SET007 Axioms: SET007+914.ax


%------------------------------------------------------------------------------
% File     : SET007+914 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Circled Sets, Circled Hull, and Circled Family
% 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    : circled1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   2 unit)
%            Number of atoms       :  468 (  28 equality)
%            Maximal formula depth :   18 (  10 average)
%            Number of connectives :  479 (  53 ~  ;   2  |; 295  &)
%                                         (   9 <=>; 120 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   40 (   5 constant; 0-4 arity)
%            Number of variables   :  123 (   0 singleton; 115 !;   8 ?)
%            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(fc1_circled1,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)
        & v3_rltopsp1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v3_rltopsp1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v2_rltopsp1(k1_rusub_5(A,B,C),A)
        & v3_rltopsp1(k1_rusub_5(A,B,C),A) ) ) )).

fof(fc2_circled1,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)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ~ v1_xboole_0(k1_circled1(A,B)) ) )).

fof(rc1_circled1,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] :
          ( m1_rlvect_2(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(A),k1_numbers)
          & v1_seq_1(B)
          & v1_circled1(B,A) ) ) )).

fof(rc2_circled1,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)
        & l2_rlvect_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] :
          ( m2_rlvect_2(C,A,B)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,u1_struct_0(A),k1_numbers)
          & v1_seq_1(C)
          & v1_circled1(C,A) ) ) )).

fof(t1_circled1,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] :
          ( ( v3_rltopsp1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v3_rltopsp1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => v3_rltopsp1(k1_rusub_5(A,B,C),A) ) ) ) )).

fof(t2_circled1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_rltopsp1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( r1_xreal_0(k18_complex1(D),np__1)
                        & r2_hidden(C,B) )
                     => r2_hidden(k3_rlvect_1(A,C,D),B) ) ) ) ) ) ) )).

fof(d1_circled1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_rltopsp1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( r1_xreal_0(k18_complex1(D),np__1)
                        & r2_hidden(C,B) )
                     => r2_hidden(k3_rlvect_1(A,C,D),B) ) ) ) ) ) ) )).

fof(t3_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( v3_rltopsp1(B,A)
               => v3_rltopsp1(k1_convex1(A,B,C),A) ) ) ) ) )).

fof(t4_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( v3_rltopsp1(B,A)
                          & v3_rltopsp1(C,A) )
                       => v3_rltopsp1(k7_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,E)),A) ) ) ) ) ) ) )).

fof(t5_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => ( ( v3_rltopsp1(B,A)
                                  & v3_rltopsp1(C,A)
                                  & v3_rltopsp1(D,A) )
                               => v3_rltopsp1(k7_rusub_4(A,k7_rusub_4(A,k1_convex1(A,B,E),k1_convex1(A,C,F)),k1_convex1(A,D,G)),A) ) ) ) ) ) ) ) ) )).

fof(t6_circled1,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) )
     => v3_rltopsp1(k3_rusub_4(A,k1_rlsub_1(A)),A) ) )).

fof(t7_circled1,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) )
     => v3_rltopsp1(k3_rusub_4(A,k2_rlsub_1(A)),A) ) )).

fof(t8_circled1,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] :
          ( ( v3_rltopsp1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v3_rltopsp1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => v3_rltopsp1(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) )).

fof(t9_circled1,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] :
          ( ( v3_rltopsp1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v3_rltopsp1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => v3_rltopsp1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) )).

fof(d2_circled1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( C = k1_circled1(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r2_hidden(D,C)
                    <=> ( v3_rltopsp1(D,A)
                        & r1_tarski(B,D) ) ) ) ) ) ) ) )).

fof(d3_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k2_circled1(A,B) = k6_setfam_1(u1_struct_0(A),k1_circled1(A,B)) ) ) )).

fof(t10_circled1,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] :
          ( 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)
               => r1_tarski(k1_circled1(A,C),k1_circled1(A,B)) ) ) ) ) )).

fof(t11_circled1,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] :
          ( 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)
               => r1_tarski(k2_circled1(A,B),k2_circled1(A,C)) ) ) ) ) )).

fof(t12_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k2_circled1(A,B)) ) ) )).

fof(t13_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v3_rltopsp1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r1_tarski(B,C)
               => r1_tarski(k2_circled1(A,B),C) ) ) ) ) )).

fof(t14_circled1,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] :
          ( ( v3_rltopsp1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k2_circled1(A,B) = B ) ) )).

fof(t15_circled1,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) )
     => k2_circled1(A,k1_pre_topc(A)) = k1_xboole_0 ) )).

fof(t16_circled1,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] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k1_convex1(A,k2_circled1(A,B),C) = k2_circled1(A,k1_convex1(A,B,C)) ) ) ) )).

fof(d4_circled1,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] :
          ( m1_rlvect_2(B,A)
         => ( v1_circled1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v2_funct_1(C)
                & k2_relat_1(C) = k10_rlvect_2(A,B)
                & ? [D] :
                    ( m2_finseq_1(D,k1_numbers)
                    & k3_finseq_1(D) = k3_finseq_1(C)
                    & k15_rvsum_1(D) = np__1
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k4_finseq_1(D))
                         => ( k1_funct_1(D,E) = k1_funct_1(B,k1_funct_1(C,E))
                            & r1_xreal_0(np__0,k1_funct_1(D,E)) ) ) ) ) ) ) ) ) )).

fof(t17_circled1,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] :
          ( m1_rlvect_2(B,A)
         => ~ ( v1_circled1(B,A)
              & k10_rlvect_2(A,B) = k1_xboole_0 ) ) ) )).

fof(t18_circled1,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] :
          ( m1_rlvect_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( v1_circled1(B,A)
                  & r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,B,C),np__0)
                  & r2_hidden(C,k10_rlvect_2(A,B)) ) ) ) ) )).

fof(t19_circled1,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] :
          ( m1_rlvect_2(B,A)
         => ~ ( v1_circled1(B,A)
              & B = k11_rlvect_2(A) ) ) ) )).

fof(t20_circled1,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] :
          ( m1_rlvect_2(B,A)
          & v1_circled1(B,A) ) ) )).

fof(t21_circled1,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ? [C] :
              ( m2_rlvect_2(C,A,B)
              & v1_circled1(C,A) ) ) ) )).

fof(d5_circled1,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] :
          ( B = k3_circled1(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ( v1_circled1(C,A)
                & m1_rlvect_2(C,A) ) ) ) ) )).

fof(d6_circled1,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( C = k4_circled1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ( v1_circled1(D,A)
                    & m2_rlvect_2(D,A,B) ) ) ) ) ) )).

fof(t22_circled1,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] :
              ( v1_circled1(C,A)
              & m1_rlvect_2(C,A)
              & k13_rlvect_2(A,C) = B
              & ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                 => ( r2_hidden(B,D)
                   => ( v1_circled1(C,A)
                      & m2_rlvect_2(C,A,D) ) ) ) ) ) ) )).

fof(t23_circled1,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & ! [D] :
                      ( ( v1_circled1(D,A)
                        & m1_rlvect_2(D,A) )
                     => ? [E] :
                          ( ~ v1_xboole_0(E)
                          & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                          & r1_tarski(k8_rlvect_2(A,B,C),E)
                          & ~ ( v1_circled1(D,A)
                              & m2_rlvect_2(D,A,E) ) ) ) ) ) ) ) )).

fof(t24_circled1,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_circled1(B,A)
            & m1_rlvect_2(B,A) )
         => ! [C] :
              ( ( v1_circled1(C,A)
                & m1_rlvect_2(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ~ r1_xreal_0(k4_real_1(D,E),np__0)
                       => k10_rlvect_2(A,k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,E,C))) = k2_rlvect_2(A,k10_rlvect_2(A,k15_rlvect_2(A,D,B)),k10_rlvect_2(A,k15_rlvect_2(A,E,C))) ) ) ) ) ) ) )).

fof(t25_circled1,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_rlvect_2(C,A)
             => ( ( v1_circled1(C,A)
                  & k10_rlvect_2(A,C) = k7_rlvect_2(A,B) )
               => ( k8_funct_2(u1_struct_0(A),k1_numbers,C,B) = np__1
                  & k13_rlvect_2(A,C) = k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,C,B)) ) ) ) ) ) )).

fof(t26_circled1,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_rlvect_2(D,A)
                 => ( ( v1_circled1(D,A)
                      & k10_rlvect_2(A,D) = k8_rlvect_2(A,B,C) )
                   => ( B = C
                      | ( k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,D,B),k8_funct_2(u1_struct_0(A),k1_numbers,D,C)) = np__1
                        & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,B))
                        & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,C))
                        & k13_rlvect_2(A,D) = k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,D,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,D,C))) ) ) ) ) ) ) ) )).

fof(t27_circled1,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_rlvect_2(C,A,k7_rlvect_2(A,B))
             => ( v1_circled1(C,A)
               => ( k8_funct_2(u1_struct_0(A),k1_numbers,C,B) = np__1
                  & k13_rlvect_2(A,C) = k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,C,B)) ) ) ) ) ) )).

fof(t28_circled1,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_rlvect_2(D,A,k8_rlvect_2(A,B,C))
                 => ( v1_circled1(D,A)
                   => ( B = C
                      | ( k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,D,B),k8_funct_2(u1_struct_0(A),k1_numbers,D,C)) = np__1
                        & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,B))
                        & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,C))
                        & k13_rlvect_2(A,D) = k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,D,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,D,C))) ) ) ) ) ) ) ) )).

fof(dt_k1_circled1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k1_circled1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k2_circled1,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)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v3_rltopsp1(k2_circled1(A,B),A)
        & m1_subset_1(k2_circled1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k3_circled1,axiom,(
    $true )).

fof(dt_k4_circled1,axiom,(
    $true )).
%------------------------------------------------------------------------------