% Mizar problem: t23_arytm_3,arytm_3,647,55 
fof(t23_arytm_3,conjecture,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( A != k1_xboole_0
           => ( k4_arytm_3(B,A) != k1_xboole_0
              & k6_ordinal3(A,k4_arytm_3(B,A)) != k1_xboole_0 ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_arytm_3,cc1_ordinal1,cc2_arytm_3,cc2_ordinal1,cc3_arytm_3,cc3_ordinal1,commutativity_k2_arytm_3,commutativity_k4_arytm_3,d4_arytm_3,dt_k15_ordinal2,dt_k1_xboole_0,dt_k2_arytm_3,dt_k4_arytm_3,dt_k5_ordinal2,dt_k6_ordinal3,dt_m1_subset_1,existence_m1_subset_1,fc1_ordinal2,fc1_xboole_0,fc2_ordinal1,fc5_arytm_3,fc6_arytm_3,rc1_arytm_3,rc1_ordinal1,rc1_xboole_0,rc2_ordinal1,rc2_xboole_0,rc3_ordinal1,redefinition_k2_arytm_3,reflexivity_r2_arytm_3,t11_arytm_3,t1_subset,t2_subset,t52_ordinal2,t6_boole,t7_boole,t8_boole]),
    [file(arytm_3,t23_arytm_3)]).

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

fof(cc1_arytm_3,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_ordinal1(B)
            & v2_ordinal1(B)
            & v3_ordinal1(B) ) ) ) ),
    file(arytm_3,cc1_arytm_3),
    []).

fof(cc1_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,cc1_ordinal1),
    []).

fof(cc2_arytm_3,axiom,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A) ) ) ),
    file(arytm_3,cc2_arytm_3),
    []).

fof(cc2_ordinal1,axiom,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_ordinal1(A) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,cc2_ordinal1),
    []).

fof(cc3_arytm_3,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k5_ordinal2)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A) ) ) ),
    file(arytm_3,cc3_arytm_3),
    []).

fof(cc3_ordinal1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(ordinal1,cc3_ordinal1),
    []).

fof(commutativity_k2_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => k2_arytm_3(A,B) = k2_arytm_3(B,A) ) ),
    file(arytm_3,k2_arytm_3),
    []).

fof(commutativity_k4_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => k4_arytm_3(A,B) = k4_arytm_3(B,A) ) ),
    file(arytm_3,k4_arytm_3),
    []).

fof(d4_arytm_3,axiom,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( m1_subset_1(C,k5_ordinal2)
             => ( C = k4_arytm_3(A,B)
              <=> ( r2_arytm_3(C,A)
                  & r2_arytm_3(C,B)
                  & ! [D] : 
                      ( ( v3_ordinal1(D)
                        & v4_ordinal2(D) )
                     => ( ( r2_arytm_3(D,A)
                          & r2_arytm_3(D,B) )
                       => r2_arytm_3(D,C) ) ) ) ) ) ) ) ),
    file(arytm_3,d4_arytm_3),
    []).

fof(dt_k15_ordinal2,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k15_ordinal2(A,B)) ) ),
    file(ordinal2,k15_ordinal2),
    []).

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

fof(dt_k2_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => v3_ordinal1(k2_arytm_3(A,B)) ) ),
    file(arytm_3,k2_arytm_3),
    []).

fof(dt_k4_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => m1_subset_1(k4_arytm_3(A,B),k5_ordinal2) ) ),
    file(arytm_3,k4_arytm_3),
    []).

fof(dt_k5_ordinal2,axiom,(
    $true ),
    file(ordinal2,k5_ordinal2),
    []).

fof(dt_k6_ordinal3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k6_ordinal3(A,B)) ) ),
    file(ordinal3,k6_ordinal3),
    []).

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

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

fof(fc1_ordinal2,axiom,
    ( v1_ordinal1(k5_ordinal2)
    & v2_ordinal1(k5_ordinal2)
    & v3_ordinal1(k5_ordinal2)
    & ~ v1_xboole_0(k5_ordinal2) ),
    file(ordinal2,fc1_ordinal2),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc2_ordinal1,axiom,
    ( v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_xboole_0(k1_xboole_0)
    & v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0)
    & v3_ordinal1(k1_xboole_0) ),
    file(ordinal1,fc2_ordinal1),
    []).

fof(fc5_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => ( v1_ordinal1(k15_ordinal2(A,B))
        & v2_ordinal1(k15_ordinal2(A,B))
        & v3_ordinal1(k15_ordinal2(A,B))
        & v4_ordinal2(k15_ordinal2(A,B)) ) ) ),
    file(arytm_3,fc5_arytm_3),
    []).

fof(fc6_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => ( v1_ordinal1(k6_ordinal3(A,B))
        & v2_ordinal1(k6_ordinal3(A,B))
        & v3_ordinal1(k6_ordinal3(A,B))
        & v4_ordinal2(k6_ordinal3(A,B)) ) ) ),
    file(arytm_3,fc6_arytm_3),
    []).

fof(rc1_arytm_3,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A) ) ),
    file(arytm_3,rc1_arytm_3),
    []).

fof(rc1_ordinal1,axiom,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc1_ordinal1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_ordinal1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc2_ordinal1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc3_ordinal1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc3_ordinal1),
    []).

fof(redefinition_k2_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => k2_arytm_3(A,B) = k15_ordinal2(A,B) ) ),
    file(arytm_3,k2_arytm_3),
    []).

fof(reflexivity_r2_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => r2_arytm_3(A,A) ) ),
    file(arytm_3,r2_arytm_3),
    []).

fof(t11_arytm_3,axiom,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r2_arytm_3(B,A)
          <=> A = k2_arytm_3(B,k6_ordinal3(A,B)) ) ) ) ),
    file(arytm_3,t11_arytm_3),
    []).

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

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

fof(t52_ordinal2,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k15_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ),
    file(ordinal2,t52_ordinal2),
    []).

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),
    []).