SET007 Axioms: SET007+742.ax


%------------------------------------------------------------------------------
% File     : SET007+742 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Linear Combinations in Real Unitary 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    : rusub_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   51 (   0 unit)
%            Number of atoms       :  655 (  45 equality)
%            Maximal formula depth :   19 (  11 average)
%            Number of connectives :  678 (  74 ~  ;   2  |; 415  &)
%                                         (  11 <=>; 176 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-3 arity)
%            Number of functors    :   31 (   2 constant; 0-5 arity)
%            Number of variables   :  184 (   0 singleton; 167 !;  17 ?)
%            Maximal term depth    :    5 (   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(t1_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( r1_rlvect_1(k1_rusub_3(A,B),C)
            <=> ? [D] :
                  ( m2_rlvect_2(D,A,B)
                  & C = k13_rlvect_2(A,D) ) ) ) ) )).

fof(t2_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( r2_hidden(C,B)
             => r1_rlvect_1(k1_rusub_3(A,B),C) ) ) ) )).

fof(t3_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => k1_rusub_3(A,k1_subset_1(u1_struct_0(A))) = k1_rusub_1(A) ) )).

fof(t4_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( k1_rusub_3(A,B) = k1_rusub_1(A)
              & B != k1_xboole_0
              & B != k7_rlvect_2(A,k1_rlvect_1(A)) ) ) ) )).

fof(t5_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( ( v1_bhsp_1(B)
            & m1_rusub_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( C = u1_struct_0(B)
               => k1_rusub_3(A,C) = B ) ) ) ) )).

fof(t6_rusub_3,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)
        & v1_bhsp_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( B = u1_struct_0(A)
           => k1_rusub_3(A,B) = A ) ) ) )).

fof(t7_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r1_tarski(B,C)
               => m1_rusub_1(k1_rusub_3(A,B),k1_rusub_3(A,C)) ) ) ) ) )).

fof(t8_rusub_3,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)
        & v1_bhsp_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( k1_rusub_3(A,B) = A
                  & r1_tarski(B,C) )
               => k1_rusub_3(A,C) = A ) ) ) ) )).

fof(t9_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k1_rusub_3(A,k4_subset_1(u1_struct_0(A),B,C)) = k1_rusub_2(A,k1_rusub_3(A,B),k1_rusub_3(A,C)) ) ) ) )).

fof(t10_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => m1_rusub_1(k1_rusub_3(A,k5_subset_1(u1_struct_0(A),B,C)),k2_rusub_2(A,k1_rusub_3(A,B),k1_rusub_3(A,C))) ) ) ) )).

fof(t11_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( v1_rlvect_3(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( r1_tarski(B,C)
                      & v1_rlvect_3(C,A)
                      & k1_rusub_3(A,C) = g1_bhsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_bhsp_1(A)) ) ) ) ) ) )).

fof(t12_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( k1_rusub_3(A,B) = A
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( r1_tarski(C,B)
                      & v1_rlvect_3(C,A)
                      & k1_rusub_3(A,C) = A ) ) ) ) ) )).

fof(d2_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( m1_rusub_3(B,A)
          <=> ( v1_rlvect_3(B,A)
              & k1_rusub_3(A,B) = g1_bhsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_bhsp_1(A)) ) ) ) ) )).

fof(t13_rusub_3,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)
        & v1_bhsp_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( v1_rlvect_3(B,A)
              & ! [C] :
                  ( m1_rusub_3(C,A)
                 => ~ r1_tarski(B,C) ) ) ) ) )).

fof(t14_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( k1_rusub_3(A,B) = A
              & ! [C] :
                  ( m1_rusub_3(C,A)
                 => ~ r1_tarski(C,B) ) ) ) ) )).

fof(t15_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( v1_rlvect_3(B,A)
              & ! [C] :
                  ( m1_rusub_3(C,A)
                 => ~ r1_tarski(B,C) ) ) ) ) )).

fof(t16_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rlvect_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(A))
                 => ~ ( r1_tarski(k2_relat_1(D),u1_struct_0(k1_rusub_3(A,C)))
                      & ! [E] :
                          ( m2_rlvect_2(E,A,C)
                         => k9_rlvect_1(A,k12_rlvect_2(A,D,B)) != k13_rlvect_2(A,E) ) ) ) ) ) ) )).

fof(t17_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rlvect_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( r1_tarski(k10_rlvect_2(A,B),u1_struct_0(k1_rusub_3(A,C)))
                  & ! [D] :
                      ( m2_rlvect_2(D,A,C)
                     => k13_rlvect_2(A,B) != k13_rlvect_2(A,D) ) ) ) ) ) )).

fof(t18_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rlvect_2(C,A)
             => ( r1_tarski(k10_rlvect_2(A,C),u1_struct_0(B))
               => ! [D] :
                    ( m1_rlvect_2(D,B)
                   => ( D = k7_relat_1(C,u1_struct_0(B))
                     => ( k10_rlvect_2(A,C) = k10_rlvect_2(B,D)
                        & k13_rlvect_2(A,C) = k13_rlvect_2(B,D) ) ) ) ) ) ) ) )).

fof(t19_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rlvect_2(C,B)
             => ? [D] :
                  ( m1_rlvect_2(D,A)
                  & k10_rlvect_2(B,C) = k10_rlvect_2(A,D)
                  & k13_rlvect_2(B,C) = k13_rlvect_2(A,D) ) ) ) ) )).

fof(t20_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rlvect_2(C,A)
             => ~ ( r1_tarski(k10_rlvect_2(A,C),u1_struct_0(B))
                  & ! [D] :
                      ( m1_rlvect_2(D,B)
                     => ~ ( k10_rlvect_2(B,D) = k10_rlvect_2(A,C)
                          & k13_rlvect_2(B,D) = k13_rlvect_2(A,C) ) ) ) ) ) ) )).

fof(t21_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_3(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_rlvect_1(k1_rusub_3(A,B),C) ) ) ) )).

fof(t22_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v1_rlvect_3(C,B)
               => ( v1_rlvect_3(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ) ) )).

fof(t23_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( v1_rlvect_3(C,A)
                  & r1_tarski(C,u1_struct_0(B)) )
               => ( v1_rlvect_3(C,B)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) ) ) ) ) )).

fof(t24_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rusub_3(C,B)
             => ? [D] :
                  ( m1_rusub_3(D,A)
                  & r1_tarski(C,D) ) ) ) ) )).

fof(t25_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_rlvect_3(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( r2_hidden(C,B)
                 => ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( D = k6_subset_1(u1_struct_0(A),B,k7_rlvect_2(A,C))
                          & r1_rlvect_1(k1_rusub_3(A,D),C) ) ) ) ) ) ) ) )).

fof(t26_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_3(B,A)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r1_xboole_0(C,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ~ ( D = k4_subset_1(u1_struct_0(A),B,C)
                        & v1_rlvect_3(D,A) ) ) ) ) ) ) )).

fof(t27_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r1_tarski(C,u1_struct_0(B))
               => m1_rusub_1(k1_rusub_3(A,C),B) ) ) ) ) )).

fof(t28_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( C = D
                   => k1_rusub_3(A,C) = k1_rusub_3(B,D) ) ) ) ) ) )).

fof(t29_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( r1_rlvect_1(k1_rusub_3(A,k7_rlvect_2(A,B)),C)
            <=> ? [D] :
                  ( m1_subset_1(D,k1_numbers)
                  & C = k3_rlvect_1(A,B,D) ) ) ) ) )).

fof(t30_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_rlvect_1(k1_rusub_3(A,k7_rlvect_2(A,B)),B) ) ) )).

fof(t31_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( r2_hidden(D,k3_rusub_1(A,B,k1_rusub_3(A,k7_rlvect_2(A,C))))
                <=> ? [E] :
                      ( m1_subset_1(E,k1_numbers)
                      & D = k4_rlvect_1(A,B,k3_rlvect_1(A,C,E)) ) ) ) ) ) )).

fof(t32_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,B,C)),D)
                <=> ? [E] :
                      ( m1_subset_1(E,k1_numbers)
                      & ? [F] :
                          ( m1_subset_1(F,k1_numbers)
                          & D = k4_rlvect_1(A,k3_rlvect_1(A,B,E),k3_rlvect_1(A,C,F)) ) ) ) ) ) ) )).

fof(t33_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,B,C)),B)
                & r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,B,C)),C) ) ) ) ) )).

fof(t34_rusub_3,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_bhsp_1(A)
        & l1_bhsp_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))
                 => ! [E] :
                      ( r2_hidden(E,k3_rusub_1(A,B,k1_rusub_3(A,k8_rlvect_2(A,C,D))))
                    <=> ? [F] :
                          ( m1_subset_1(F,k1_numbers)
                          & ? [G] :
                              ( m1_subset_1(G,k1_numbers)
                              & E = k4_rlvect_1(A,k4_rlvect_1(A,B,k3_rlvect_1(A,C,F)),k3_rlvect_1(A,D,G)) ) ) ) ) ) ) ) )).

fof(t35_rusub_3,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_bhsp_1(A)
        & l1_bhsp_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))
                 => ! [E] :
                      ( r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),E)
                    <=> ? [F] :
                          ( m1_subset_1(F,k1_numbers)
                          & ? [G] :
                              ( m1_subset_1(G,k1_numbers)
                              & ? [H] :
                                  ( m1_subset_1(H,k1_numbers)
                                  & E = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,F),k3_rlvect_1(A,C,G)),k3_rlvect_1(A,D,H)) ) ) ) ) ) ) ) ) )).

fof(t36_rusub_3,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_bhsp_1(A)
        & l1_bhsp_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_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),B)
                    & r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),C)
                    & r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),D) ) ) ) ) ) )).

fof(t37_rusub_3,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_bhsp_1(A)
        & l1_bhsp_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))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( r2_hidden(F,k3_rusub_1(A,B,k1_rusub_3(A,k9_rlvect_2(A,C,D,E))))
                        <=> ? [G] :
                              ( m1_subset_1(G,k1_numbers)
                              & ? [H] :
                                  ( m1_subset_1(H,k1_numbers)
                                  & ? [I] :
                                      ( m1_subset_1(I,k1_numbers)
                                      & F = k4_rlvect_1(A,k4_rlvect_1(A,k4_rlvect_1(A,B,k3_rlvect_1(A,C,G)),k3_rlvect_1(A,D,H)),k3_rlvect_1(A,E,I)) ) ) ) ) ) ) ) ) ) )).

fof(t38_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_rlvect_1(k1_rusub_3(A,k7_rlvect_2(A,C)),B)
               => ( B = k1_rlvect_1(A)
                  | k1_rusub_3(A,k7_rlvect_2(A,B)) = k1_rusub_3(A,k7_rlvect_2(A,C)) ) ) ) ) ) )).

fof(t39_rusub_3,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_bhsp_1(A)
        & l1_bhsp_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))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( ( v1_rlvect_3(k8_rlvect_2(A,B,C),A)
                          & r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,D,E)),B)
                          & r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,D,E)),C) )
                       => ( B = C
                          | ( k1_rusub_3(A,k8_rlvect_2(A,D,E)) = k1_rusub_3(A,k8_rlvect_2(A,B,C))
                            & v1_rlvect_3(k8_rlvect_2(A,D,E),A)
                            & D != E ) ) ) ) ) ) ) ) )).

fof(t40_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( r1_rlvect_1(k1_rusub_1(A),B)
        <=> B = k1_rlvect_1(A) ) ) )).

fof(t41_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rusub_1(C,A)
             => ! [D] :
                  ( m1_rusub_1(D,A)
                 => ( m1_rusub_1(B,D)
                   => m1_rusub_1(k2_rusub_2(A,B,C),D) ) ) ) ) ) )).

fof(t42_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rusub_1(C,A)
             => ! [D] :
                  ( m1_rusub_1(D,A)
                 => ( ( m1_rusub_1(B,C)
                      & m1_rusub_1(B,D) )
                   => m1_rusub_1(B,k2_rusub_2(A,C,D)) ) ) ) ) ) )).

fof(t43_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rusub_1(C,A)
             => ! [D] :
                  ( m1_rusub_1(D,A)
                 => ( ( m1_rusub_1(B,D)
                      & m1_rusub_1(C,D) )
                   => m1_rusub_1(k1_rusub_2(A,B,C),D) ) ) ) ) ) )).

fof(t44_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rusub_1(C,A)
             => ! [D] :
                  ( m1_rusub_1(D,A)
                 => ( m1_rusub_1(B,C)
                   => m1_rusub_1(B,k1_rusub_2(A,C,D)) ) ) ) ) ) )).

fof(t45_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_1(B,A)
         => ! [C] :
              ( m1_rusub_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( m1_rusub_1(B,C)
                   => r1_tarski(k3_rusub_1(A,D,B),k3_rusub_1(A,D,C)) ) ) ) ) ) )).

fof(dt_m1_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_rusub_3(B,A)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(existence_m1_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ? [B] : m1_rusub_3(B,A) ) )).

fof(dt_k1_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_bhsp_1(k1_rusub_3(A,B))
        & m1_rusub_1(k1_rusub_3(A,B),A) ) ) )).

fof(d1_rusub_3,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_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_bhsp_1(C)
                & m1_rusub_1(C,A) )
             => ( C = k1_rusub_3(A,B)
              <=> u1_struct_0(C) = a_2_0_rusub_3(A,B) ) ) ) ) )).

fof(fraenkel_a_2_0_rusub_3,axiom,(
    ! [A,B,C] :
      ( ( ~ 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_bhsp_1(B)
        & l1_bhsp_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_0_rusub_3(B,C))
      <=> ? [D] :
            ( m2_rlvect_2(D,B,C)
            & A = k13_rlvect_2(B,D) ) ) ) )).
%------------------------------------------------------------------------------