SET007 Axioms: SET007+443.ax


%------------------------------------------------------------------------------
% File     : SET007+443 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Monoid of Endomorphisms of Universal and Many Sorted Algebras
% 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    : endalg [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   45 (   5 unit)
%            Number of atoms       :  367 (  28 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  387 (  65 ~  ;   0  |; 216  &)
%                                         (  13 <=>;  93 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   41 (   1 propositional; 0-4 arity)
%            Number of functors    :   30 (   1 constant; 0-6 arity)
%            Number of variables   :   98 (   0 singleton;  95 !;   3 ?)
%            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(fc1_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ( ~ v3_struct_0(k3_endalg(A))
        & v1_vectsp_1(k3_endalg(A)) ) ) )).

fof(fc2_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ( ~ v3_struct_0(k3_endalg(A))
        & v2_group_1(k3_endalg(A))
        & v4_group_1(k3_endalg(A))
        & v1_vectsp_1(k3_endalg(A))
        & v6_vectsp_1(k3_endalg(A))
        & v8_vectsp_1(k3_endalg(A)) ) ) )).

fof(fc3_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( ~ v3_struct_0(k6_endalg(A,B))
        & v1_vectsp_1(k6_endalg(A,B)) ) ) )).

fof(fc4_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( ~ v3_struct_0(k6_endalg(A,B))
        & v2_group_1(k6_endalg(A,B))
        & v4_group_1(k6_endalg(A,B))
        & v1_vectsp_1(k6_endalg(A,B))
        & v6_vectsp_1(k6_endalg(A,B))
        & v8_vectsp_1(k6_endalg(A,B)) ) ) )).

fof(rc1_endalg,axiom,(
    ? [A] :
      ( l1_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v8_vectsp_1(A) ) )).

fof(rc2_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v8_vectsp_1(B)
        & l1_vectsp_1(B) )
     => ? [C] :
          ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
          & v1_relat_1(C)
          & v1_funct_1(C)
          & ~ v1_xboole_0(C)
          & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
          & v1_partfun1(C,u1_struct_0(A),u1_struct_0(B))
          & v1_group_6(C,A,B)
          & v1_endalg(C,A,B) ) ) )).

fof(d1_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
         => ( B = k1_endalg(A)
          <=> ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                  & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
               => ( r2_hidden(C,B)
                <=> r1_alg_1(A,A,C) ) ) ) ) ) )).

fof(t1_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r1_tarski(k1_endalg(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) )).

fof(t2_endalg,axiom,(
    $true )).

fof(t3_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r2_hidden(k6_partfun1(u1_struct_0(A)),k1_endalg(A)) ) )).

fof(t4_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
             => r2_hidden(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),k1_endalg(A)) ) ) ) )).

fof(d2_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A))
            & m2_relset_1(B,k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A)) )
         => ( B = k2_endalg(A)
          <=> ! [C] :
                ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
               => ! [D] :
                    ( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
                   => k2_binop_1(k1_endalg(A),k1_endalg(A),k1_endalg(A),B,C,D) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,D) ) ) ) ) ) )).

fof(d3_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( v1_vectsp_1(B)
            & l1_vectsp_1(B) )
         => ( B = k3_endalg(A)
          <=> ( u1_struct_0(B) = k1_endalg(A)
              & u1_group_1(B) = k2_endalg(A)
              & u1_vectsp_1(B) = k6_partfun1(u1_struct_0(A)) ) ) ) ) )).

fof(t5_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_endalg(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k3_endalg(A)))
             => ! [D] :
                  ( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
                 => ! [E] :
                      ( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
                     => ( ( B = D
                          & C = E )
                       => k1_group_1(k3_endalg(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,E) ) ) ) ) ) ) )).

fof(t6_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => k6_partfun1(u1_struct_0(A)) = k2_group_1(k3_endalg(A)) ) )).

fof(d4_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))
             => ( C = k4_endalg(A,B)
              <=> ( ! [D] :
                      ( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),C)
                     => m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
                  & ! [D] :
                      ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))
                     => ( r2_hidden(D,C)
                      <=> r1_msualg_3(A,B,B,D) ) ) ) ) ) ) ) )).

fof(t7_endalg,axiom,(
    $true )).

fof(t8_endalg,axiom,(
    $true )).

fof(t9_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => r1_tarski(k4_endalg(A,B),k4_card_3(k12_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)))) ) ) )).

fof(t10_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => r2_hidden(k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)),k4_endalg(A,B)) ) ) )).

fof(t11_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m2_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
             => ! [D] :
                  ( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
                 => r2_hidden(k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),D,C),k4_endalg(A,B)) ) ) ) ) )).

fof(t12_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m3_pboole(B,u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)))
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
             => ( B = k10_pboole(k1_tarski(np__0),C)
               => r2_hidden(B,k4_endalg(k7_msualg_1(A),k10_msualg_1(A))) ) ) ) ) )).

fof(d5_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B))
                & m2_relset_1(C,k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B)) )
             => ( C = k5_endalg(A,B)
              <=> ! [D] :
                    ( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
                   => ! [E] :
                        ( m2_autalg_1(E,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
                       => k2_binop_1(k4_endalg(A,B),k4_endalg(A,B),k4_endalg(A,B),C,D,E) = k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),D,E) ) ) ) ) ) ) )).

fof(d6_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v1_vectsp_1(C)
                & l1_vectsp_1(C) )
             => ( C = k6_endalg(A,B)
              <=> ( u1_struct_0(C) = k4_endalg(A,B)
                  & u1_group_1(C) = k5_endalg(A,B)
                  & u1_vectsp_1(C) = k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ) ) )).

fof(t13_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_endalg(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k6_endalg(A,B)))
                 => ! [E] :
                      ( m2_autalg_1(E,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
                     => ! [F] :
                          ( m2_autalg_1(F,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
                         => ( ( C = E
                              & D = F )
                           => k1_group_1(k6_endalg(A,B),C,D) = k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),E,F) ) ) ) ) ) ) ) )).

fof(t14_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)) = k2_group_1(k6_endalg(A,B)) ) ) )).

fof(t15_endalg,axiom,(
    $true )).

fof(t16_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
         => m3_pboole(k10_pboole(k1_tarski(np__0),B),u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A))) ) ) )).

fof(d7_endalg,axiom,(
    $true )).

fof(d8_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v1_endalg(C,A,B)
              <=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_group_1(A)) = k2_group_1(B) ) ) ) ) )).

fof(d9_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v8_vectsp_1(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_endalg(A,B,C)
              <=> v2_funct_1(C) ) ) ) ) )).

fof(d10_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v8_vectsp_1(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r2_endalg(A,B,C)
              <=> k2_relat_1(C) = u1_struct_0(B) ) ) ) ) )).

fof(d11_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v8_vectsp_1(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_endalg(A,B,C)
              <=> ( r2_endalg(A,B,C)
                  & r1_endalg(A,B,C) ) ) ) ) ) )).

fof(t17_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ( v1_funct_1(k6_partfun1(u1_struct_0(A)))
        & v1_funct_2(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A))
        & v1_group_6(k6_partfun1(u1_struct_0(A)),A,A)
        & v1_endalg(k6_partfun1(u1_struct_0(A)),A,A)
        & m2_relset_1(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A)) ) ) )).

fof(d12_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v8_vectsp_1(B)
            & l1_vectsp_1(B) )
         => ( r4_endalg(A,B)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v1_group_6(C,A,B)
                & v1_endalg(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
                & r3_endalg(A,B,C) ) ) ) ) )).

fof(t18_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_relat_1(B) = k1_endalg(A)
              & ! [C] :
                  ( r2_hidden(C,k1_endalg(A))
                 => k1_funct_1(B,C) = k10_pboole(k1_tarski(np__0),C) ) )
           => ( v1_funct_1(B)
              & v1_funct_2(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A))))
              & v1_group_6(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
              & v1_endalg(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
              & m2_relset_1(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))) ) ) ) ) )).

fof(t19_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A))))
            & v1_group_6(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
            & v1_endalg(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
            & m2_relset_1(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))) )
         => ( ! [C] :
                ( r2_hidden(C,k1_endalg(A))
               => k1_funct_1(B,C) = k10_pboole(k1_tarski(np__0),C) )
           => r3_endalg(k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)),B) ) ) ) )).

fof(t20_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r4_endalg(k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A))) ) )).

fof(reflexivity_r4_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v8_vectsp_1(B)
        & l1_vectsp_1(B) )
     => r4_endalg(A,A) ) )).

fof(dt_k1_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => m1_fraenkel(k1_endalg(A),u1_struct_0(A),u1_struct_0(A)) ) )).

fof(dt_k2_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ( v1_funct_1(k2_endalg(A))
        & v1_funct_2(k2_endalg(A),k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A))
        & m2_relset_1(k2_endalg(A),k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A)) ) ) )).

fof(dt_k3_endalg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ( v1_vectsp_1(k3_endalg(A))
        & l1_vectsp_1(k3_endalg(A)) ) ) )).

fof(dt_k4_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => m1_autalg_1(k4_endalg(A,B),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) )).

fof(dt_k5_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v1_funct_1(k5_endalg(A,B))
        & v1_funct_2(k5_endalg(A,B),k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B))
        & m2_relset_1(k5_endalg(A,B),k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B)) ) ) )).

fof(dt_k6_endalg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v1_vectsp_1(k6_endalg(A,B))
        & l1_vectsp_1(k6_endalg(A,B)) ) ) )).
%------------------------------------------------------------------------------