SET007 Axioms: SET007+895.ax


%------------------------------------------------------------------------------
% File     : SET007+895 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Connected Subsets of the Real Line
% 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    : rcomp_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  115 (   0 unit)
%            Number of atoms       :  890 (  99 equality)
%            Maximal formula depth :   31 (  10 average)
%            Number of connectives :  874 (  99 ~  ;  31  |; 367  &)
%                                         (  10 <=>; 367 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   46 (   0 propositional; 1-5 arity)
%            Number of functors    :   52 (   6 constant; 0-4 arity)
%            Number of variables   :  336 (   0 singleton; 320 !;  16 ?)
%            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_rcomp_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_subset_1(A)) ) )).

fof(cc1_rcomp_3,axiom,(
    ! [A] :
      ( m1_topmetr(A,k8_metric_1)
     => v1_borsuk_6(A) ) )).

fof(fc2_rcomp_3,axiom,(
    ! [A] :
      ( ( v1_borsuk_6(A)
        & l1_struct_0(A) )
     => ( v1_membered(u1_struct_0(A))
        & v2_membered(u1_struct_0(A)) ) ) )).

fof(rc1_rcomp_3,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A)
      & v1_membered(A)
      & v2_membered(A)
      & v1_seq_4(A)
      & v2_seq_4(A) ) )).

fof(fc3_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k1_rcomp_2(A,B))
        & v2_membered(k1_rcomp_2(A,B))
        & v1_seq_4(k1_rcomp_2(A,B))
        & v2_seq_4(k1_rcomp_2(A,B))
        & v3_seq_4(k1_rcomp_2(A,B)) ) ) )).

fof(fc4_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k2_rcomp_2(A,B))
        & v2_membered(k2_rcomp_2(A,B))
        & v1_seq_4(k2_rcomp_2(A,B))
        & v2_seq_4(k2_rcomp_2(A,B))
        & v3_seq_4(k2_rcomp_2(A,B)) ) ) )).

fof(fc5_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k2_rcomp_1(A,B))
        & v2_membered(k2_rcomp_1(A,B))
        & v1_seq_4(k2_rcomp_1(A,B))
        & v2_seq_4(k2_rcomp_1(A,B))
        & v3_seq_4(k2_rcomp_1(A,B))
        & v3_rcomp_1(k2_rcomp_1(A,B)) ) ) )).

fof(fc6_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k1_rcomp_1(A,B))
        & v2_membered(k1_rcomp_1(A,B))
        & v1_seq_4(k1_rcomp_1(A,B))
        & v2_seq_4(k1_rcomp_1(A,B))
        & v3_seq_4(k1_rcomp_1(A,B))
        & v1_rcomp_1(k1_rcomp_1(A,B))
        & v2_rcomp_1(k1_rcomp_1(A,B))
        & v1_jct_misc(k1_rcomp_1(A,B)) ) ) )).

fof(fc7_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k1_rcomp_2(A,B))
        & v2_membered(k1_rcomp_2(A,B))
        & v1_seq_4(k1_rcomp_2(A,B))
        & v2_seq_4(k1_rcomp_2(A,B))
        & v3_seq_4(k1_rcomp_2(A,B))
        & v1_jct_misc(k1_rcomp_2(A,B)) ) ) )).

fof(fc8_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k2_rcomp_2(A,B))
        & v2_membered(k2_rcomp_2(A,B))
        & v1_seq_4(k2_rcomp_2(A,B))
        & v2_seq_4(k2_rcomp_2(A,B))
        & v3_seq_4(k2_rcomp_2(A,B))
        & v1_jct_misc(k2_rcomp_2(A,B)) ) ) )).

fof(fc9_rcomp_3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k2_rcomp_1(A,B))
        & v2_membered(k2_rcomp_1(A,B))
        & v1_seq_4(k2_rcomp_1(A,B))
        & v2_seq_4(k2_rcomp_1(A,B))
        & v3_seq_4(k2_rcomp_1(A,B))
        & v3_rcomp_1(k2_rcomp_1(A,B))
        & v1_jct_misc(k2_rcomp_1(A,B)) ) ) )).

fof(rc2_rcomp_3,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & v1_seq_4(A)
      & v2_seq_4(A)
      & v3_seq_4(A)
      & v3_rcomp_1(A)
      & v1_jct_misc(A) ) )).

fof(fc10_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ v1_xboole_0(k2_limfunc1(A))
        & v1_membered(k2_limfunc1(A))
        & v2_membered(k2_limfunc1(A))
        & v1_seq_4(k2_limfunc1(A))
        & ~ v2_seq_4(k2_limfunc1(A))
        & v2_rcomp_1(k2_limfunc1(A))
        & v1_jct_misc(k2_limfunc1(A)) ) ) )).

fof(fc11_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ v1_xboole_0(k12_prob_1(A))
        & v1_membered(k12_prob_1(A))
        & v2_membered(k12_prob_1(A))
        & v1_seq_4(k12_prob_1(A))
        & ~ v2_seq_4(k12_prob_1(A))
        & v3_rcomp_1(k12_prob_1(A))
        & v1_jct_misc(k12_prob_1(A)) ) ) )).

fof(fc12_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ v1_xboole_0(k3_limfunc1(A))
        & v1_membered(k3_limfunc1(A))
        & v2_membered(k3_limfunc1(A))
        & ~ v1_seq_4(k3_limfunc1(A))
        & v2_seq_4(k3_limfunc1(A))
        & v2_rcomp_1(k3_limfunc1(A))
        & v1_jct_misc(k3_limfunc1(A)) ) ) )).

fof(fc13_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ v1_xboole_0(k4_limfunc1(A))
        & v1_membered(k4_limfunc1(A))
        & v2_membered(k4_limfunc1(A))
        & ~ v1_seq_4(k4_limfunc1(A))
        & v2_seq_4(k4_limfunc1(A))
        & v3_rcomp_1(k4_limfunc1(A))
        & v1_jct_misc(k4_limfunc1(A)) ) ) )).

fof(fc14_rcomp_3,axiom,
    ( ~ v1_xboole_0(k2_subset_1(k1_numbers))
    & v1_membered(k2_subset_1(k1_numbers))
    & v2_membered(k2_subset_1(k1_numbers))
    & ~ v1_seq_4(k2_subset_1(k1_numbers))
    & ~ v2_seq_4(k2_subset_1(k1_numbers))
    & v2_rcomp_1(k2_subset_1(k1_numbers))
    & v3_rcomp_1(k2_subset_1(k1_numbers))
    & v1_jct_misc(k2_subset_1(k1_numbers)) )).

fof(rc3_rcomp_3,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & ~ v3_seq_4(A)
      & v2_rcomp_1(A)
      & v3_rcomp_1(A)
      & v1_jct_misc(A) ) )).

fof(fc15_rcomp_3,axiom,(
    ! [A] :
      ( ( v2_topalg_2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topalg_2))) )
     => ( v1_pre_topc(k3_pre_topc(k3_topalg_2,A))
        & v2_pre_topc(k3_pre_topc(k3_topalg_2,A))
        & v1_borsuk_6(k3_pre_topc(k3_topalg_2,A))
        & v3_topalg_2(k3_pre_topc(k3_topalg_2,A)) ) ) )).

fof(fc16_rcomp_3,axiom,(
    ! [A] :
      ( ( v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( v1_membered(k5_toprealb(A))
        & v2_membered(k5_toprealb(A))
        & v2_topalg_2(k5_toprealb(A)) ) ) )).

fof(fc17_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ v3_struct_0(k4_topmetr(A,A))
        & v1_pre_topc(k4_topmetr(A,A))
        & v2_pre_topc(k4_topmetr(A,A))
        & v2_compts_1(k4_topmetr(A,A))
        & v6_group_1(k4_topmetr(A,A))
        & v3_realset2(k4_topmetr(A,A))
        & v10_pscomp_1(k4_topmetr(A,A))
        & v1_borsuk_6(k4_topmetr(A,A)) ) ) )).

fof(rc4_rcomp_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v3_pre_topc(B,A)
          & v4_pre_topc(B,A)
          & v2_connsp_1(B,A) ) ) )).

fof(rc5_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v1_connsp_1(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v3_pre_topc(B,A)
          & v4_pre_topc(B,A)
          & v2_connsp_1(B,A) ) ) )).

fof(rc6_rcomp_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
          & ~ v1_xboole_0(B)
          & v1_tops_2(B,A)
          & v2_tops_2(B,A)
          & v1_rcomp_3(B,A) ) ) )).

fof(t1_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A)
        & v2_seq_4(A) )
     => ! [B] :
          ( ( v2_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( r1_tarski(A,B)
           => r2_hidden(k3_seq_4(A),B) ) ) ) )).

fof(t2_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A)
        & v1_seq_4(A) )
     => ! [B] :
          ( ( v2_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( r1_tarski(A,B)
           => r2_hidden(k2_seq_4(A),B) ) ) ) )).

fof(t3_rcomp_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => k7_pscomp_1(k4_subset_1(k1_numbers,A,B)) = k4_subset_1(k1_numbers,k7_pscomp_1(A),k7_pscomp_1(B)) ) ) )).

fof(t4_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k4_pscomp_1(k1_rcomp_2(A,B)) = A ) ) ) )).

fof(t5_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k3_pscomp_1(k1_rcomp_2(A,B)) = B ) ) ) )).

fof(t6_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k4_pscomp_1(k2_rcomp_2(A,B)) = A ) ) ) )).

fof(t7_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k3_pscomp_1(k2_rcomp_2(A,B)) = B ) ) ) )).

fof(t8_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( k1_rcomp_1(A,B) = k1_rcomp_1(C,D)
                   => ( ( ~ r1_xreal_0(A,B)
                        & ~ r1_xreal_0(C,D) )
                      | ( A = C
                        & B = D ) ) ) ) ) ) ) )).

fof(t9_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( k2_rcomp_1(A,B) = k2_rcomp_1(C,D)
                   => ( ( r1_xreal_0(B,A)
                        & r1_xreal_0(D,C) )
                      | ( A = C
                        & B = D ) ) ) ) ) ) ) )).

fof(t10_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( k2_rcomp_2(A,B) = k2_rcomp_2(C,D)
                   => ( ( r1_xreal_0(B,A)
                        & r1_xreal_0(D,C) )
                      | ( A = C
                        & B = D ) ) ) ) ) ) ) )).

fof(t11_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( k1_rcomp_2(A,B) = k1_rcomp_2(C,D)
                   => ( ( r1_xreal_0(B,A)
                        & r1_xreal_0(D,C) )
                      | ( A = C
                        & B = D ) ) ) ) ) ) ) )).

fof(t12_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_tarski(k1_rcomp_2(A,B),k1_rcomp_1(C,D))
                   => ( r1_xreal_0(B,A)
                      | ( r1_xreal_0(C,A)
                        & r1_xreal_0(B,D) ) ) ) ) ) ) ) )).

fof(t13_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_tarski(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
                   => ( r1_xreal_0(B,A)
                      | ( r1_xreal_0(C,A)
                        & r1_xreal_0(B,D) ) ) ) ) ) ) ) )).

fof(t14_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_tarski(k2_rcomp_2(A,B),k1_rcomp_1(C,D))
                   => ( r1_xreal_0(B,A)
                      | ( r1_xreal_0(C,A)
                        & r1_xreal_0(B,D) ) ) ) ) ) ) ) )).

fof(t15_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_tarski(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
                   => ( r1_xreal_0(B,A)
                      | ( r1_xreal_0(C,A)
                        & r1_xreal_0(B,D) ) ) ) ) ) ) ) )).

fof(t16_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k3_subset_1(k1_numbers,k1_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k12_prob_1(A),k4_limfunc1(B)) ) ) )).

fof(t17_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k3_subset_1(k1_numbers,k2_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B)) ) ) )).

fof(t18_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k3_subset_1(k1_numbers,k1_rcomp_2(A,B)) = k4_subset_1(k1_numbers,k12_prob_1(A),k3_limfunc1(B)) ) ) )).

fof(t19_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k3_subset_1(k1_numbers,k2_rcomp_2(A,B)) = k4_subset_1(k1_numbers,k2_limfunc1(A),k4_limfunc1(B)) ) ) )).

fof(t20_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k5_subset_1(k1_numbers,k1_rcomp_1(A,B),k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B))) = k2_tarski(A,B) ) ) ) )).

fof(t21_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_pscomp_1(k2_limfunc1(A)) = A ) )).

fof(t22_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_pscomp_1(k12_prob_1(A)) = A ) )).

fof(t23_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k4_pscomp_1(k3_limfunc1(A)) = A ) )).

fof(t24_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k4_pscomp_1(k4_limfunc1(A)) = A ) )).

fof(t25_rcomp_3,axiom,(
    ! [A] :
      ( ( v3_seq_4(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( ( r2_hidden(k4_pscomp_1(A),A)
          & r2_hidden(k3_pscomp_1(A),A) )
       => A = k1_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) )).

fof(t26_rcomp_3,axiom,(
    ! [A] :
      ( ( v3_seq_4(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( ~ r2_hidden(k4_pscomp_1(A),A)
       => r1_tarski(A,k2_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A))) ) ) )).

fof(t27_rcomp_3,axiom,(
    ! [A] :
      ( ( v3_seq_4(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( r2_hidden(k3_pscomp_1(A),A)
       => ( r2_hidden(k4_pscomp_1(A),A)
          | A = k2_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) ) )).

fof(t28_rcomp_3,axiom,(
    ! [A] :
      ( ( v3_seq_4(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( ~ r2_hidden(k3_pscomp_1(A),A)
       => r1_tarski(A,k1_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A))) ) ) )).

fof(t29_rcomp_3,axiom,(
    ! [A] :
      ( ( v3_seq_4(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( r2_hidden(k4_pscomp_1(A),A)
       => ( r2_hidden(k3_pscomp_1(A),A)
          | A = k1_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) ) )).

fof(t30_rcomp_3,axiom,(
    ! [A] :
      ( ( v3_seq_4(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ~ ( ~ r2_hidden(k4_pscomp_1(A),A)
          & ~ r2_hidden(k3_pscomp_1(A),A)
          & ~ r1_tarski(A,k2_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A))) ) ) )).

fof(t31_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_seq_4(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ~ ( ~ r2_hidden(k4_pscomp_1(A),A)
          & ~ r2_hidden(k3_pscomp_1(A),A)
          & A != k2_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) )).

fof(t32_rcomp_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_seq_4(A)
       => r1_tarski(A,k2_limfunc1(k3_pscomp_1(A))) ) ) )).

fof(t33_rcomp_3,axiom,(
    ! [A] :
      ( ( v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( ( v1_seq_4(A)
          & r2_hidden(k3_pscomp_1(A),A) )
       => ( v2_seq_4(A)
          | A = k2_limfunc1(k3_pscomp_1(A)) ) ) ) )).

fof(t34_rcomp_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_seq_4(A)
       => ( r2_hidden(k3_pscomp_1(A),A)
          | r1_tarski(A,k12_prob_1(k3_pscomp_1(A))) ) ) ) )).

fof(t35_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( v1_seq_4(A)
       => ( v2_seq_4(A)
          | r2_hidden(k3_pscomp_1(A),A)
          | A = k12_prob_1(k3_pscomp_1(A)) ) ) ) )).

fof(t36_rcomp_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v2_seq_4(A)
       => r1_tarski(A,k3_limfunc1(k4_pscomp_1(A))) ) ) )).

fof(t37_rcomp_3,axiom,(
    ! [A] :
      ( ( v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( ( v2_seq_4(A)
          & r2_hidden(k4_pscomp_1(A),A) )
       => ( v1_seq_4(A)
          | A = k3_limfunc1(k4_pscomp_1(A)) ) ) ) )).

fof(t38_rcomp_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v2_seq_4(A)
       => ( r2_hidden(k4_pscomp_1(A),A)
          | r1_tarski(A,k4_limfunc1(k4_pscomp_1(A))) ) ) ) )).

fof(t39_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( v2_seq_4(A)
       => ( v1_seq_4(A)
          | r2_hidden(k4_pscomp_1(A),A)
          | A = k4_limfunc1(k4_pscomp_1(A)) ) ) ) )).

fof(t40_rcomp_3,axiom,(
    ! [A] :
      ( ( v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ~ ( ~ v1_seq_4(A)
          & ~ v2_seq_4(A)
          & A != k1_numbers ) ) )).

fof(t41_rcomp_3,axiom,(
    ! [A] :
      ( ( v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ~ ( ~ v1_xboole_0(A)
          & A != k1_numbers
          & ! [B] :
              ( v1_xreal_0(B)
             => A != k2_limfunc1(B) )
          & ! [B] :
              ( v1_xreal_0(B)
             => A != k12_prob_1(B) )
          & ! [B] :
              ( v1_xreal_0(B)
             => A != k3_limfunc1(B) )
          & ! [B] :
              ( v1_xreal_0(B)
             => A != k4_limfunc1(B) )
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => ~ ( r1_xreal_0(B,C)
                      & A = k1_rcomp_1(B,C) ) ) )
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & A = k1_rcomp_2(B,C) ) ) )
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & A = k2_rcomp_2(B,C) ) ) )
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & A = k2_rcomp_1(B,C) ) ) ) ) ) )).

fof(t42_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_jct_misc(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ~ ( ~ r2_hidden(A,B)
              & ~ r1_xreal_0(A,k4_pscomp_1(B))
              & ~ r1_xreal_0(k3_pscomp_1(B),A) ) ) ) )).

fof(t43_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_seq_4(A)
        & v1_jct_misc(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_seq_4(B)
            & v1_jct_misc(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( r1_xreal_0(k4_pscomp_1(A),k4_pscomp_1(B))
              & r1_xreal_0(k3_pscomp_1(B),k3_pscomp_1(A)) )
           => ( ( k4_pscomp_1(A) = k4_pscomp_1(B)
                & r2_hidden(k4_pscomp_1(B),B)
                & ~ r2_hidden(k4_pscomp_1(A),A) )
              | ( k3_pscomp_1(A) = k3_pscomp_1(B)
                & r2_hidden(k3_pscomp_1(B),B)
                & ~ r2_hidden(k3_pscomp_1(A),A) )
              | r1_tarski(B,A) ) ) ) ) )).

fof(t44_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( ( r1_xreal_0(A,B)
                  & C = k1_rcomp_1(A,B) )
               => k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) )).

fof(t45_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k2_rcomp_1(A,B)
               => ( r1_xreal_0(B,A)
                  | k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ) )).

fof(t46_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k1_rcomp_2(A,B)
               => ( r1_xreal_0(B,A)
                  | k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ) )).

fof(t47_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k2_rcomp_2(A,B)
               => ( r1_xreal_0(B,A)
                  | k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ) )).

fof(t48_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k1_rcomp_1(A,B)
               => k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) )).

fof(t49_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k2_rcomp_1(A,B)
               => k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) )).

fof(t50_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k1_rcomp_2(A,B)
               => k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) )).

fof(t51_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
             => ( C = k2_rcomp_2(A,B)
               => k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) )).

fof(t52_rcomp_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( A = B
           => ( v2_connsp_1(A,k3_topalg_2)
            <=> v1_jct_misc(B) ) ) ) ) )).

fof(t53_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
               => ( v3_seq_4(C)
                  & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) ) ) ) ) ) )).

fof(t54_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_xreal_0(A,B)
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
                       => ( ( E = k1_rcomp_2(C,D)
                            & r1_xreal_0(D,B) )
                         => ( r1_xreal_0(C,A)
                            | k1_tops_1(k4_topmetr(A,B),E) = k2_rcomp_1(C,D) ) ) ) ) ) ) ) ) )).

fof(t55_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_xreal_0(A,B)
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
                       => ( ( E = k2_rcomp_2(C,D)
                            & r1_xreal_0(A,C) )
                         => ( r1_xreal_0(B,D)
                            | k1_tops_1(k4_topmetr(A,B),E) = k2_rcomp_1(C,D) ) ) ) ) ) ) ) ) )).

fof(t56_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
                 => ( C = D
                   => ( v2_connsp_1(C,k4_topmetr(A,B))
                    <=> v1_jct_misc(D) ) ) ) ) ) ) )).

fof(t57_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] :
                ( ( v3_pre_topc(C,k4_topmetr(A,B))
                  & v2_connsp_1(C,k4_topmetr(A,B))
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))) )
               => ~ ( ~ v1_xboole_0(C)
                    & C != k1_rcomp_1(A,B)
                    & ! [D] :
                        ( v1_xreal_0(D)
                       => ~ ( ~ r1_xreal_0(D,A)
                            & r1_xreal_0(D,B)
                            & C = k1_rcomp_2(A,D) ) )
                    & ! [D] :
                        ( v1_xreal_0(D)
                       => ~ ( r1_xreal_0(A,D)
                            & ~ r1_xreal_0(B,D)
                            & C = k2_rcomp_2(D,B) ) )
                    & ! [D] :
                        ( v1_xreal_0(D)
                       => ! [E] :
                            ( v1_xreal_0(E)
                           => ~ ( r1_xreal_0(A,D)
                                & ~ r1_xreal_0(E,D)
                                & r1_xreal_0(E,B)
                                & C = k2_rcomp_1(D,E) ) ) ) ) ) ) ) ) )).

fof(t58_rcomp_3,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_pre_topc(A,B)
          <=> r1_compts_1(A,B,k2_pre_topc(A)) ) ) ) )).

fof(t60_rcomp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_realset2(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_pre_topc(A,C)
               => r2_hidden(k1_struct_0(A,B),C) ) ) ) ) )).

fof(d1_rcomp_3,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_rcomp_3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v2_connsp_1(C,A) ) ) ) ) ) )).

fof(d2_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                  & v1_tops_2(C,k4_topmetr(A,B))
                  & v1_rcomp_3(C,k4_topmetr(A,B))
                  & r1_xreal_0(A,B) )
               => ! [D] :
                    ( m2_finseq_1(D,k1_pcomps_1(k1_numbers))
                   => ( m1_rcomp_3(D,A,B,C)
                    <=> ( r1_tarski(k5_relset_1(k5_numbers,k1_pcomps_1(k1_numbers),D),C)
                        & k5_setfam_1(k1_numbers,k5_relset_1(k5_numbers,k1_pcomps_1(k1_numbers),D)) = k1_rcomp_1(A,B)
                        & ! [E] :
                            ( v4_ordinal2(E)
                           => ( r1_xreal_0(np__1,E)
                             => ( ~ ( r1_xreal_0(E,k3_finseq_1(D))
                                    & v1_xboole_0(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)) )
                                & ( r1_xreal_0(k2_xcmplx_0(E,np__1),k3_finseq_1(D))
                                 => ( r1_xreal_0(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__1))))
                                    & r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__1))))
                                    & ~ r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__1)))) ) )
                                & ( r1_xreal_0(k2_xcmplx_0(E,np__2),k3_finseq_1(D))
                                 => r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__2)))) ) ) ) )
                        & ( r2_hidden(k1_rcomp_1(A,B),C)
                         => D = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)) )
                        & ( ~ r2_hidden(k1_rcomp_1(A,B),C)
                         => ( ? [E] :
                                ( v1_xreal_0(E)
                                & ~ r1_xreal_0(E,A)
                                & r1_xreal_0(E,B)
                                & k1_funct_1(D,np__1) = k1_rcomp_2(A,E) )
                            & ? [E] :
                                ( v1_xreal_0(E)
                                & r1_xreal_0(A,E)
                                & ~ r1_xreal_0(B,E)
                                & k1_funct_1(D,k3_finseq_1(D)) = k2_rcomp_2(E,B) )
                            & ! [E] :
                                ( v4_ordinal2(E)
                               => ~ ( ~ r1_xreal_0(E,np__1)
                                    & ~ r1_xreal_0(k3_finseq_1(D),E)
                                    & ! [F] :
                                        ( v1_xreal_0(F)
                                       => ! [G] :
                                            ( v1_xreal_0(G)
                                           => ~ ( r1_xreal_0(A,F)
                                                & ~ r1_xreal_0(G,F)
                                                & r1_xreal_0(G,B)
                                                & k1_funct_1(D,E) = k2_rcomp_1(F,G) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t63_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                  & v1_tops_2(C,k4_topmetr(A,B))
                  & v1_rcomp_3(C,k4_topmetr(A,B))
                  & r1_xreal_0(A,B)
                  & r2_hidden(k1_rcomp_1(A,B),C) )
               => m1_rcomp_3(k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)),A,B,C) ) ) ) ) )).

fof(t64_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,A)))))
         => ! [C] :
              ( m1_rcomp_3(C,A,A,B)
             => ( ( r1_pre_topc(k4_topmetr(A,A),B)
                  & v1_tops_2(B,k4_topmetr(A,A))
                  & v1_rcomp_3(B,k4_topmetr(A,A)) )
               => C = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,A)) ) ) ) ) )).

fof(t65_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B) )
                   => r1_xreal_0(np__1,k3_finseq_1(D)) ) ) ) ) ) )).

fof(t66_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B)
                      & k3_finseq_1(D) = np__1 )
                   => D = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)) ) ) ) ) ) )).

fof(t67_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                     => ! [F] :
                          ( m1_rcomp_3(F,A,B,E)
                         => ( ( r1_pre_topc(k4_topmetr(A,B),E)
                              & v1_tops_2(E,k4_topmetr(A,B))
                              & v1_rcomp_3(E,k4_topmetr(A,B))
                              & r1_xreal_0(A,B)
                              & r2_hidden(C,k4_finseq_1(F))
                              & r2_hidden(D,k4_finseq_1(F)) )
                           => ( r1_xreal_0(D,C)
                              | r1_xreal_0(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,C)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,D))) ) ) ) ) ) ) ) ) )).

fof(t68_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                     => ! [F] :
                          ( m1_rcomp_3(F,A,B,E)
                         => ( ( r1_pre_topc(k4_topmetr(A,B),E)
                              & v1_tops_2(E,k4_topmetr(A,B))
                              & v1_rcomp_3(E,k4_topmetr(A,B))
                              & r1_xreal_0(A,B)
                              & r2_hidden(C,k4_finseq_1(F))
                              & r2_hidden(D,k4_finseq_1(F)) )
                           => ( r1_xreal_0(D,C)
                              | r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,C)),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,D))) ) ) ) ) ) ) ) ) )).

fof(t69_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                 => ! [E] :
                      ( m1_rcomp_3(E,A,B,D)
                     => ~ ( r1_pre_topc(k4_topmetr(A,B),D)
                          & v1_tops_2(D,k4_topmetr(A,B))
                          & v1_rcomp_3(D,k4_topmetr(A,B))
                          & r1_xreal_0(A,B)
                          & r1_xreal_0(np__1,C)
                          & r1_xreal_0(k2_xcmplx_0(C,np__1),k3_finseq_1(E))
                          & v1_xboole_0(k2_rcomp_1(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,k2_xcmplx_0(C,np__1))),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,C)))) ) ) ) ) ) ) )).

fof(t70_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B) )
                   => k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,np__1)) = A ) ) ) ) ) )).

fof(t71_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B) )
                   => r2_hidden(A,k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,np__1)) ) ) ) ) ) )).

fof(t72_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B) )
                   => k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k3_finseq_1(D))) = B ) ) ) ) ) )).

fof(t73_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B) )
                   => r2_hidden(B,k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k3_finseq_1(D))) ) ) ) ) ) )).

fof(d3_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                      & v1_tops_2(C,k4_topmetr(A,B))
                      & v1_rcomp_3(C,k4_topmetr(A,B))
                      & r1_xreal_0(A,B) )
                   => ! [E] :
                        ( m2_finseq_1(E,k1_numbers)
                       => ( m2_rcomp_3(E,A,B,C,D)
                        <=> ( k3_finseq_1(E) = k1_nat_1(k3_finseq_1(D),np__1)
                            & k2_seq_1(k5_numbers,k1_numbers,E,np__1) = A
                            & k2_seq_1(k5_numbers,k1_numbers,E,k3_finseq_1(E)) = B
                            & ! [F] :
                                ( v4_ordinal2(F)
                               => ( r1_xreal_0(np__1,F)
                                 => ( r1_xreal_0(k3_finseq_1(E),k2_xcmplx_0(F,np__1))
                                    | r2_hidden(k2_seq_1(k5_numbers,k1_numbers,E,k2_xcmplx_0(F,np__1)),k2_rcomp_1(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(F,np__1))),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,F)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t74_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ! [E] :
                      ( m2_rcomp_3(E,A,B,C,D)
                     => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                          & v1_tops_2(C,k4_topmetr(A,B))
                          & v1_rcomp_3(C,k4_topmetr(A,B))
                          & r1_xreal_0(A,B) )
                       => r1_xreal_0(np__2,k3_finseq_1(E)) ) ) ) ) ) ) )).

fof(t75_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ! [E] :
                      ( m2_rcomp_3(E,A,B,C,D)
                     => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                          & v1_tops_2(C,k4_topmetr(A,B))
                          & v1_rcomp_3(C,k4_topmetr(A,B))
                          & r1_xreal_0(A,B)
                          & k3_finseq_1(D) = np__1 )
                       => E = k10_finseq_1(A,B) ) ) ) ) ) ) )).

fof(t76_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                 => ! [E] :
                      ( m1_rcomp_3(E,A,B,D)
                     => ! [F] :
                          ( m2_rcomp_3(F,A,B,D,E)
                         => ~ ( r1_pre_topc(k4_topmetr(A,B),D)
                              & v1_tops_2(D,k4_topmetr(A,B))
                              & v1_rcomp_3(D,k4_topmetr(A,B))
                              & r1_xreal_0(A,B)
                              & r1_xreal_0(np__1,C)
                              & ~ r1_xreal_0(k3_finseq_1(F),k2_xcmplx_0(C,np__1))
                              & r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,C)),k2_seq_1(k5_numbers,k1_numbers,F,k2_xcmplx_0(C,np__1))) ) ) ) ) ) ) ) )).

fof(t77_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                 => ! [E] :
                      ( m1_rcomp_3(E,A,B,D)
                     => ! [F] :
                          ( m2_rcomp_3(F,A,B,D,E)
                         => ~ ( r1_pre_topc(k4_topmetr(A,B),D)
                              & v1_tops_2(D,k4_topmetr(A,B))
                              & v1_rcomp_3(D,k4_topmetr(A,B))
                              & r1_xreal_0(A,B)
                              & ~ r1_xreal_0(C,np__1)
                              & r1_xreal_0(C,k3_finseq_1(E))
                              & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,F,C),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,C))) ) ) ) ) ) ) ) )).

fof(t78_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                 => ! [E] :
                      ( m1_rcomp_3(E,A,B,D)
                     => ! [F] :
                          ( m2_rcomp_3(F,A,B,D,E)
                         => ( ( r1_pre_topc(k4_topmetr(A,B),D)
                              & v1_tops_2(D,k4_topmetr(A,B))
                              & v1_rcomp_3(D,k4_topmetr(A,B))
                              & r1_xreal_0(A,B)
                              & r1_xreal_0(np__1,C) )
                           => ( r1_xreal_0(k3_finseq_1(E),C)
                              | r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,F,C),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,k2_xcmplx_0(C,np__1)))) ) ) ) ) ) ) ) ) )).

fof(t79_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
             => ! [D] :
                  ( m1_rcomp_3(D,A,B,C)
                 => ! [E] :
                      ( m2_rcomp_3(E,A,B,C,D)
                     => ( ( r1_pre_topc(k4_topmetr(A,B),C)
                          & v1_tops_2(C,k4_topmetr(A,B))
                          & v1_rcomp_3(C,k4_topmetr(A,B)) )
                       => ( r1_xreal_0(B,A)
                          | v1_seqm_3(E) ) ) ) ) ) ) ) )).

fof(t80_rcomp_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
                 => ! [E] :
                      ( m1_rcomp_3(E,A,B,D)
                     => ! [F] :
                          ( m2_rcomp_3(F,A,B,D,E)
                         => ( ( r1_pre_topc(k4_topmetr(A,B),D)
                              & v1_tops_2(D,k4_topmetr(A,B))
                              & v1_rcomp_3(D,k4_topmetr(A,B))
                              & r1_xreal_0(A,B)
                              & r1_xreal_0(np__1,C) )
                           => ( r1_xreal_0(k3_finseq_1(F),C)
                              | r1_tarski(k1_rcomp_1(k2_seq_1(k5_numbers,k1_numbers,F,C),k2_seq_1(k5_numbers,k1_numbers,F,k2_xcmplx_0(C,np__1))),k1_funct_1(E,C)) ) ) ) ) ) ) ) ) )).

fof(dt_m1_rcomp_3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) )
     => ! [D] :
          ( m1_rcomp_3(D,A,B,C)
         => m2_finseq_1(D,k1_pcomps_1(k1_numbers)) ) ) )).

fof(existence_m1_rcomp_3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) )
     => ? [D] : m1_rcomp_3(D,A,B,C) ) )).

fof(dt_m2_rcomp_3,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
        & m1_rcomp_3(D,A,B,C) )
     => ! [E] :
          ( m2_rcomp_3(E,A,B,C,D)
         => m2_finseq_1(E,k1_numbers) ) ) )).

fof(existence_m2_rcomp_3,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
        & m1_rcomp_3(D,A,B,C) )
     => ? [E] : m2_rcomp_3(E,A,B,C,D) ) )).

fof(t59_rcomp_3,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_pre_topc(A,B)
                  & C = k3_tex_1(u1_struct_0(A),B,a_2_0_rcomp_3(A,B)) )
               => r1_pre_topc(A,C) ) ) ) ) )).

fof(t61_rcomp_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_pre_topc(A,B)
                  & v1_finset_1(B) )
               => ! [D] :
                    ( ( C = k3_tex_1(u1_struct_0(A),B,a_2_1_rcomp_3(A,B))
                      & D = a_2_2_rcomp_3(A,C) )
                   => r5_orders_1(k1_wellord2(D),D) ) ) ) ) ) )).

fof(t62_rcomp_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B,C] :
          ( C = a_2_3_rcomp_3(A,B)
         => ! [D] :
              ( ( r7_orders_1(k1_wellord2(C),D)
                & r2_hidden(D,k3_relat_1(k1_wellord2(C))) )
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                 => ( r2_hidden(E,D)
                   => ! [F] :
                        ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [G] :
                            ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( r2_hidden(F,D)
                                & r2_hidden(G,D)
                                & r1_tarski(E,k4_subset_1(u1_struct_0(A),F,G))
                                & E != F
                                & E != G ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_0_rcomp_3,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(B)
        & v1_finset_1(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
     => ( r2_hidden(A,a_2_0_rcomp_3(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
            & A = D
            & r2_hidden(D,C)
            & ? [E] :
                ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
                & r2_hidden(E,C)
                & r1_tarski(D,E)
                & D != E ) ) ) ) )).

fof(fraenkel_a_2_1_rcomp_3,axiom,(
    ! [A,B,C] :
      ( ( v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
     => ( r2_hidden(A,a_2_1_rcomp_3(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
            & A = D
            & r2_hidden(D,C)
            & ? [E] :
                ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
                & r2_hidden(E,C)
                & r1_tarski(D,E)
                & D != E ) ) ) ) )).

fof(fraenkel_a_2_2_rcomp_3,axiom,(
    ! [A,B,C] :
      ( ( v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
     => ( r2_hidden(A,a_2_2_rcomp_3(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
            & A = D
            & r1_pre_topc(B,D)
            & r1_tarski(D,C) ) ) ) )).

fof(fraenkel_a_2_3_rcomp_3,axiom,(
    ! [A,B,C] :
      ( ( v2_pre_topc(B)
        & l1_pre_topc(B) )
     => ( r2_hidden(A,a_2_3_rcomp_3(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
            & A = D
            & r1_pre_topc(B,D)
            & r1_tarski(D,C) ) ) ) )).
%------------------------------------------------------------------------------