reserve X,X1 for set,
  r,s for Real,
  z for Complex,
  RNS for RealNormSpace,
  CNS, CNS1,CNS2 for ComplexNormSpace;

theorem
  for X be ComplexNormSpace, x,y be Point of X holds ||.x-y.|| > 0 iff x <> y
proof
  let X be ComplexNormSpace;
  let x, y be Point of X;
  0 < ||.x-y.|| implies x-y <> 0.X by NORMSP_0:def 6;
  hence 0 < ||.x-y.|| implies x <> y by RLVECT_1:15;
  now
    assume x <> y;
    then 0 <> ||.x-y.|| by CLVECT_1:112;
    hence 0 < ||.x-y.|| by CLVECT_1:105;
  end;
  hence thesis;
end;
