theorem Th2:
  for a,b being Real holds b < a & a <= 0 implies |.a.| < |.b.|
  proof
    let a,b be Real;
    assume that
A1: b < a and
A2: a <= 0;
A3: |.b.| = -b by A1,A2,ABSVALUE:def 1;
    per cases by A2;
    suppose a = 0;
      hence thesis by A1;
    end;
    suppose a < 0;
      then |.a.| = -a by ABSVALUE:def 1;
      hence thesis by A1,A3,XREAL_1:24;
    end;
  end;
