
theorem DIO:
  for a,b be non zero Real holds a - b = a*b iff 1/b - 1/a = 1
  proof
    let a,b be non zero Real;
    1*a/(a*b) = 1/b & 1*b/(a*b) = 1/a by XCMPLX_1:91; then
    (1/b - 1/a)*(a*b) = (a-b)/(a*b)*(a*b)
    .= a - b by XCMPLX_1:87;
    hence thesis by XCMPLX_1:7;
  end;
