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

theorem
  x is Real or y is Real implies |.|.x.|-|.y.|.| <= |.x-y.|
proof
A1: |.y.|-|.x.| = -(|.x.|-|.y.|) by XXREAL_3:26;
  assume
A2: x is Real or y is Real;
  then
A3: |.x.|-|.y.| <= |.x-y.| by Th20;
  y - x = -(x - y) by XXREAL_3:26;
  then
A4: |.y-x.| = |.x-y.| by Th18;
  |.y.|-|.x.| <= |.y-x.| by A2,Th20;
  then -|.x-y.| <= -(-(|.x.|-|.y.|)) by A4,A1,XXREAL_3:38;
  hence thesis by A3,Th12;
end;
