
theorem
  for a,b be Real st a*b is negative holds |.a - b.| > |.a + b.|
  proof
    let a,b be Real such that
    A1: a*b is negative;
    A2: |.a-b.| = a-b or |.a-b.| = -(a-b) by ABSVALUE:def 1;
    per cases;
    suppose
      a = 0;
      hence thesis by A1;
    end;
    suppose
      B1: a > 0; then
      B2: b < 0 by A1; then
      (a-b) + 2*b < (a-b) + 0 & -(a-b) + 2*a > -(a-b) + 0 by B1,XREAL_1:6; then
      a + b < a - b & -(a + b) < --(a - b) by XREAL_1:24;
      hence thesis by B1,B2,A2,ABSVALUE:def 1;
    end;
    suppose
      B1: a < 0; then
      B2: b > 0 by A1; then
      (a-b) + 2*b > (a-b) + 0 & (b-a) + 2*a < (b-a) + 0 by B1,XREAL_1:6; then
      -(a+b) < -(a-b) & (a+b) < (b-a) by XREAL_1:24;
      hence thesis by B1,B2,A2,ABSVALUE:def 1;
    end;
  end;
