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

theorem
   |.x-y.| <= |.x.| + |.y.|
proof
  reconsider x,y as R_eal by XXREAL_0:def 1;
  per cases;
  suppose x = +infty or x = -infty;
    then |.x.| + |.y.| = +infty by Th19,XXREAL_3:def 2;
   hence thesis by XXREAL_0:3;
  end;
  suppose
A1: x <> +infty & x <> -infty;
    then reconsider a = x as Element of REAL by XXREAL_0:14;
      per cases;
      suppose
A2:     y = +infty;
        x - y = -infty by A1,A2,XXREAL_3:13;
        hence thesis by A2,Th19,XXREAL_3:def 2;
      end;
      suppose
A3:    y = -infty;
        x - y = +infty by A1,A3,XXREAL_3:14;
        hence thesis by A3,Th19,XXREAL_3:def 2;
      end;
      suppose
        y <> +infty & y <> -infty;
        then reconsider b = y as Element of REAL by XXREAL_0:14;
        |.x.|=|.a qua Complex.| & |.y.|=|.b qua Complex.| by Th1;
        then
A4:    |.x.|+|.y.|=|.a qua Complex.|+|.b qua Complex.| by SUPINF_2:1;
        x - y = a - b by SUPINF_2:3;
        then |.x-y.| = |.a-b qua Complex.| by Th1;
        hence thesis by A4,COMPLEX1:57;
      end;
    end;
end;
