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

theorem
  |.z1.| + |.z2.| >= (|.z1 + z2.|+|.z2 - z1.|)/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.|+|.z2 - z1.| <= 2*(|.z1.| + |.z2.|) by QUATERNI:83;
  hence thesis by XREAL_1:79;
end;
