
theorem ABS1:
  for a,b be Real holds
    |.|.a.| - |.b.|.| = |.a + b.| or |.|.a.| - |. b.|.| = |.a - b.|
  proof
    let a,b be Real;
    per cases;
    suppose
      A1: a >= 0;
      per cases;
      suppose b >= 0; then
        |.a.| = a & |.b.| = b by A1,ABSVALUE:def 1;
        hence thesis;
      end;
      suppose b < 0; then
        |.a.| = a & |.b.| = -b by A1,ABSVALUE:def 1;
        hence thesis;
      end;
    end;
    suppose
      A1: a < 0;
      per cases;
      suppose b >= 0; then
        |.a.| = -a & |.b.| = b by A1,ABSVALUE:def 1; then
        |.|.a.| - |.b.|.| = |.-(a + b).|;
        hence thesis by COMPLEX1:52;
      end;
      suppose b < 0; then
        |.a.| = -a & |.b.| = -b by A1,ABSVALUE:def 1; then
        |.|.a.| - |.b.|.| =|.-(a - b).|;
        hence thesis by COMPLEX1:52;
      end;
    end;
  end;
