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.| = 0 iff z1 = z2
proof
  thus |.z1 - z2.| = 0 implies z1 = z2 by Lm38,Th59;
  assume z1 = z2; then
  z1 - z2 = [*Rea z1-Rea z1, Im1 z1-Im1 z1, Im2 z1-Im2 z1, Im3 z1-Im3 z1*]
    by Lm36
    .= 0 by Lm6;
  hence thesis by Th58;
end;
