reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;
reserve z,z1,z2,z3,z4 for Quaternion;
 reserve x for Real;

theorem
  |.z1.| - |.z2.| <= |.z1 + z2.|
proof z1 = z1 + z2 - z2 by Lm34;
  then |.z1.| <= |.z1 + z2.| + |.z2.| by Th73;
  hence thesis by XREAL_1:20;
end;
