
theorem Th0:
  for a be Real holds a is weightless iff a = sgn a
  proof
    let a be Real;
    A1: a is weightless implies a = sgn a
    proof
      assume
      a is weightless; then
      a = 0 or a = |.1.| or -a = |.1.| by ABSVALUE:1;
      hence thesis by ABSVALUE:def 2;
    end;
    a = sgn a implies a is weightless
    proof
      assume that
      A1: a = sgn a;
      a > 0 or a = 0 or a < 0; then
      sgn a = 0 or sgn a = 1 or sgn a = -1 by ABSVALUE:def 2;
      hence thesis by A1;
    end;
    hence thesis by A1;
  end;
