
theorem
  for a,b be non zero Real holds
    sgn (1/a - 1/b) = sgn (b - a) iff sgn b = sgn a
  proof
    let a,b be non zero Real;
    A1: sgn b = sgn a implies sgn (1/a - 1/b) = sgn (b - a)
    proof
      assume sgn a = sgn b; then
      (a is positive & b is positive) or (a is negative & b is negative);
      hence thesis by OPR,NPR;
    end;
    sgn b <> sgn a implies sgn (1/a - 1/b) <> sgn (b - a)
    proof
      assume sgn b <> sgn a; then
      per cases by XXREAL_0:1;
      suppose
        sgn b > sgn a; then
        b is positive & a is negative by SGNZ;
        hence thesis;
      end;
      suppose
        sgn b < sgn a; then
        b is negative & a is positive by SGNZ;
        hence thesis;
      end;
    end;
    hence thesis by A1;
  end;
