reserve x,y,w,z for ExtReal,
  a for Real;

theorem Th18:
  |.x.| = |.-x.|
proof
  reconsider x as R_eal by XXREAL_0:def 1;
  per cases;
  suppose 0 < x;
    then |.-x.| = -(-x) by Def1
      .= x;
    hence thesis;
  end;
  suppose x < 0;
    then |.x.| = -x by Def1;
    hence thesis;
  end;
  suppose x = 0;
    hence thesis;
  end;
end;
