
theorem SGNZ:
  for a,b be non zero Real st sgn a > sgn b holds a is positive & b is negative
  proof
    let a,b be non zero Real;
    assume
    A1: sgn a > sgn b;
    (a > 0 or a < 0) & (b > 0 or b < 0); then
    (sgn a = 1 or sgn a = -1) & (sgn b = 1 or sgn b = -1) by ABSVALUE:def 2;
    hence thesis by A1;
  end;
