theorem Th1:
  |.r1-r2.| > s implies r1+s < r2 or r2+s < r1
proof
  assume
A1: |.r1-r2.| > s;
  now
    per cases;
    case
      r1 < r2;
      then r1 - r2 < 0 by XREAL_1:49;
      then |.r1-r2.| = -(r1-r2) by ABSVALUE:def 1
        .= r2 - r1;
      hence r1+s < r2 by A1,XREAL_1:20;
    end;
    case
      r2 <= r1;
      then r1 - r2 >= 0 by XREAL_1:48;
      then |.r1-r2.| = r1 - r2 by ABSVALUE:def 1;
      hence r2+s < r1 by A1,XREAL_1:20;
    end;
  end;
  hence thesis;
end;
