Modulus in Terms of Conjugate

Theorem
Let $z = a + i b$ be a complex number.

Let $\left\vert{z}\right\vert$ be the modulus of $z$.

Let $\overline z$ be the conjugate of $z$.

Then $\left\vert{z}\right\vert^2 = z \overline z$.

Proof
Let $z = a + i b$.

Then: