
theorem LmABS:
  for a,b be Real holds
    |.|.a.| - |.b.|.| = |.a + b.| iff  |.a - b.| = |.a.| + |.b.|
  proof
    let a,b be Real;
      per cases;
      suppose
        B1: a >= 0;
        per cases;
        suppose b >= 0; then
          |.a.| = a & |.b.| = b by B1,ABSVALUE:def 1;
          hence thesis;
        end;
        suppose b < 0; then
          |.a.| = a & |.b.| = -b by B1,ABSVALUE:def 1;
          hence thesis;
        end;
      end;
      suppose
        B1: a < 0;
        per cases;
        suppose b >= 0; then
          C1: |.a.| = -a & |.b.| = b by B1,ABSVALUE:def 1; then
          C2: |.|.a.| - |.b.|.| = |.-(a + b).|;
          |.a - b.| = |.-(|.a.|+|.b.|).| by C1
          .= |.|.a.| + |.b.|.| by COMPLEX1:52;
        hence thesis by C2,COMPLEX1:52;
      end;
      suppose b < 0; then
        C1: |.a.| = -a & |.b.| = -b by B1,ABSVALUE:def 1; then
        C2: |.|.a.| - |.b.|.| =|.-(a - b).|;
        |.a + b.| = |.-(|.a.|+|.b.|).| by C1
        .= |.|.a.| + |.b.|.| by COMPLEX1:52;
        hence thesis by COMPLEX1:52,C2;
      end;
    end;
  end;
