% Mizar problem: t9_wsierp_1,wsierp_1,209,47 
fof(t9_wsierp_1,conjecture,(
    ! [A] : 
      ( v1_int_1(A)
     => ! [B] : 
          ( v1_int_1(B)
         => ! [C] : 
              ( v1_int_1(C)
             => ( ( r2_int_1(A,B)
                  & r2_int_1(A,C) )
               => r2_int_1(A,k2_xcmplx_0(B,C)) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[cc1_rat_1,cc2_rat_1,cc2_xreal_0,cc4_int_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,d9_int_1,dt_k2_xcmplx_0,dt_k3_xcmplx_0,fc1_int_1,fc1_rat_1,fc2_int_1,fc2_rat_1,fc3_xreal_0,fc4_rat_1,fc4_xreal_0,fc6_rat_1,fc7_rat_1,fc9_rat_1,rc1_xreal_0,rc2_int_1,rc2_rat_1,reflexivity_r2_int_1]),
    [file(wsierp_1,t9_wsierp_1)]).

fof(cc1_rat_1,axiom,(
    ! [A] : 
      ( v1_rat_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(rat_1,cc1_rat_1),
    []).

fof(cc2_rat_1,axiom,(
    ! [A] : 
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v1_rat_1(A) ) ) ),
    file(rat_1,cc2_rat_1),
    []).

fof(cc2_xreal_0,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => v1_xcmplx_0(A) ) ),
    file(xreal_0,cc2_xreal_0),
    []).

fof(cc4_int_1,axiom,(
    ! [A] : 
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(int_1,cc4_int_1),
    []).

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_xcmplx_0,axiom,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ),
    file(xcmplx_0,k3_xcmplx_0),
    []).

fof(d9_int_1,axiom,(
    ! [A] : 
      ( v1_int_1(A)
     => ! [B] : 
          ( v1_int_1(B)
         => ( r2_int_1(A,B)
          <=> ? [C] : 
                ( v1_int_1(C)
                & B = k3_xcmplx_0(A,C) ) ) ) ) ),
    file(int_1,d9_int_1),
    []).

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

fof(dt_k3_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k3_xcmplx_0),
    []).

fof(fc1_int_1,axiom,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_int_1(k2_xcmplx_0(A,B)) ) ) ),
    file(int_1,fc1_int_1),
    []).

fof(fc1_rat_1,axiom,(
    ! [A,B] : 
      ( ( v1_rat_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ),
    file(rat_1,fc1_rat_1),
    []).

fof(fc2_int_1,axiom,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_int_1(k3_xcmplx_0(A,B)) ) ) ),
    file(int_1,fc2_int_1),
    []).

fof(fc2_rat_1,axiom,(
    ! [A,B] : 
      ( ( v1_rat_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ),
    file(rat_1,fc2_rat_1),
    []).

fof(fc3_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ),
    file(xreal_0,fc3_xreal_0),
    []).

fof(fc4_rat_1,axiom,(
    ! [A,B] : 
      ( ( v1_rat_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ),
    file(rat_1,fc4_rat_1),
    []).

fof(fc4_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ),
    file(xreal_0,fc4_xreal_0),
    []).

fof(fc6_rat_1,axiom,(
    ! [A,B] : 
      ( ( v1_rat_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ),
    file(rat_1,fc6_rat_1),
    []).

fof(fc7_rat_1,axiom,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ),
    file(rat_1,fc7_rat_1),
    []).

fof(fc9_rat_1,axiom,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ),
    file(rat_1,fc9_rat_1),
    []).

fof(rc1_xreal_0,axiom,(
    ? [A] : 
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(xreal_0,rc1_xreal_0),
    []).

fof(rc2_int_1,axiom,(
    ? [A] : v1_int_1(A) ),
    file(int_1,rc2_int_1),
    []).

fof(rc2_rat_1,axiom,(
    ? [A] : v1_rat_1(A) ),
    file(rat_1,rc2_rat_1),
    []).

fof(reflexivity_r2_int_1,axiom,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => r2_int_1(A,A) ) ),
    file(int_1,r2_int_1),
    []).