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

theorem
  |.z1+z2+z3.| <= |.z1.| + |.z2.|+|.z3.|
proof
  |.z1 + z2+z3.|= |.z1 + (z2+z3).| by QUATERN2:2;
  then |.z1 + z2+z3.|<= |.z1.| + |.z2+z3.| by QUATERNI:79;
  then A1: |.z1 + z2+z3.|-|.z1.| <= |.z2+z3.| by XREAL_1:20;
  |.z2+z3.|<=|.z2.|+|.z3.| by QUATERNI:79; then
  (|.z1 + z2+z3.|-|.z1.|)+|.z2+z3.| <=
   |.z2+z3.| +(|.z2.|+|.z3.|) by A1,XREAL_1:7;
  then (|.z1 + z2+z3.|-|.z1.|) <= |.z2+z3.| +(|.z2.|+|.z3.|)-|.z2+z3.| by
XREAL_1:19;
  then |.z1 + z2+z3.|<= |.z2.|+|.z3.|+|.z1.| by XREAL_1:20;
  hence thesis;
end;
