Absolute Value Equals Square Root of Square

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Theorem

Let $x \in \R$ be a real number.

Let $|x|$ be its absolute value.


Then $|x| = \sqrt{x^2}$, where:

$x^2$ is the square of $x$
$\sqrt{x^2}$ is the square root of $x^2$.


Proof

Note that by Square of Real Number is Positive, indeed $x^2 \ge 0$.