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

theorem
  |.z1.| - |.z2.| <= (|.z1 + z2.|+|.z1-z2.|)/2
proof
  |.z1.| - |.z2.| <= |.z1 + z2.| & |.z1.| - |.z2.| <= |.z1 - z2.| by
QUATERNI:81,82; then
  (|.z1.| - |.z2.|) +(|.z1.| - |.z2.|) <= |.z1 + z2.| + |.z1 - z2.| by
XREAL_1:7;
  then 2*(|.z1.| - |.z2.|) <=|.z1 + z2.| + |.z1 - z2.|;
  hence thesis by XREAL_1:77;
end;
