Open Real Interval is Homeomorphic to Real Number Line/Proof 2

Proof
Let $I := \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ denote the open real interval from $-\dfrac \pi 2$ to $\dfrac \pi 2$.

Consider the real function $f: I \to \R$ defined as:
 * $\forall x \in I: \map f x = \tan x$

Then we have:
 * $\forall x \in \R: \map {f^{-1} } x = \arctan x$

From Homeomorphism Relation is Equivalence it follows that $I$ and $\R$ are homeomorphic.

Then by Open Real Intervals are Homeomorphic, $I$ is homeomorphic to every other open real interval.