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 Th48:
  for X being positive-implicative BCK-Algebra_with_Condition(S)
  holds for x,y being Element of X holds x = (x\y)*(x\(x\y))
proof
  let X be positive-implicative BCK-Algebra_with_Condition(S);
  for x,y being Element of X holds x = (x\y)*(x\(x\y))
  proof
    let x,y be Element of X;
    (x\y)\x = (x\x)\y by BCIALG_1:7
      .= y` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
    then x\y <= x;
    then x = (x\y)*x by Th45;
    hence thesis by Th47;
  end;
  hence thesis;
end;
