
theorem ADB:
  for a,b be heavy positive Real holds a+b-1 > a/b > 1/(a+b-1)
  proof
    let a,b be heavy positive Real;
    a > 1 & b > 1 by TA1; then
    a + b > 1 + 1 by XREAL_1:8; then
    (a+b) - 1 > 2 - 1 by XREAL_1:9; then
    reconsider c = a + b - 1 as heavy positive Real by TA1;
    reconsider k = a/b as positive Real;
    (k+1)*b > (k+1)*1 by TA1,XREAL_1:68; then
    (k+1)*b - 1 > (k+1) - 1 by XREAL_1:9; then
    A1: k*b + b - 1 > k;
    (k"+1)*a > (k"+1)*1 by TA1,XREAL_1:68; then
    (k"+1)*a -1 > (k"+1) -1 by XREAL_1:9; then
    k"*a + a - 1 > k"; then
    (b/a)*a + a - 1 > k" by XCMPLX_1:213; then
    (b + a - 1)"" > k" by XCMPLX_1:87;
    hence thesis by A1,XCMPLX_1:87,XREAL_1:91;
  end;
