reserve X for non empty BCIStr_1;
reserve d for Element of X;
reserve n,m,k for Nat;
reserve f for sequence of  the carrier of X;

theorem
  for X being commutative BCK-Algebra_with_Condition(S), a being Element
of X st a is greatest holds for x,y being Element of X holds x*y = a\((a\x)\y)
proof
  let X be commutative BCK-Algebra_with_Condition(S);
  let a be Element of X;
  assume
A1: a is greatest;
  for x,y being Element of X holds x*y = a\((a\x)\y)
  proof
    let x,y be Element of X;
A2: (x*y)<=a by A1,BCIALG_2:def 5;
    a\((a\x)\y) = a\(a\(x*y)) by Th11
      .= (x*y)\((x*y)\a) by Def9
      .= (x*y)\0.X by A2;
    hence thesis by BCIALG_1:2;
  end;
  hence thesis;
end;
