reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;

theorem Th18:
  for X being BCK-algebra of i,j,m,n holds for k being Element of
  NAT holds X is BCK-algebra of i,j+k,m+k,n
proof
  let X be BCK-algebra of i,j,m,n;
  let k be Element of NAT;
  for x,y being Element of X holds Polynom (i,j+k,x,y) = Polynom (m+k,n,y, x)
  proof
    let x,y be Element of X;
A1: (Polynom (m,n,y,x),(y\x)) to_power k = ((((y,(y\x)) to_power (m+1)),(y
    \x)) to_power k,(x\y)) to_power n by BCIALG_2:11
      .= (((y,(y\x)) to_power (m+1+k)),(x\y)) to_power n by BCIALG_2:10
      .= Polynom (m+k,n,y,x);
    (Polynom (i,j,x,y),(y\x)) to_power k = Polynom (i,j+k,x,y) by BCIALG_2:10;
    hence thesis by A1,Def3;
  end;
  hence thesis by Def3;
end;
