Condition for Denesting of Square Root/Lemma

Theorem
Let $a, b, c, d \in \Q_{\ge 0}$.

Suppose $\sqrt b \notin \Q$.

Then:


 * $\sqrt {a + \sqrt b} = \sqrt {c + \sqrt d} \implies a = c, b = d$

Proof
$b \ne d$.

Then:

But this contradicts our assertion that $\sqrt b \notin \Q$.

Hence our supposition that $b \ne d$ must be false.

Therefore we must have $b = d$.

Consequently:
 * $a - c = \sqrt d - \sqrt b = 0$

This implies $a = c$.