
theorem abMinab01:
  for a, b being Element of [.0,1.] holds
    a + b - a * b in [.0,1.]
  proof
    let a, b be Element of [.0,1.];
S1: 1 - b in [.0,1.] by OpIn01; then
    a * (1 - b) in [.0,1.] by Lemma1; then
B1: a * (1 - b) >= 0 by XXREAL_1:1;
a0: b >= 0 by XXREAL_1:1;
S2: a <= 1 & b <= 1 by XXREAL_1:1;
    0 <= 1 - b <= 1 by S1,XXREAL_1:1; then
    a * (1 - b) <= 1 * (1 - b) by S2,XREAL_1:64; then
    a * (1 - b) + b <= 1 - b + b by XREAL_1:6;
    hence thesis by XXREAL_1:1,a0,B1;
  end;
