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 Th37:
  for X being BCK-Algebra_with_Condition(S) holds for x,y being
  Element of X holds x <= x*y & y <= x*y
proof
  let X be BCK-Algebra_with_Condition(S);
  for x,y being Element of X holds x <= x*y & y <= x*y
  proof
    let x,y be Element of X;
A1: (x\x)\y = y` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
A2: (y\y)\x = x` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
A3: (x\x)\y = x\(x*y) by Th11;
    (y\y)\x = y\(y*x) by Th11
      .= y\(x*y) by Th6;
    hence thesis by A3,A1,A2;
  end;
  hence thesis;
end;
