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:79,80;
  then |.z1 + z2.|+|.z1 - z2.| <= |.z1.| + |.z2.|+(|.z1.| + |.z2.|) by
XREAL_1:7;
  then |.z1 + z2.|+|.z1 - z2.| <= 2*(|.z1.| + |.z2.|);
  hence thesis by XREAL_1:79;
end;
