SET007 Axioms: SET007+746.ax


%------------------------------------------------------------------------------
% File     : SET007+746 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Convex Sets and Convex Combinations
% 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    : convex1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   55 (   0 unit)
%            Number of atoms       :  513 (  59 equality)
%            Maximal formula depth :   19 (  10 average)
%            Number of connectives :  537 (  79 ~  ;  10  |; 254  &)
%                                         (  11 <=>; 183 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   0 propositional; 1-3 arity)
%            Number of functors    :   45 (   5 constant; 0-4 arity)
%            Number of variables   :  188 (   0 singleton; 179 !;   9 ?)
%            Maximal term depth    :    6 (   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_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_convex1(B,A) ) ) )).

fof(rc2_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v1_xboole_0(B)
          & v1_membered(B)
          & v2_membered(B)
          & v3_membered(B)
          & v4_membered(B)
          & v5_membered(B)
          & v1_convex1(B,A) ) ) )).

fof(d2_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_convex1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ( ( r2_hidden(C,B)
                            & r2_hidden(D,B) )
                         => ( r1_xreal_0(E,np__0)
                            | r1_xreal_0(np__1,E)
                            | r2_hidden(k2_rlvect_1(A,k3_rlvect_1(A,C,E),k3_rlvect_1(A,D,k5_real_1(np__1,E))),B) ) ) ) ) ) ) ) ) )).

fof(t1_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
             => ( v1_convex1(B,A)
               => v1_convex1(k1_convex1(A,B,C),A) ) ) ) ) )).

fof(t2_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_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)))
             => ( ( v1_convex1(B,A)
                  & v1_convex1(C,A) )
               => v1_convex1(k7_rusub_4(A,B,C),A) ) ) ) ) )).

fof(t3_convex1,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)))
             => ( ( v1_convex1(B,A)
                  & v1_convex1(C,A) )
               => v1_convex1(k1_rusub_5(A,B,C),A) ) ) ) ) )).

fof(t4_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_convex1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ~ r1_xreal_0(np__1,C)
                    & ~ r1_tarski(k6_rusub_4(A,k1_convex1(A,B,C),k1_convex1(A,B,k5_real_1(np__1,C))),B) ) ) ) ) ) )).

fof(t5_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_convex1(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ~ r1_xreal_0(np__1,C)
                    & ~ r1_tarski(k7_rusub_4(A,k1_convex1(A,B,k5_real_1(np__1,C)),k1_convex1(A,B,C)),B) ) ) ) ) ) )).

fof(t6_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_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)))
             => ( ( v1_convex1(B,A)
                  & v1_convex1(C,A) )
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => v1_convex1(k7_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,k5_real_1(np__1,D))),A) ) ) ) ) ) )).

fof(t7_convex1,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,u1_struct_0(A))
             => ( v1_convex1(B,A)
              <=> v1_convex1(k5_rusub_4(A,B,C),A) ) ) ) ) )).

fof(t8_convex1,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) )
     => v1_convex1(k3_rusub_4(A,k1_rlsub_1(A)),A) ) )).

fof(t9_convex1,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) )
     => v1_convex1(k3_rusub_4(A,k2_rlsub_1(A)),A) ) )).

fof(t10_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( B = k1_xboole_0
           => v1_convex1(B,A) ) ) ) )).

fof(t11_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_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)
                     => ( ( v1_convex1(B,A)
                          & v1_convex1(C,A) )
                       => v1_convex1(k7_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,E)),A) ) ) ) ) ) ) )).

fof(t12_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => r1_tarski(k1_convex1(A,B,k3_real_1(C,D)),k6_rusub_4(A,k1_convex1(A,B,C),k1_convex1(A,B,D))) ) ) ) ) )).

fof(t13_convex1,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)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r1_xreal_0(np__0,C)
                      & r1_xreal_0(np__0,D)
                      & v1_convex1(B,A) )
                   => k7_rusub_4(A,k1_convex1(A,B,C),k1_convex1(A,B,D)) = k1_convex1(A,B,k3_real_1(C,D)) ) ) ) ) ) )).

fof(t14_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_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)
                             => ( ( v1_convex1(B,A)
                                  & v1_convex1(C,A)
                                  & v1_convex1(D,A) )
                               => v1_convex1(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(t15_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v1_convex1(C,A) ) )
           => v1_convex1(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t16_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_rusub_4(B,A)
           => v1_convex1(B,A) ) ) ) )).

fof(d3_convex1,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)
         => ( v2_convex1(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(t21_convex1,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)
         => ~ ( v2_convex1(B,A)
              & k10_rlvect_2(A,B) = k1_xboole_0 ) ) ) )).

fof(t22_convex1,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))
             => ~ ( v2_convex1(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(t23_convex1,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)
         => ~ ( v2_convex1(B,A)
              & B = k11_rlvect_2(A) ) ) ) )).

fof(t24_convex1,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))
             => ( v2_convex1(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(t25_convex1,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))
                 => ( v2_convex1(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(t26_convex1,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_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m2_rlvect_2(E,A,k9_rlvect_2(A,B,C,D))
                     => ( v2_convex1(E,A)
                       => ( B = C
                          | C = D
                          | D = B
                          | ( k3_real_1(k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,E,B),k8_funct_2(u1_struct_0(A),k1_numbers,E,C)),k8_funct_2(u1_struct_0(A),k1_numbers,E,D)) = np__1
                            & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,B))
                            & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))
                            & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))
                            & k13_rlvect_2(A,E) = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,E,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))),k3_rlvect_1(A,D,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))) ) ) ) ) ) ) ) ) )).

fof(t27_convex1,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)
             => ( ( v2_convex1(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 ) ) ) ) )).

fof(t28_convex1,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)
                 => ( ( v2_convex1(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)) ) ) ) ) ) ) ) )).

fof(t29_convex1,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_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_rlvect_2(E,A)
                     => ( ( v2_convex1(E,A)
                          & k10_rlvect_2(A,E) = k9_rlvect_2(A,B,C,D) )
                       => ( B = C
                          | C = D
                          | D = B
                          | ( k3_real_1(k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,E,B),k8_funct_2(u1_struct_0(A),k1_numbers,E,C)),k8_funct_2(u1_struct_0(A),k1_numbers,E,D)) = np__1
                            & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,B))
                            & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))
                            & r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))
                            & k13_rlvect_2(A,E) = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,E,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))),k3_rlvect_1(A,D,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))) ) ) ) ) ) ) ) ) )).

fof(d4_convex1,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 = k2_convex1(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r2_hidden(D,C)
                    <=> ( v1_convex1(D,A)
                        & r1_tarski(B,D) ) ) ) ) ) ) ) )).

fof(d5_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_convex1(A,B) = k6_setfam_1(u1_struct_0(A),k2_convex1(A,B)) ) ) )).

fof(t30_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_convex1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r1_tarski(B,C)
               => r1_tarski(k3_convex1(A,B),C) ) ) ) ) )).

fof(t31_convex1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C,D] :
          ~ ( v2_funct_1(A)
            & k2_relat_1(A) = k1_enumset1(B,C,D)
            & B != C
            & C != D
            & D != B
            & A != k11_finseq_1(B,C,D)
            & A != k11_finseq_1(B,D,C)
            & A != k11_finseq_1(C,B,D)
            & A != k11_finseq_1(C,D,B)
            & A != k11_finseq_1(D,B,C)
            & A != k11_finseq_1(D,C,B) ) ) )).

fof(t32_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k1_convex1(A,B,np__1) = B ) ) )).

fof(t33_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k1_convex1(A,B,C) = k1_xboole_0 ) ) ) )).

fof(t34_convex1,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))) )
         => k1_convex1(A,B,np__0) = k7_rlvect_2(A,k1_rlvect_1(A)) ) ) )).

fof(t35_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k6_rusub_4(A,B,k7_rlvect_2(A,k1_rlvect_1(A))) = B ) ) )).

fof(t36_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & l1_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)))
                 => k6_rusub_4(A,k6_rusub_4(A,B,C),D) = k6_rusub_4(A,B,k6_rusub_4(A,C,D)) ) ) ) ) )).

fof(t37_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => k1_convex1(A,k1_convex1(A,B,D),C) = k1_convex1(A,B,k4_real_1(C,D)) ) ) ) ) )).

fof(t38_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
                 => k1_convex1(A,k6_rusub_4(A,B,C),D) = k6_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,D)) ) ) ) ) )).

fof(t39_convex1,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(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r1_tarski(B,C)
                   => r1_tarski(k1_convex1(A,B,D),k1_convex1(A,C,D)) ) ) ) ) ) )).

fof(t40_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k6_rusub_4(A,B,C) = k1_xboole_0 ) ) ) )).

fof(dt_k1_convex1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_numbers) )
     => m1_subset_1(k1_convex1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k2_convex1,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(k2_convex1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k3_convex1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_convex1(k3_convex1(A,B),A)
        & m1_subset_1(k3_convex1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(d1_convex1,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_numbers)
             => k1_convex1(A,B,C) = a_3_0_convex1(A,B,C) ) ) ) )).

fof(t17_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( B = a_3_1_convex1(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(t18_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( B = a_3_2_convex1(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(t19_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( B = a_3_3_convex1(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(t20_convex1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( B = a_3_4_convex1(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(fraenkel_a_3_0_convex1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l2_rlvect_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_0_convex1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k3_rlvect_1(B,E,D)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_3_1_convex1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_1_convex1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_xreal_0(D,k1_bhsp_1(B,E,C)) ) ) ) )).

fof(fraenkel_a_3_2_convex1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_2_convex1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ~ r1_xreal_0(k1_bhsp_1(B,E,C),D) ) ) ) )).

fof(fraenkel_a_3_3_convex1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_3_convex1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_xreal_0(k1_bhsp_1(B,E,C),D) ) ) ) )).

fof(fraenkel_a_3_4_convex1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_4_convex1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ~ r1_xreal_0(D,k1_bhsp_1(B,E,C)) ) ) ) )).
%------------------------------------------------------------------------------