
theorem LABS:
  for a,b be Real holds
    |.a + b.| = |.a.| + |.b.| iff |.a - b.| = |.|.a.| - |.b.|.|
  proof
    let a,b be Real;
    reconsider c = -b  as Real;
    |.|.a.| - |.c.|.| = |.a + c.| iff  |.a - c.| = |.a.| + |.c.| by LmABS;
    hence thesis by COMPLEX1:52;
  end;
