Normal function of euclidian integers

This is to show that the normal function of euclidian integers preserves the multiplicative property. To start lets define the euclidian integers:

Z [i] = {a + b i ∣ a, b \in Z}

Then the normal function is defined as

N (a + b i) = (a + b i) (a - b i) = a^{2} + b^{2}

Now to prove the corresponding homomorphism preserves multiplication:

N (a + b i) N (c + d i) = (a^{2} + b^{2}) (c^{2} + d^{2}) = (a^{2} c^{2} + a^{2} d^{2} + b^{2} c^{2} + b^{2} d^{2})

And from the other side

N ((a + b i) (c + d i)) = N (a c + a d i + c b i - d b) = N ((a c - d v) + (a d + c b) i) = (a c - d v)^{2} + (a d + c b)^{2}