
theorem HamCoIn01:
  for a,b being Element of [.0,1.] holds
    (a + b - 2 * a * b) / (1 - a * b) in [.0,1.]
  proof
    let a,b be Element of [.0,1.];
    0 <= a & b <= 1 by XXREAL_1:1; then
S1: a * b <= a by XREAL_1:153;
S2: 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
S5: a * b - a * b <= a + b - a * b - a * b by XREAL_1:9,19;
    1 - b in [.0,1.] by OpIn01; then
    1 - b >= 0 by XXREAL_1:1; then
    a * (1 - b) <= 1 * (1 - b) by S2,XREAL_1:64; then
    a - a * b + b <= 1 - b + b by XREAL_1:6; then
    a + b - a * b - a * b <= 1 - a * b by XREAL_1:9; then
B9: (a + b - 2 * a * b) / (1 - a * b) <= 1 by XREAL_1:183,S5;
    a * b in [.0,1.] by Lemma1; then
    a * b <= 1 by XXREAL_1:1; then
    1 - a * b >= 1 - 1 by XREAL_1:10;
    hence thesis by XXREAL_1:1,B9,S5;
  end;
