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

theorem
  0 <= x * y implies |.x+y.| = |.x.|+|.y.|
proof
  assume
A1: 0 <= x * y;
  per cases by A1;
  suppose (0 <= x or 0 < x) & (0 <= y or 0 < y);
    then |.x.| = x & |.y.| = y by Def1;
    hence thesis by Def1;
  end;
  suppose
A2: (x <= 0 or x < 0) & (y <= 0 or y < 0);
    then
A3: |.x+y.| = -(x + y) by Th7
      .= -x -y by XXREAL_3:25;
    |.x.| = -x by A2,Th7;
    hence thesis by A2,A3,Th7;
  end;
end;
