% Mizar problem: t16_arytm_3,arytm_3,416,60 
fof(t16_arytm_3,conjecture,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ( ( r2_arytm_3(A,B)
                  & r2_arytm_3(A,k1_arytm_3(B,C)) )
               => r2_arytm_3(A,C) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[cc1_ordinal1,cc2_ordinal1,commutativity_k1_arytm_3,commutativity_k2_arytm_3,d2_arytm_3,dt_k14_ordinal2,dt_k15_ordinal2,dt_k1_arytm_3,dt_k2_arytm_3,dt_k5_ordinal3,fc3_arytm_3,fc4_arytm_3,fc5_arytm_3,rc1_ordinal1,redefinition_k1_arytm_3,redefinition_k2_arytm_3,reflexivity_r2_arytm_3,t65_ordinal3,t76_ordinal3,t9_arytm_3]),
    [file(arytm_3,t16_arytm_3)]).

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

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

fof(commutativity_k1_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => k1_arytm_3(A,B) = k1_arytm_3(B,A) ) ),
    file(arytm_3,k1_arytm_3),
    []).

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(d2_arytm_3,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_arytm_3(A,B)
          <=> ? [C] : 
                ( v3_ordinal1(C)
                & B = k15_ordinal2(A,C) ) ) ) ) ),
    file(arytm_3,d2_arytm_3),
    []).

fof(dt_k14_ordinal2,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k14_ordinal2(A,B)) ) ),
    file(ordinal2,k14_ordinal2),
    []).

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_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => v3_ordinal1(k1_arytm_3(A,B)) ) ),
    file(arytm_3,k1_arytm_3),
    []).

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_k5_ordinal3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => v3_ordinal1(k5_ordinal3(A,B)) ) ),
    file(ordinal3,k5_ordinal3),
    []).

fof(fc3_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => ( v1_ordinal1(k14_ordinal2(A,B))
        & v2_ordinal1(k14_ordinal2(A,B))
        & v3_ordinal1(k14_ordinal2(A,B))
        & v4_ordinal2(k14_ordinal2(A,B)) ) ) ),
    file(arytm_3,fc3_arytm_3),
    []).

fof(fc4_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => ( v1_ordinal1(k5_ordinal3(A,B))
        & v2_ordinal1(k5_ordinal3(A,B))
        & v3_ordinal1(k5_ordinal3(A,B))
        & v4_ordinal2(k5_ordinal3(A,B)) ) ) ),
    file(arytm_3,fc4_arytm_3),
    []).

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

fof(redefinition_k1_arytm_3,axiom,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A)
        & v3_ordinal1(B)
        & v4_ordinal2(B) )
     => k1_arytm_3(A,B) = k14_ordinal2(A,B) ) ),
    file(arytm_3,k1_arytm_3),
    []).

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(t65_ordinal3,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k5_ordinal3(k14_ordinal2(A,B),A) = B ) ) ),
    file(ordinal3,t65_ordinal3),
    []).

fof(t76_ordinal3,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => k5_ordinal3(k15_ordinal2(A,B),k15_ordinal2(C,B)) = k15_ordinal2(k5_ordinal3(A,C),B) ) ) ) ),
    file(ordinal3,t76_ordinal3),
    []).

fof(t9_arytm_3,axiom,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r2_arytm_3(A,B)
          <=> ? [C] : 
                ( v3_ordinal1(C)
                & v4_ordinal2(C)
                & B = k2_arytm_3(A,C) ) ) ) ) ),
    file(arytm_3,t9_arytm_3),
    []).