% Mizar problem: t11_euclmetr,euclmetr,309,49 
fof(t11_euclmetr,conjecture,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_euclmetr(A)
      <=> v1_conaffm(A) ) ) ),
    inference(mizar_bg_added,[status(thm)],[abstractness_v1_analoaf,cc1_analmetr,d3_euclmetr,dt_g1_analoaf,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k3_analmetr,dt_l1_analmetr,dt_l1_analoaf,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_analoaf,dt_u1_struct_0,existence_l1_analmetr,existence_l1_analoaf,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc2_analmetr,fc3_analmetr,fc4_analmetr,free_g1_analoaf,l19_euclmetr,rc2_analmetr,redefinition_m2_relset_1,reflexivity_r1_tarski,t3_subset]),
    [file(euclmetr,t11_euclmetr)]).

fof(abstractness_v1_analoaf,axiom,(
    ! [A] : 
      ( l1_analoaf(A)
     => ( v1_analoaf(A)
       => A = g1_analoaf(u1_struct_0(A),u1_analoaf(A)) ) ) ),
    file(analoaf,v1_analoaf),
    []).

fof(cc1_analmetr,axiom,(
    ! [A] : 
      ( l1_analmetr(A)
     => ( ( ~ v3_struct_0(A)
          & v3_analmetr(A) )
       => ( ~ v3_struct_0(A)
          & v2_analmetr(A) ) ) ) ),
    file(analmetr,cc1_analmetr),
    []).

fof(d3_euclmetr,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_euclmetr(A)
      <=> v4_aff_2(k3_analmetr(A)) ) ) ),
    file(euclmetr,d3_euclmetr),
    []).

fof(dt_g1_analoaf,axiom,(
    ! [A,B] : 
      ( m1_relset_1(B,k2_zfmisc_1(A,A),k2_zfmisc_1(A,A))
     => ( v1_analoaf(g1_analoaf(A,B))
        & l1_analoaf(g1_analoaf(A,B)) ) ) ),
    file(analoaf,g1_analoaf),
    []).

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

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

fof(dt_k3_analmetr,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_analmetr(A) )
     => ( v1_analoaf(k3_analmetr(A))
        & l1_analoaf(k3_analmetr(A)) ) ) ),
    file(analmetr,k3_analmetr),
    []).

fof(dt_l1_analmetr,axiom,(
    ! [A] : 
      ( l1_analmetr(A)
     => l1_analoaf(A) ) ),
    file(analmetr,l1_analmetr),
    []).

fof(dt_l1_analoaf,axiom,(
    ! [A] : 
      ( l1_analoaf(A)
     => l1_struct_0(A) ) ),
    file(analoaf,l1_analoaf),
    []).

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_analoaf,axiom,(
    ! [A] : 
      ( l1_analoaf(A)
     => m2_relset_1(u1_analoaf(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) ) ),
    file(analoaf,u1_analoaf),
    []).

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

fof(existence_l1_analmetr,axiom,(
    ? [A] : l1_analmetr(A) ),
    file(analmetr,l1_analmetr),
    []).

fof(existence_l1_analoaf,axiom,(
    ? [A] : l1_analoaf(A) ),
    file(analoaf,l1_analoaf),
    []).

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(fc2_analmetr,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_analmetr(A) )
     => ( ~ v3_struct_0(k3_analmetr(A))
        & v1_analoaf(k3_analmetr(A)) ) ) ),
    file(analmetr,fc2_analmetr),
    []).

fof(fc3_analmetr,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_analmetr(A)
        & l1_analmetr(A) )
     => ( ~ v3_struct_0(k3_analmetr(A))
        & ~ v3_realset2(k3_analmetr(A))
        & v1_analoaf(k3_analmetr(A))
        & v1_diraf(k3_analmetr(A)) ) ) ),
    file(analmetr,fc3_analmetr),
    []).

fof(fc4_analmetr,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( ~ v3_struct_0(k3_analmetr(A))
        & ~ v3_realset2(k3_analmetr(A))
        & v1_analoaf(k3_analmetr(A))
        & v1_diraf(k3_analmetr(A))
        & v2_diraf(k3_analmetr(A)) ) ) ),
    file(analmetr,fc4_analmetr),
    []).

fof(free_g1_analoaf,axiom,(
    ! [A,B] : 
      ( m1_relset_1(B,k2_zfmisc_1(A,A),k2_zfmisc_1(A,A))
     => ! [C,D] : 
          ( g1_analoaf(A,B) = g1_analoaf(C,D)
         => ( A = C
            & B = D ) ) ) ),
    file(analoaf,g1_analoaf),
    []).

fof(l19_euclmetr,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v1_conaffm(A)
      <=> v4_aff_2(k3_analmetr(A)) ) ) ),
    file(euclmetr,l19_euclmetr),
    []).

fof(rc2_analmetr,axiom,(
    ? [A] : 
      ( l1_analmetr(A)
      & ~ v3_struct_0(A) ) ),
    file(analmetr,rc2_analmetr),
    []).

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