Why $|z_1z_2|^2 = |z_1|^2|z_2|^2$ implies $|z_1z_2| = |z_1||z_2|$

I have this proof and I couldn't comprehend the final result. I do understand the steps though but why $|z_1z_2|^2 = |z_1|^2|z_2|^2 \implies |z_1z_2| = |z_1||z_2|$. The vise verse is lucid to me

For any two complex numbers, $z_1$ and $z_2$ $$ |z_1 z_2| = |z_1||z_2| $$ Proof: $$ \begin{align*} |z_1 z_2|^2 &= (z_1z_2)\overline{(z_1z_2)} \\ &= z_1z_2\overline{z}_1\overline{z}_2 \\ &= z_1\overline{z}_1z_2\overline{z}_2 \\ &= |z_1|^2|z_2|^2 \end{align*} $$ From this it follows that $$ |z_1 z_2| = |z_1||z_2| \qquad \text{QED} $$

Solution 1:

As per Stephen Donovan's comment, all of the moduli are positive real number which negates $\pm$ from resulting when you square root. Continuing from the proof: $$\therefore \sqrt{|z_{1}z_{2}|^{2}} = \sqrt{|z_{1}|^{2}|z_{2}|^{2}}$$ $$|\pm z_{1}z_{2}| = |\pm z_{1}|^{2*0.5}|\pm z_{2}|^{2*0.5}$$ $$|z_{1}z_{2}|=|\pm z_{1}||\pm z_{2}|$$ $$|z_{1}z_{2}|=|z_{1}||z_{2}|$$