
theorem OPR:
  for a,b be positive Real holds sgn (1/a - 1/b) = sgn (b - a)
  proof
    let a,b be positive Real;
    A1: sgn (a*b) = 1 by ABSVALUE:def 2;
    ((1/a)*a)*b = (1/a)*(a*b) & (1/b)*(a*b) = ((1/b)*b)*a &
      (1/b)*b = 1 & (1/a)*a = 1 by XCMPLX_1:87; then
    (1/a)*(a*b) = b & (1/b)*(a*b) = a; then
    sgn (b - a) = sgn ((1/a - 1/b)*(a*b))
    .= (sgn (1/a - 1/b)) * sgn (a*b) by ABSVALUE:18;
    hence thesis by A1;
  end;
