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

theorem
  |.z1.| - |.z2.| <= |.z1 - z.| + |.z - z2.|
proof
  |.z1.| - |.z2.| <= |.z1 - z2.| & |.z1 - z2.| <= |.z1 - z.| + |.z - z2.|
    by QUATERNI:82,85; then
  |.z1.| - |.z2.| + |.z1 - z2.| <= |.z1 - z2.| + (|.z1 - z.| + |.z - z2.|)
    by XREAL_1:7; then
  |.z1.| - |.z2.| <= |.z1 - z2.| + |.z1 - z.| + |.z - z2.|-|.z1 - z2.| by
XREAL_1:19;
  hence thesis;
end;
