
theorem HamIn01:
  for a, b being Element of [.0,1.] holds
    (a * b) / (a + b - a * b) in [.0,1.]
  proof
    let a, b be Element of [.0,1.];
A1: a * b in [.0,1.] by Lemma1;
    a + b - a * b in [.0,1.] by abMinab01; then
A2: a + b - a * b >= 0 by XXREAL_1:1;
T0: a * b >= 0 by A1,XXREAL_1:1;
    0 <= a & b <= 1 by XXREAL_1:1; then
S1: a * b <= a by XREAL_1:153;
    0 <= b & a <= 1 by XXREAL_1:1; then
    a * b <= b by XREAL_1:153; then
    a * b + a * b <= a + b by S1,XREAL_1:7; then
    a * b <= a + b - a * b by XREAL_1:19; then
    (a * b) / (a + b - a * b) <= 1 by XREAL_1:183,T0;
    hence thesis by T0,A2,XXREAL_1:1;
  end;
