% Mizar problem: t55_rfunct_1,rfunct_1,924,50 
fof(t55_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) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_seq_1(C) )
             => ( k3_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,B)) = k1_rfunct_1(k3_seq_1(A,C),B)
                & k4_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,B)) = k1_rfunct_1(k4_seq_1(A,C),B) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k3_seq_1,commutativity_k5_seq_1,dt_k1_rfunct_1,dt_k3_rfunct_1,dt_k3_seq_1,dt_k4_seq_1,dt_k5_seq_1,fc1_rfunct_1,fc2_rfunct_1,fc3_seq_1,fc4_seq_1,fc5_seq_1,rc1_funct_1,rc1_seq_1,t22_rfunct_1,t26_rfunct_1,t47_rfunct_1]),
    [file(rfunct_1,t55_rfunct_1)]).

fof(commutativity_k3_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) )
     => k3_seq_1(A,B) = k3_seq_1(B,A) ) ),
    file(seq_1,k3_seq_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_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_k3_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(k3_seq_1(A,B))
        & v1_funct_1(k3_seq_1(A,B)) ) ) ),
    file(seq_1,k3_seq_1),
    []).

fof(dt_k4_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(k4_seq_1(A,B))
        & v1_funct_1(k4_seq_1(A,B)) ) ) ),
    file(seq_1,k4_seq_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(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(fc3_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(k3_seq_1(A,B))
        & v1_funct_1(k3_seq_1(A,B))
        & v1_seq_1(k3_seq_1(A,B)) ) ) ),
    file(seq_1,fc3_seq_1),
    []).

fof(fc4_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(k4_seq_1(A,B))
        & v1_funct_1(k4_seq_1(A,B))
        & v1_seq_1(k4_seq_1(A,B)) ) ) ),
    file(seq_1,fc4_seq_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(t22_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) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_seq_1(C) )
             => k5_seq_1(k3_seq_1(A,B),C) = k3_seq_1(k5_seq_1(A,C),k5_seq_1(B,C)) ) ) ) ),
    file(rfunct_1,t22_rfunct_1),
    []).

fof(t26_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) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_seq_1(C) )
             => k5_seq_1(k4_seq_1(A,B),C) = k4_seq_1(k5_seq_1(A,C),k5_seq_1(B,C)) ) ) ) ),
    file(rfunct_1,t26_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),
    []).