SET007 Axioms: SET007+257.ax


%------------------------------------------------------------------------------
% File     : SET007+257 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Normed Space
% 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    : normsp_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   75 (  17 unit)
%            Number of atoms       :  695 (  53 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  682 (  62 ~  ;   0  |; 438  &)
%                                         (  13 <=>; 169 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   37 (   4 constant; 0-5 arity)
%            Number of variables   :  193 (   0 singleton; 180 !;  13 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_normsp_1,axiom,(
    ? [A] :
      ( l1_normsp_1(A)
      & v1_normsp_1(A) ) )).

fof(rc2_normsp_1,axiom,(
    ? [A] :
      ( l1_normsp_1(A)
      & ~ v3_struct_0(A) ) )).

fof(rc3_normsp_1,axiom,(
    ? [A] :
      ( l1_normsp_1(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v7_rlvect_1(A)
      & v1_normsp_1(A)
      & v2_normsp_1(A) ) )).

fof(d1_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_normsp_1(A,B) = k2_seq_1(u1_struct_0(A),k1_numbers,u1_normsp_1(A),B) ) ) )).

fof(d2_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_normsp_1(A) )
     => ( v2_normsp_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( k1_normsp_1(A,B) = np__0
                       => B = k1_rlvect_1(A) )
                      & ( B = k1_rlvect_1(A)
                       => k1_normsp_1(A,B) = np__0 )
                      & k1_normsp_1(A,k3_rlvect_1(A,B,D)) = k4_real_1(k18_complex1(D),k1_normsp_1(A,B))
                      & r1_xreal_0(k1_normsp_1(A,k2_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))) ) ) ) ) ) ) )).

fof(t1_normsp_1,axiom,(
    $true )).

fof(t2_normsp_1,axiom,(
    $true )).

fof(t3_normsp_1,axiom,(
    $true )).

fof(t4_normsp_1,axiom,(
    $true )).

fof(t5_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => k1_normsp_1(A,k1_rlvect_1(A)) = np__0 ) )).

fof(t6_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_normsp_1(A,k5_rlvect_1(A,B)) = k1_normsp_1(A,B) ) ) )).

fof(t7_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))) ) ) ) )).

fof(t8_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_xreal_0(np__0,k1_normsp_1(A,B)) ) ) )).

fof(t9_normsp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_rlvect_1(C)
                & v2_normsp_1(C)
                & l1_normsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(C))
                     => r1_xreal_0(k1_normsp_1(C,k4_rlvect_1(C,k3_rlvect_1(C,D,A),k3_rlvect_1(C,E,B))),k3_real_1(k4_real_1(k18_complex1(A),k1_normsp_1(C,D)),k4_real_1(k18_complex1(B),k1_normsp_1(C,E)))) ) ) ) ) ) )).

fof(t10_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k1_normsp_1(A,k6_rlvect_1(A,B,C)) = np__0
              <=> B = C ) ) ) ) )).

fof(t11_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_normsp_1(A,k6_rlvect_1(A,B,C)) = k1_normsp_1(A,k6_rlvect_1(A,C,B)) ) ) ) )).

fof(t12_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k5_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C)),k1_normsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) )).

fof(t13_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k18_complex1(k5_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))),k1_normsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) )).

fof(t14_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,k6_rlvect_1(A,B,D)),k1_normsp_1(A,k6_rlvect_1(A,D,C)))) ) ) ) ) )).

fof(t15_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & k1_normsp_1(A,k6_rlvect_1(A,B,C)) = np__0 ) ) ) ) )).

fof(t16_normsp_1,axiom,(
    $true )).

fof(t17_normsp_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ( ( v1_funct_1(A)
                  & v1_funct_2(A,k5_numbers,u1_struct_0(B))
                  & m2_relset_1(A,k5_numbers,u1_struct_0(B)) )
              <=> ( k1_relat_1(A) = k5_numbers
                  & ! [D] :
                      ( r2_hidden(D,k5_numbers)
                     => m1_subset_1(k1_funct_1(A,D),u1_struct_0(B)) ) ) ) ) ) ) )).

fof(t18_normsp_1,axiom,(
    $true )).

fof(t19_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,k5_numbers,u1_struct_0(A))
              & m2_relset_1(C,k5_numbers,u1_struct_0(A))
              & k2_relat_1(C) = k1_struct_0(A,B) ) ) ) )).

fof(t20_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ~ ( ? [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_relat_1(B) != k1_struct_0(A,C) ) ) ) ) )).

fof(t21_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & k2_relat_1(B) = k1_struct_0(A,C) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) ) ) ) ) )).

fof(t22_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,D)) ) ) ) ) ) )).

fof(t23_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,D)) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,D) ) ) ) ) ) )).

fof(t24_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ~ ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,D) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) )).

fof(t25_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,u1_struct_0(A))
          & m2_relset_1(B,k5_numbers,u1_struct_0(A))
          & k2_relat_1(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) )).

fof(d3_normsp_1,axiom,(
    $true )).

fof(d4_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v5_seqm_3(B)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) )).

fof(t26_normsp_1,axiom,(
    $true )).

fof(t27_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v5_seqm_3(B)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & k2_relat_1(B) = k1_struct_0(A,C) ) ) ) ) )).

fof(d5_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k3_normsp_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k4_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) )).

fof(d6_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k4_normsp_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) )).

fof(d7_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k5_normsp_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) )).

fof(d8_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k6_normsp_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k3_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) )).

fof(d9_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_normsp_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & r1_xreal_0(D,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) )).

fof(t28_normsp_1,axiom,(
    $true )).

fof(t29_normsp_1,axiom,(
    $true )).

fof(t30_normsp_1,axiom,(
    $true )).

fof(t31_normsp_1,axiom,(
    $true )).

fof(t32_normsp_1,axiom,(
    $true )).

fof(t33_normsp_1,axiom,(
    $true )).

fof(t34_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => v4_normsp_1(k3_normsp_1(A,B,C),A) ) ) ) ) )).

fof(t35_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => v4_normsp_1(k4_normsp_1(A,B,C),A) ) ) ) ) )).

fof(t36_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v4_normsp_1(C,A)
               => v4_normsp_1(k5_normsp_1(A,C,B),A) ) ) ) ) )).

fof(t37_normsp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & v2_normsp_1(B)
            & l1_normsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( v4_normsp_1(C,B)
               => v4_normsp_1(k6_normsp_1(B,C,A),B) ) ) ) ) )).

fof(d10_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k7_normsp_1(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k1_normsp_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) )).

fof(t38_normsp_1,axiom,(
    $true )).

fof(t39_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_normsp_1(B,A)
           => v4_seq_2(k7_normsp_1(A,B)) ) ) ) )).

fof(d11_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_normsp_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k8_normsp_1(A,B)
                <=> ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ? [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(E,F)
                                  & r1_xreal_0(D,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ) )).

fof(t40_normsp_1,axiom,(
    $true )).

fof(t41_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(C,A)
                  & k8_normsp_1(A,C) = B )
               => ( v4_seq_2(k7_normsp_1(A,k5_normsp_1(A,C,B)))
                  & k2_seq_2(k7_normsp_1(A,k5_normsp_1(A,C,B))) = np__0 ) ) ) ) ) )).

fof(t42_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => k8_normsp_1(A,k3_normsp_1(A,B,C)) = k4_rlvect_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) )).

fof(t43_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => k8_normsp_1(A,k4_normsp_1(A,B,C)) = k6_rlvect_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) )).

fof(t44_normsp_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v4_normsp_1(C,A)
               => k8_normsp_1(A,k5_normsp_1(A,C,B)) = k6_rlvect_1(A,k8_normsp_1(A,C),B) ) ) ) ) )).

fof(t45_normsp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & v2_normsp_1(B)
            & l1_normsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( v4_normsp_1(C,B)
               => k8_normsp_1(B,k6_normsp_1(B,C,A)) = k3_rlvect_1(B,k8_normsp_1(B,C),A) ) ) ) ) )).

fof(dt_l1_normsp_1,axiom,(
    ! [A] :
      ( l1_normsp_1(A)
     => l2_rlvect_1(A) ) )).

fof(existence_l1_normsp_1,axiom,(
    ? [A] : l1_normsp_1(A) )).

fof(abstractness_v1_normsp_1,axiom,(
    ! [A] :
      ( l1_normsp_1(A)
     => ( v1_normsp_1(A)
       => A = g1_normsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_normsp_1(A)) ) ) )).

fof(redefinition_v3_normsp_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => ( v3_normsp_1(B,A)
      <=> v5_seqm_3(B) ) ) )).

fof(dt_k1_normsp_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_normsp_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k1_normsp_1(A,B),k1_numbers) ) )).

fof(dt_k2_normsp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) )).

fof(redefinition_k2_normsp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k3_normsp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k3_normsp_1(A,B,C))
        & v1_funct_2(k3_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k3_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k4_normsp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k4_normsp_1(A,B,C))
        & v1_funct_2(k4_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k4_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k5_normsp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_funct_1(k5_normsp_1(A,B,C))
        & v1_funct_2(k5_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k5_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k6_normsp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k1_numbers) )
     => ( v1_funct_1(k6_normsp_1(A,B,C))
        & v1_funct_2(k6_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k6_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k7_normsp_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k7_normsp_1(A,B))
        & v1_funct_2(k7_normsp_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k7_normsp_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k8_normsp_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => m1_subset_1(k8_normsp_1(A,B),u1_struct_0(A)) ) )).

fof(dt_u1_normsp_1,axiom,(
    ! [A] :
      ( l1_normsp_1(A)
     => ( v1_funct_1(u1_normsp_1(A))
        & v1_funct_2(u1_normsp_1(A),u1_struct_0(A),k1_numbers)
        & m2_relset_1(u1_normsp_1(A),u1_struct_0(A),k1_numbers) ) ) )).

fof(dt_g1_normsp_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(k1_numbers,A),A)
        & m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A)
        & v1_funct_1(E)
        & v1_funct_2(E,A,k1_numbers)
        & m1_relset_1(E,A,k1_numbers) )
     => ( v1_normsp_1(g1_normsp_1(A,B,C,D,E))
        & l1_normsp_1(g1_normsp_1(A,B,C,D,E)) ) ) )).

fof(free_g1_normsp_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(k1_numbers,A),A)
        & m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A)
        & v1_funct_1(E)
        & v1_funct_2(E,A,k1_numbers)
        & m1_relset_1(E,A,k1_numbers) )
     => ! [F,G,H,I,J] :
          ( g1_normsp_1(A,B,C,D,E) = g1_normsp_1(F,G,H,I,J)
         => ( A = F
            & B = G
            & C = H
            & D = I
            & E = J ) ) ) )).
%------------------------------------------------------------------------------