% Mizar problem: t19_metric_3,metric_3,552,48 
fof(t19_metric_3,conjecture,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] : 
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] : 
                      ( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                     => ! [F] : 
                          ( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                         => ( k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) = 0
                          <=> E = F ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc6_membered,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k4_metric_1,connectedness_r1_xreal_0,d1_binop_1,d7_metric_3,dt_k10_mcart_1,dt_k11_mcart_1,dt_k1_binop_1,dt_k1_funct_1,dt_k1_metric_1,dt_k1_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_metric_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_real_1,dt_k4_mcart_1,dt_k4_metric_1,dt_k4_tarski,dt_k4_zfmisc_1,dt_k7_metric_3,dt_k8_mcart_1,dt_k9_mcart_1,dt_l1_metric_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,existence_l1_metric_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_struct_0,fc1_subset_1,fc2_membered,fc4_subset_1,fc6_membered,fc6_subset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,redefinition_k1_metric_1,redefinition_k3_real_1,redefinition_k4_metric_1,redefinition_m2_relset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,t1_arithm,t1_metric_1,t1_subset,t29_xreal_1,t2_metric_1,t2_subset,t3_subset,t4_subset,t5_metric_1,t5_subset,t60_mcart_1,t6_boole,t7_boole,t8_boole,t9_xreal_1]),
    [file(metric_3,t19_metric_3)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc10_membered,axiom,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => v1_xcmplx_0(B) ) ) ),
    file(membered,cc10_membered),
    []).

fof(cc11_membered,axiom,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) ),
    file(membered,cc11_membered),
    []).

fof(cc12_membered,axiom,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc12_membered),
    []).

fof(cc13_membered,axiom,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc13_membered),
    []).

fof(cc14_membered,axiom,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v4_ordinal2(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc14_membered),
    []).

fof(cc15_membered,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_membered(A)
        & v2_membered(A)
        & v3_membered(A)
        & v4_membered(A)
        & v5_membered(A) ) ) ),
    file(membered,cc15_membered),
    []).

fof(cc16_membered,axiom,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_membered(B) ) ) ),
    file(membered,cc16_membered),
    []).

fof(cc17_membered,axiom,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B) ) ) ) ),
    file(membered,cc17_membered),
    []).

fof(cc18_membered,axiom,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B) ) ) ) ),
    file(membered,cc18_membered),
    []).

fof(cc19_membered,axiom,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B) ) ) ) ),
    file(membered,cc19_membered),
    []).

fof(cc1_membered,axiom,(
    ! [A] : 
      ( v5_membered(A)
     => v4_membered(A) ) ),
    file(membered,cc1_membered),
    []).

fof(cc1_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    []).

fof(cc20_membered,axiom,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B)
            & v5_membered(B) ) ) ) ),
    file(membered,cc20_membered),
    []).

fof(cc2_membered,axiom,(
    ! [A] : 
      ( v4_membered(A)
     => v3_membered(A) ) ),
    file(membered,cc2_membered),
    []).

fof(cc3_membered,axiom,(
    ! [A] : 
      ( v3_membered(A)
     => v2_membered(A) ) ),
    file(membered,cc3_membered),
    []).

fof(cc4_membered,axiom,(
    ! [A] : 
      ( v2_membered(A)
     => v1_membered(A) ) ),
    file(membered,cc4_membered),
    []).

fof(cc6_membered,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_membered(A)
        & v2_membered(A) ) ) ),
    file(membered,cc6_membered),
    []).

fof(commutativity_k2_xcmplx_0,axiom,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ),
    file(xcmplx_0,k2_xcmplx_0),
    []).

fof(commutativity_k3_real_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k3_real_1(A,B) = k3_real_1(B,A) ) ),
    file(real_1,k3_real_1),
    []).

fof(commutativity_k4_metric_1,axiom,(
    ! [A,B,C] : 
      ( ( v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_metric_1(A,B,C) = k4_metric_1(A,C,B) ) ),
    file(metric_1,k4_metric_1),
    []).

fof(connectedness_r1_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) ),
    file(xreal_0,r1_xreal_0),
    []).

fof(d1_binop_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ),
    file(binop_1,d1_binop_1),
    []).

fof(d7_metric_3,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] : 
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] : 
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers)
                        & m2_relset_1(E,k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers) )
                     => ( E = k7_metric_3(A,B,C,D)
                      <=> ! [F] : 
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] : 
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] : 
                                    ( m1_subset_1(H,u1_struct_0(B))
                                   => ! [I] : 
                                        ( m1_subset_1(I,u1_struct_0(B))
                                       => ! [J] : 
                                            ( m1_subset_1(J,u1_struct_0(C))
                                           => ! [K] : 
                                                ( m1_subset_1(K,u1_struct_0(C))
                                               => ! [L] : 
                                                    ( m1_subset_1(L,u1_struct_0(D))
                                                   => ! [M] : 
                                                        ( m1_subset_1(M,u1_struct_0(D))
                                                       => ! [N] : 
                                                            ( m1_subset_1(N,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                                                           => ! [O] : 
                                                                ( m1_subset_1(O,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                                                               => ( ( N = k4_mcart_1(F,H,J,L)
                                                                    & O = k4_mcart_1(G,I,K,M) )
                                                                 => k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),E,N,O) = k3_real_1(k3_real_1(k4_metric_1(A,F,G),k4_metric_1(B,H,I)),k3_real_1(k4_metric_1(C,J,K),k4_metric_1(D,L,M))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file(metric_3,d7_metric_3),
    []).

fof(dt_k10_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k10_mcart_1(A,B,C,D,E),C) ) ),
    file(mcart_1,k10_mcart_1),
    []).

fof(dt_k11_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k11_mcart_1(A,B,C,D,E),D) ) ),
    file(mcart_1,k11_mcart_1),
    []).

fof(dt_k1_binop_1,axiom,(
    $true ),
    file(binop_1,k1_binop_1),
    []).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    []).

fof(dt_k1_metric_1,axiom,(
    ! [A,B,C,D,E] : 
      ( ( v1_funct_1(C)
        & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B) )
     => m1_subset_1(k1_metric_1(A,B,C,D,E),k1_numbers) ) ),
    file(metric_1,k1_metric_1),
    []).

fof(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_metric_1,axiom,(
    ! [A,B,C] : 
      ( ( l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k2_metric_1(A,B,C),k1_numbers) ) ),
    file(metric_1,k2_metric_1),
    []).

fof(dt_k2_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k2_xcmplx_0),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k3_real_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ),
    file(real_1,k3_real_1),
    []).

fof(dt_k4_mcart_1,axiom,(
    $true ),
    file(mcart_1,k4_mcart_1),
    []).

fof(dt_k4_metric_1,axiom,(
    ! [A,B,C] : 
      ( ( v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_metric_1(A,B,C),k1_numbers) ) ),
    file(metric_1,k4_metric_1),
    []).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    []).

fof(dt_k4_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k4_zfmisc_1),
    []).

fof(dt_k7_metric_3,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C)
        & ~ v3_struct_0(D)
        & v6_metric_1(D)
        & v7_metric_1(D)
        & v8_metric_1(D)
        & v9_metric_1(D)
        & l1_metric_1(D) )
     => ( v1_funct_1(k7_metric_3(A,B,C,D))
        & v1_funct_2(k7_metric_3(A,B,C,D),k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers)
        & m2_relset_1(k7_metric_3(A,B,C,D),k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers) ) ) ),
    file(metric_3,k7_metric_3),
    []).

fof(dt_k8_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k8_mcart_1(A,B,C,D,E),A) ) ),
    file(mcart_1,k8_mcart_1),
    []).

fof(dt_k9_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k9_mcart_1(A,B,C,D,E),B) ) ),
    file(mcart_1,k9_mcart_1),
    []).

fof(dt_l1_metric_1,axiom,(
    ! [A] : 
      ( l1_metric_1(A)
     => l1_struct_0(A) ) ),
    file(metric_1,l1_metric_1),
    []).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    []).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    []).

fof(existence_l1_metric_1,axiom,(
    ? [A] : l1_metric_1(A) ),
    file(metric_1,l1_metric_1),
    []).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    []).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(fc1_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc2_membered,axiom,
    ( ~ v1_xboole_0(k1_numbers)
    & v1_membered(k1_numbers)
    & v2_membered(k1_numbers) ),
    file(membered,fc2_membered),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

fof(fc6_membered,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_membered(k1_xboole_0)
    & v2_membered(k1_xboole_0)
    & v3_membered(k1_xboole_0)
    & v4_membered(k1_xboole_0)
    & v5_membered(k1_xboole_0) ),
    file(membered,fc6_membered),
    []).

fof(fc6_subset_1,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D) )
     => ~ v1_xboole_0(k4_zfmisc_1(A,B,C,D)) ) ),
    file(subset_1,fc6_subset_1),
    []).

fof(rc1_membered,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & v3_membered(A)
      & v4_membered(A)
      & v5_membered(A) ) ),
    file(membered,rc1_membered),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc3_struct_0,axiom,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    []).

fof(rc5_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B) ) ) ),
    file(struct_0,rc5_struct_0),
    []).

fof(redefinition_k1_metric_1,axiom,(
    ! [A,B,C,D,E] : 
      ( ( v1_funct_1(C)
        & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B) )
     => k1_metric_1(A,B,C,D,E) = k1_binop_1(C,D,E) ) ),
    file(metric_1,k1_metric_1),
    []).

fof(redefinition_k3_real_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ),
    file(real_1,k3_real_1),
    []).

fof(redefinition_k4_metric_1,axiom,(
    ! [A,B,C] : 
      ( ( v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_metric_1(A,B,C) = k2_metric_1(A,B,C) ) ),
    file(metric_1,k4_metric_1),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(reflexivity_r1_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) ),
    file(xreal_0,r1_xreal_0),
    []).

fof(spc0_boole,axiom,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    []).

fof(t1_arithm,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k2_xcmplx_0(A,0) = A ) ),
    file(arithm,t1_arithm),
    []).

fof(t1_metric_1,axiom,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_metric_1(A,B,B) = 0 )
      <=> v6_metric_1(A) ) ) ),
    file(metric_1,t1_metric_1),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t29_xreal_1,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(0,A)
              & r1_xreal_0(0,B)
              & k2_xcmplx_0(A,B) = 0 )
           => A = 0 ) ) ) ),
    file(xreal_1,t29_xreal_1),
    []).

fof(t2_metric_1,axiom,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_metric_1(A,B,C) = 0
                 => B = C ) ) )
      <=> v7_metric_1(A) ) ) ),
    file(metric_1,t2_metric_1),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_metric_1,axiom,(
    ! [A] : 
      ( ( v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(0,k4_metric_1(A,B,C)) ) ) ) ),
    file(metric_1,t5_metric_1),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

fof(t60_mcart_1,axiom,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => E = k4_mcart_1(k8_mcart_1(A,B,C,D,E),k9_mcart_1(A,B,C,D,E),k10_mcart_1(A,B,C,D,E),k11_mcart_1(A,B,C,D,E)) ) ) ),
    file(mcart_1,t60_mcart_1),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).

fof(t9_xreal_1,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)
                      & r1_xreal_0(C,D) )
                   => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ),
    file(xreal_1,t9_xreal_1),
    []).