
theorem ABS:
  for a,b be Real holds |.a + b.| = |.a.| + |.b.| or |.a - b.| = |.a.| + |. b.|
  proof
    let a,b be Real;
    per cases;
    suppose (a >= 0 & b >= 0) or (a <= 0 & b <= 0); then
      a*b >= 0;
      hence thesis by ABSVALUE:11;
    end;
    suppose (a > 0 & b < 0) or (a < 0 & b > 0); then
      a*(-b) >= 0; then
      |.a + - b.| = |.a.|+ |.-b.| by ABSVALUE:11;
      hence thesis by COMPLEX1:52;
  end;
end;
