reserve z1,z2,z3,z4,z for Quaternion;

theorem
  |.|.z1.| - |.z2.|.| <= |.z1.| + |.z2.|
proof
  |.|.z1.| - |.z2.|.| <= |.z1 - z2.| & |.z1 - z2.| <= |.z1.| + |.z2.| by
QUATERNI:80,86; then
  |.|.z1.| - |.z2.|.|+ |.z1 - z2.| <= |.z1 - z2.|+(|.z1.| + |.z2.|) by
XREAL_1:7;
  then |.|.z1.| - |.z2.|.| <= |.z1 - z2.|+|.z1.| + |.z2.|- |.z1 - z2.| by
XREAL_1:19;
  hence thesis;
end;
