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 Th17:
  for X being BCK-algebra of i,j,m,n holds for k being Element of
  NAT holds X is BCK-algebra of i+k,j,m,n+k
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+k,j,x,y) = Polynom (m,n+k,y, x)
  proof
    let x,y be Element of X;
A1: (Polynom (m,n,y,x),(x\y)) to_power k = Polynom (m,n+k,y,x) by BCIALG_2:10;
    (Polynom (i,j,x,y),(x\y)) to_power k = ((((x,(x\y)) to_power (i+1)),(x\
    y)) to_power k,(y\x)) to_power j by BCIALG_2:11
      .= ((x,(x\y)) to_power (i+1+k),(y\x)) to_power j by BCIALG_2:10
      .= Polynom (i+k,j,x,y);
    hence thesis by A1,Def3;
  end;
  hence thesis by Def3;
end;
