
theorem
  for a,b be non zero Real holds
  (|.|.a.| - |.b.|.| = |.a + b.| & |.a - b.| = |.a.| + |. b.|)
  iff not (|.|.a.| - |.b.|.| = |.a - b.| & |.a + b.| = |.a.| + |. b.|)
  proof
    let a,b be non zero Real;
    A1: |.|.a.| - |.b.|.| = |.a + b.| iff  |.a - b.| = |.a.| + |.b.| by LmABS;
    A2: |.|.a.| - |.b.|.| = |.a - b.| iff  |.a + b.| = |.a.| + |.b.|
    proof
      reconsider c = -b as non zero Real;
      |.|.a.| - |.c.|.| = |.a + c.| iff  |.a - c.| = |.a.| + |.c.| by LmABS;
      hence thesis by COMPLEX1:52;
    end;
    |.a + b.| = |.a.| + |.b.| iff not |.a - b.| = |.a.| + |.b.|
    proof
      (|.a.| - |.b.|) + 0 < (|.a.| - |.b.|) + 2*|.b.| &
      (|.b.| - |.a.|) + 0 < (|.b.| - |.a.|) + 2*|.a.|
      by XREAL_1:6; then
      B0: |.a.| + |.b.| > |.a.| - |.b.| & |.a.| + |.b.| > -(|.a.| - |.b.|);
      per cases by ABS;
      suppose |.a + b.| = |.a.| + |.b.|;
        hence thesis by A2,B0,ABSVALUE:1;
      end;
      suppose |.a - b.| = |.a.| + |. b.|;
        hence thesis by A1,B0,ABSVALUE:1;
      end;
    end;
    hence thesis by LmABS,A2;
  end;
