% Mizar problem: t51_rfunct_1,rfunct_1,886,22 
fof(t51_rfunct_1,conjecture,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_seq_1(B) )
         => k3_rfunct_1(k1_rfunct_1(A,B)) = k1_rfunct_1(k7_relat_1(B,k1_relat_1(k3_rfunct_1(B))),A) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k5_seq_1,dt_k1_relat_1,dt_k1_rfunct_1,dt_k3_rfunct_1,dt_k5_seq_1,dt_k7_relat_1,fc1_rfunct_1,fc2_rfunct_1,fc2_seq_1,fc4_funct_1,fc5_seq_1,rc1_funct_1,rc1_seq_1,t42_rfunct_1,t43_rfunct_1,t47_rfunct_1]),
    [file(rfunct_1,t51_rfunct_1)]).

fof(commutativity_k5_seq_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => k5_seq_1(A,B) = k5_seq_1(B,A) ) ),
    file(seq_1,k5_seq_1),
    []).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    []).

fof(dt_k1_rfunct_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k1_rfunct_1(A,B))
        & v1_funct_1(k1_rfunct_1(A,B)) ) ) ),
    file(rfunct_1,k1_rfunct_1),
    []).

fof(dt_k3_rfunct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k3_rfunct_1(A))
        & v1_funct_1(k3_rfunct_1(A)) ) ) ),
    file(rfunct_1,k3_rfunct_1),
    []).

fof(dt_k5_seq_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k5_seq_1(A,B))
        & v1_funct_1(k5_seq_1(A,B)) ) ) ),
    file(seq_1,k5_seq_1),
    []).

fof(dt_k7_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(A)
     => v1_relat_1(k7_relat_1(A,B)) ) ),
    file(relat_1,k7_relat_1),
    []).

fof(fc1_rfunct_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k1_rfunct_1(A,B))
        & v1_funct_1(k1_rfunct_1(A,B))
        & v1_seq_1(k1_rfunct_1(A,B)) ) ) ),
    file(rfunct_1,fc1_rfunct_1),
    []).

fof(fc2_rfunct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k3_rfunct_1(A))
        & v1_funct_1(k3_rfunct_1(A))
        & v1_seq_1(k3_rfunct_1(A)) ) ) ),
    file(rfunct_1,fc2_rfunct_1),
    []).

fof(fc2_seq_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_seq_1(k7_relat_1(A,B)) ) ) ),
    file(seq_1,fc2_seq_1),
    []).

fof(fc4_funct_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) ),
    file(funct_1,fc4_funct_1),
    []).

fof(fc5_seq_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k5_seq_1(A,B))
        & v1_funct_1(k5_seq_1(A,B))
        & v1_seq_1(k5_seq_1(A,B)) ) ) ),
    file(seq_1,fc5_seq_1),
    []).

fof(rc1_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    []).

fof(rc1_seq_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_seq_1(A) ) ),
    file(seq_1,rc1_seq_1),
    []).

fof(t42_rfunct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => k3_rfunct_1(k3_rfunct_1(A)) = k7_relat_1(A,k1_relat_1(k3_rfunct_1(A))) ) ),
    file(rfunct_1,t42_rfunct_1),
    []).

fof(t43_rfunct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_seq_1(B) )
         => k3_rfunct_1(k5_seq_1(A,B)) = k5_seq_1(k3_rfunct_1(A),k3_rfunct_1(B)) ) ) ),
    file(rfunct_1,t43_rfunct_1),
    []).

fof(t47_rfunct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_seq_1(B) )
         => k1_rfunct_1(A,B) = k5_seq_1(A,k3_rfunct_1(B)) ) ) ),
    file(rfunct_1,t47_rfunct_1),
    []).