
theorem ReichenbachIn01:
  for a,b being Element of [.0,1.] holds
    1 - a + a * b in [.0,1.]
  proof
    let a,b be Element of [.0,1.];
    1 - a in [.0,1.] by FUZNORM1:7; then
A0: 1 - a >= 0 by XXREAL_1:1;
    a * b in [.0,1.] by FUZNORM1:3; then
a1: a * b >= 0 by XXREAL_1:1;
    b <= 1 by XXREAL_1:1; then
a2: b - 1 <= 1 - 1 by XREAL_1:9;
    a >= 0 by XXREAL_1:1; then
    a * (b - 1) <= 0 by a2; then
    1 + (a * b - a) <= 1 + 0 by XREAL_1:7;
    hence thesis by a1,A0,XXREAL_1:1;
  end;
