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

theorem Th4:
  x <> 0 implies 0 < |.x.|
proof
  assume
A1: x <> 0;
  per cases;
  suppose
    0 <= x;
    hence thesis by A1,Def1;
  end;
  suppose
A2: not 0 <= x;
    then 0 < -x;
    hence thesis by A2,Def1;
  end;
end;
