Fermat's Two Squares Theorem/Uniqueness Lemma/Proof 1

Proof
Suppose:
 * $p = a^2 + b^2 = c^2 + d^2$

where $a, b, c, d \in \Z_{>0}$.

We have that:

$p$ is prime.

From Euclid's Lemma for Prime Divisors:
 * $p \divides \paren {a c + b d}$

or:
 * $p \divides \paren {a d + b c}$

, suppose $p \divides \paren {a c + b d}$.

By :
 * $(1): \quad a c + b d \ge p$

We have:

Hence:
 * $c = a$

and:
 * $b = d$