,k1_nat_1(k2_nat_1(np__2,C),np__1)) = u1_struct_0(A)
                    & k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)),B,k1_nat_1(k2_nat_1(np__2,C),np__2)) = k1_scmring1(A) ) ) ) ) ) ) )).

fof(t1_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => k3_ami_2 != k1_scmring1(A) ) )).

fof(t2_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = k3_ami_2
          <=> A = np__0 ) ) ) )).

fof(t3_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = u1_struct_0(B)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & A = k1_nat_1(k2_nat_1(np__2,C),np__1) ) ) ) ) )).

fof(t4_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = k1_scmring1(B)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & A = k1_nat_1(k2_nat_1(np__2,C),np__2) ) ) ) ) )).

fof(t5_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k2_ami_2)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = u1_struct_0(B) ) ) )).

fof(t6_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k3_ami_2)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = k1_scmring1(B) ) ) )).

fof(t7_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => k5_card_3(np__0,k4_card_3(k2_scmring1(A))) = k3_ami_2 ) )).

fof(t8_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k2_ami_2)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => k5_card_3(A,k4_card_3(k2_scmring1(B))) = u1_struct_0(B) ) ) )).

fof(t9_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k3_ami_2)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_scmring1(B)
            & l1_struct_0(B) )
         => k5_card_3(A,k4_card_3(k2_scmring1(B))) = k1_scmring1(B) ) ) )).

fof(d4_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => k3_scmring1(A,B) = k1_funct_1(B,np__0) ) ) )).

fof(d5_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k3_ami_2)
             => k4_scmring1(A,B,C) = k1_funct_4(B,k3_cqc_lang(np__0,C)) ) ) ) )).

fof(t10_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k3_ami_2)
             => k1_funct_1(k4_scmring1(A,B,C),np__0) = C ) ) ) )).

fof(t11_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k3_ami_2)
             => ! [D] :
                  ( m2_subset_1(D,k5_numbers,k2_ami_2)
                 => k1_funct_1(k4_scmring1(A,B,C),D) = k1_funct_1(B,D) ) ) ) ) )).

fof(t12_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k3_ami_2)
             => ! [D] :
                  ( m2_subset_1(D,k5_numbers,k3_ami_2)
                 => k1_funct_1(k4_scmring1(A,B,C),D) = k1_funct_1(B,D) ) ) ) ) )).

fof(d6_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k2_ami_2)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k5_scmring1(A,B,C,D) = k1_funct_4(B,k3_cqc_lang(C,D)) ) ) ) ) )).

fof(t13_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k2_ami_2)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k1_funct_1(k5_scmring1(A,B,C,D),np__0) = k1_funct_1(B,np__0) ) ) ) ) )).

fof(t14_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k2_ami_2)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k1_funct_1(k5_scmring1(A,B,C,D),C) = D ) ) ) ) )).

fof(t15_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k2_ami_2)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m2_subset_1(E,k5_numbers,k2_ami_2)
                     => ( E != C
                       => k1_funct_1(k5_scmring1(A,B,C,D),E) = k1_funct_1(B,E) ) ) ) ) ) ) )).

fof(t16_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k2_ami_2)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m2_subset_1(E,k5_numbers,k3_ami_2)
                     => k1_funct_1(k5_scmring1(A,B,C,D),E) = k1_funct_1(B,E) ) ) ) ) ) )).

fof(d7_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
                & ? [D] :
                    ( m2_subset_1(D,k5_numbers,k2_ami_2)
                    & ? [E] :
                        ( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
                        & B = k4_tarski(E,k2_finseq_4(k2_ami_2,C,D)) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
               => ( C = k7_scmring1(A,B)
                <=> ? [D] :
                      ( m2_finseq_1(D,k2_ami_2)
                      & D = k2_mcart_1(B)
                      & C = k4_finseq_4(k5_numbers,k2_ami_2,D,np__1) ) ) ) ) ) ) )).

fof(d8_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
                & ? [D] :
                    ( m2_subset_1(D,k5_numbers,k2_ami_2)
                    & ? [E] :
                        ( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
                        & B = k4_tarski(E,k2_finseq_4(k2_ami_2,C,D)) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
               => ( C = k8_scmring1(A,B)
                <=> ? [D] :
                      ( m2_finseq_1(D,k2_ami_2)
                      & D = k2_mcart_1(B)
                      & C = k4_finseq_4(k5_numbers,k2_ami_2,D,np__2) ) ) ) ) ) ) )).

fof(t17_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_scmring1(B))
             => ! [D] :
                  ( m2_subset_1(D,k5_numbers,k2_ami_2)
                 => ! [E] :
                      ( m2_subset_1(E,k5_numbers,k2_ami_2)
                     => ( C = k4_tarski(A,k2_finseq_4(k2_ami_2,D,E))
                       => ( k7_scmring1(B,C) = D
                          & k8_scmring1(B,C) = E ) ) ) ) ) ) ) )).

fof(d9_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k3_ami_2)
                & ? [D] :
                    ( m2_subset_1(D,k5_numbers,k1_gr_cy_1(np__8))
                    & B = k4_tarski(D,k12_finseq_1(k3_ami_2,C)) ) )
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k3_ami_2)
               => ( C = k9_scmring1(A,B)
                <=> ? [D] :
                      ( m2_finseq_1(D,k3_ami_2)
                      & D = k2_mcart_1(B)
                      & C = k4_finseq_4(k5_numbers,k3_ami_2,D,np__1) ) ) ) ) ) ) )).

fof(t18_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_scmring1(B))
             => ! [D] :
                  ( m2_subset_1(D,k5_numbers,k3_ami_2)
                 => ( C = k4_tarski(A,k12_finseq_1(k3_ami_2,D))
                   => k9_scmring1(B,C) = D ) ) ) ) ) )).

fof(d10_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k3_ami_2)
                & ? [D] :
                    ( m2_subset_1(D,k5_numbers,k2_ami_2)
                    & ? [E] :
                        ( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
                        & B = k4_tarski(E,k2_finseq_4(k5_numbers,C,D)) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k3_ami_2)
               => ( C = k10_scmring1(A,B)
                <=> ? [D] :
                      ( m2_subset_1(D,k5_numbers,k3_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & k2_finseq_4(k5_numbers,D,E) = k2_mcart_1(B)
                          & C = k4_finseq_4(k5_numbers,k5_numbers,k2_finseq_4(k5_numbers,D,E),np__1) ) ) ) ) ) ) ) )).

fof(d11_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k3_ami_2)
                & ? [D] :
                    ( m2_subset_1(D,k5_numbers,k2_ami_2)
                    & ? [E] :
                        ( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
                        & B = k4_tarski(E,k2_finseq_4(k5_numbers,C,D)) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
               => ( C = k11_scmring1(A,B)
                <=> ? [D] :
                      ( m2_subset_1(D,k5_numbers,k3_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & k2_finseq_4(k5_numbers,D,E) = k2_mcart_1(B)
                          & C = k4_finseq_4(k5_numbers,k5_numbers,k2_finseq_4(k5_numbers,D,E),np__2) ) ) ) ) ) ) ) )).

fof(t19_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_scmring1(B))
             => ! [D] :
                  ( m2_subset_1(D,k5_numbers,k3_ami_2)
                 => ! [E] :
                      ( m2_subset_1(E,k5_numbers,k2_ami_2)
                     => ( C = k4_tarski(A,k2_finseq_4(k5_numbers,D,E))
                       => ( k10_scmring1(B,C) = D
                          & k11_scmring1(B,C) = E ) ) ) ) ) ) ) )).

fof(d12_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ? [E] :
                        ( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
                        & B = k4_tarski(E,k12_scmring1(A,C,D)) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
               => ( C = k13_scmring1(A,B)
                <=> ? [D] :
                      ( m2_finseq_1(D,k2_xboole_0(k2_ami_2,u1_struct_0(A)))
                      & D = k2_mcart_1(B)
                      & C = k4_finseq_4(k5_numbers,k2_xboole_0(k2_ami_2,u1_struct_0(A)),D,np__1) ) ) ) ) ) ) )).

fof(d13_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_scmring1(A))
         => ( ? [C] :
                ( m2_subset_1(C,k5_numbers,k2_ami_2)
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ? [E] :
                        ( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
                        & B = k4_tarski(E,k12_scmring1(A,C,D)) ) ) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k14_scmring1(A,B)
                <=> ? [D] :
                      ( m2_finseq_1(D,k2_xboole_0(k2_ami_2,u1_struct_0(A)))
                      & D = k2_mcart_1(B)
                      & C = k4_finseq_4(k5_numbers,k2_xboole_0(k2_ami_2,u1_struct_0(A)),D,np__2) ) ) ) ) ) ) )).

fof(t20_scmring1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_scmring1(B))
             => ! [D] :
                  ( m2_subset_1(D,k5_numbers,k2_ami_2)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( C = k4_tarski(A,k12_scmring1(B,D,E))
                       => ( k13_scmring1(B,C) = D
                          & k14_scmring1(B,C) = E ) ) ) ) ) ) ) )).

fof(d14_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_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,k1_scmring1(A))
         => ! [C] :
              ( m1_subset_1(C,k4_card_3(k2_scmring1(A)))
             => ( ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k2_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & B = k4_tarski(np__1,k2_finseq_4(k2_ami_2,D,E)) ) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k6_scmring1(A,C,k8_scmring1(A,B))),k15_ami_2(k3_scmring1(A,C))) )
                & ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k2_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & B = k4_tarski(np__2,k2_finseq_4(k2_ami_2,D,E)) ) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k4_rlvect_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) )
                & ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k2_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & B = k4_tarski(np__3,k2_finseq_4(k2_ami_2,D,E)) ) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k6_rlvect_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) )
                & ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k2_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & B = k4_tarski(np__4,k2_finseq_4(k2_ami_2,D,E)) ) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k1_group_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) )
                & ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k3_ami_2)
                      & B = k4_tarski(np__6,k12_finseq_1(k3_ami_2,D)) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,C,k9_scmring1(A,B)) )
                & ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k3_ami_2)
                      & ? [E] :
                          ( m2_subset_1(E,k5_numbers,k2_ami_2)
                          & B = k4_tarski(np__7,k2_finseq_4(k5_numbers,D,E)) ) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,C,k2_cqc_lang(k3_ami_2,k6_scmring1(A,C,k11_scmring1(A,B)),k1_rlvect_1(A),k10_scmring1(A,B),k15_ami_2(k3_scmring1(A,C)))) )
                & ( ? [D] :
                      ( m2_subset_1(D,k5_numbers,k2_ami_2)
                      & ? [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                          & B = k4_tarski(np__5,k12_scmring1(A,D,E)) ) )
                 => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k13_scmring1(A,B),k14_scmring1(A,B)),k15_ami_2(k3_scmring1(A,C))) )
                & ( ( ! [D] :
                        ( m2_subset_1(D,k5_numbers,k2_ami_2)
                       => ! [E] :
                            ( m2_subset_1(E,k5_numbers,k2_ami_2)
                           => B != k4_tarski(np__1,k2_finseq_4(k2_ami_2,D,E)) ) )
                    & ! [D] :
                        ( m2_subset_1(D,k5_numbers,k2_ami_2)
                       => ! [E] :
                            ( m2_subset_1(E,k5_numbers,k2_ami_2)
                           => B != k4_tarski(np__2,k2_finseq_4(k2_ami_2,D,E)) ) )
                    & ! [D] :
                        ( m2_subset_1(D,k5_numbers,k2_ami_2)
                       => ! [E] :
                            ( m2_subset_1(E,k5_numbers,k2_ami_2)
                           => B != k4_tarski(np__3,k2_finseq_4(k2_ami_2,D,E)) ) )
                    & ! [D] :
                        ( m2_subset_1(D,k5_numbers,k2_ami_2)
                       => ! [E] :
                            ( m2_subset_1(E,k5_numbers,k2_ami_2)
                           => B != k4_tarski(np__4,k2_finseq_4(k2_ami_2,D,E)) ) )
                    & ! [D] :
                        ( m2_subset_1(D,k5_numbers,k3_ami_2)
                       => B != k4_tarski(np__6,k12_finseq_1(k3_ami_2,D)) )
                    & ! [D] :
                        ( m2_subset_1(D,k5_numbers,k3_ami_2)
                       => ! [E] :
                            ( m2_subset_1(E,k5_numbers,k2_ami_2)
                           => B != k4_tarski(np__7,k2_finseq_4(k5_numbers,D,E)) ) )
                    & ! [D] :
                        ( m2_subset_1(D,k5_numbers,k2_ami_2)
                       => ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => B != k4_tarski(np__5,k12_scmring1(A,D,E)) ) ) )
                 => k15_scmring1(A,B,C) = C ) ) ) ) ) )).

fof(d15_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
            & m2_relset_1(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) )
         => ( B = k16_scmring1(A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_scmring1(A))
               => ! [D] :
                    ( m1_subset_1(D,k4_card_3(k2_scmring1(A)))
                   => k8_funct_2(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)),k1_cat_2(k1_scmring1(A),k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))),B,C),D) = k15_scmring1(A,C,D) ) ) ) ) ) )).

fof(dt_k1_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => m1_subset_1(k1_scmring1(A),k1_zfmisc_1(k2_zfmisc_1(k1_gr_cy_1(np__8),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),k5_numbers))))) ) )).

fof(dt_k2_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ( v1_funct_1(k2_scmring1(A))
        & v1_funct_2(k2_scmring1(A),k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)))
        & m2_relset_1(k2_scmring1(A),k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2))) ) ) )).

fof(dt_k3_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k4_card_3(k2_scmring1(A))) )
     => m2_subset_1(k3_scmring1(A,B),k5_numbers,k3_ami_2) ) )).

fof(dt_k4_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k4_card_3(k2_scmring1(A)))
        & m1_subset_1(C,k3_ami_2) )
     => m1_subset_1(k4_scmring1(A,B,C),k4_card_3(k2_scmring1(A))) ) )).

fof(dt_k5_scmring1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k4_card_3(k2_scmring1(A)))
        & m1_subset_1(C,k2_ami_2)
        & m1_subset_1(D,u1_struct_0(A)) )
     => m1_subset_1(k5_scmring1(A,B,C,D),k4_card_3(k2_scmring1(A))) ) )).

fof(dt_k6_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k4_card_3(k2_scmring1(A)))
        & m1_subset_1(C,k2_ami_2) )
     => m1_subset_1(k6_scmring1(A,B,C),u1_struct_0(A)) ) )).

fof(redefinition_k6_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_scmring1(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k4_card_3(k2_scmring1(A)))
        & m1_subset_1(C,k2_ami_2) )
     => k6_scmring1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k7_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m2_subset_1(k7_scmring1(A,B),k5_numbers,k2_ami_2) ) )).

fof(dt_k8_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m2_subset_1(k8_scmring1(A,B),k5_numbers,k2_ami_2) ) )).

fof(dt_k9_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m2_subset_1(k9_scmring1(A,B),k5_numbers,k3_ami_2) ) )).

fof(dt_k10_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m2_subset_1(k10_scmring1(A,B),k5_numbers,k3_ami_2) ) )).

fof(dt_k11_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m2_subset_1(k11_scmring1(A,B),k5_numbers,k2_ami_2) ) )).

fof(dt_k12_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k2_ami_2)
        & m1_subset_1(C,u1_struct_0(A)) )
     => m2_finseq_1(k12_scmring1(A,B,C),k2_xboole_0(k2_ami_2,u1_struct_0(A))) ) )).

fof(redefinition_k12_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k2_ami_2)
        & m1_subset_1(C,u1_struct_0(A)) )
     => k12_scmring1(A,B,C) = k10_finseq_1(B,C) ) )).

fof(dt_k13_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m2_subset_1(k13_scmring1(A,B),k5_numbers,k2_ami_2) ) )).

fof(dt_k14_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_scmring1(A)) )
     => m1_subset_1(k14_scmring1(A,B),u1_struct_0(A)) ) )).

fof(dt_k15_scmring1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_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,k1_scmring1(A))
        & m1_subset_1(C,k4_card_3(k2_scmring1(A))) )
     => m1_subset_1(k15_scmring1(A,B,C),k4_card_3(k2_scmring1(A))) ) )).

fof(dt_k16_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_scmring1(A)
        & l3_vectsp_1(A) )
     => ( v1_funct_1(k16_scmring1(A))
        & v1_funct_2(k16_scmring1(A),k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
        & m2_relset_1(k16_scmring1(A),k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) ) ) )).

fof(d1_scmring1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => k1_scmring1(A) = k2_xboole_0(k2_xboole_0(k2_xboole_0(k2_xboole_0(k1_tarski(k4_tarski(np__0,k1_xboole_0)),a_0_0_scmring1),a_0_1_scmring1),a_0_2_scmring1),a_1_0_scmring1(A)) ) )).

fof(fraenkel_a_0_0_scmring1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_scmring1)
    <=> ? [B,C,D] :
          ( m2_subset_1(B,k5_numbers,k1_gr_cy_1(np__8))
          & m2_subset_1(C,k5_numbers,k2_ami_2)
          & m2_subset_1(D,k5_numbers,k2_ami_2)
          & A = k4_tarski(B,k2_finseq_4(k2_ami_2,C,D))
          & r2_hidden(B,k2_enumset1(np__1,np__2,np__3,np__4)) ) ) )).

fof(fraenkel_a_0_1_scmring1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_scmring1)
    <=> ? [B] :
          ( m2_subset_1(B,k5_numbers,k3_ami_2)
          & A = k4_tarski(np__6,k12_finseq_1(k3_ami_2,B)) ) ) )).

fof(fraenkel_a_0_2_scmring1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_scmring1)
    <=> ? [B,C] :
          ( m2_subset_1(B,k5_numbers,k3_ami_2)
          & m2_subset_1(C,k5_numbers,k2_ami_2)
          & A = k4_tarski(np__7,k2_finseq_4(k5_numbers,B,C)) ) ) )).

fof(fraenkel_a_1_0_scmring1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_struct_0(B) )
     => ( r2_hidden(A,a_1_0_scmring1(B))
      <=> ? [C,D] :
            ( m2_subset_1(C,k5_numbers,k2_ami_2)
            & m1_subset_1(D,u1_struct_0(B))
            & A = k4_tarski(np__5,k10_finseq_1(C,D)) ) ) ) )).
%------------------------------------------------------------------------------