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

theorem Th20:
  x is Real or y is Real implies |.x.|-|.y.| <= |.x-y.|
proof
  assume
A1: x is Real or y is Real;
  reconsider x,y as R_eal by XXREAL_0:def 1;
  per cases by A1;
  suppose y is Real;
    then (x - y) + y = x by XXREAL_3:22;
    then |.x.| <= |.x-y.| + |.y.| by Th13;
   hence thesis by XXREAL_3:42;
  end;
  suppose
    x is Real;
    then reconsider a = x as Real;
A2: |.x.| = |.a.|;
      per cases;
      suppose
A3:      y = +infty or y = -infty;
         |.y.| = +infty by Th19,A3;
         then |.x.| - |.y.| = -infty by A2,XXREAL_3:13;
         then |.x.|-|.y.| <= |.x-y.|;
        hence thesis;
      end;
      suppose
        y <> +infty & y <> -infty;
        then reconsider b = y as Element of REAL by XXREAL_0:14;
        x - y = a - b by SUPINF_2:3;
        then
A4:     |.x-y.| = |.a-b qua Complex.| by Th1;
        |.y.| = |.b qua Complex.| by Th1;
        then |.x.|-|.y.| = |.a qua Complex.|-|.b qua Complex.|
           by A2,SUPINF_2:3;
        hence thesis by A4,COMPLEX1:59;
      end;
    end;
end;
