Floor of Root of Floor equals Floor of Root/Proof 2

Proof
The square root is defined on the interval $\hointr 0 \to$.

We have that Square Root is Strictly Increasing.

From Continuity of Root Function, the square root is continuous.

So we can apply McEliece's Theorem (Integer Functions) to get the statement is equivalent to proving $\sqrt x \in \Z \implies x \in \Z$.

Let $y = \sqrt x$ such that $y \in \Z$.

By definition of the square root:
 * $x = y^2$.

From Integer Multiplication is Closed:
 * $x \in \Z$

as required.