
theorem Lemacik:
  for x, y being Element of [.0,1.] holds
    x * y = x + y implies x = 0
  proof
    let x, y be Element of [.0,1.];
    assume x * y = x + y; then
z1: y * (1 - x) = -x;
    assume x <> 0; then
    0 < x <= 1 by XXREAL_1:1; then
A4: -x < -0 by XREAL_1:25;
    1 - x in [.0,1.] by OpIn01; then
A2: 0 <= 1 - x & 1 - x <= 1 by XXREAL_1:1;
    y < 0 by A4,A2,z1;
    hence thesis by XXREAL_1:1;
  end;
