Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 2

Theorem
Let $\struct{R, +, \circ}$ be a ring with unity $1_R$.

Let $\iota_{_R} : R \to R$ be the identity mapping on $R$.

For all $y \in R$, let $y * \iota_{_R} : R \to R$ be the mapping defined by:
 * $\forall z \in R: \map {\paren{y * \iota_{_R}}} z = y * \map {\iota_{_R}} z$

For all $y \in R$, let $\iota_{_R} * y : R \to R$ be the mapping defined by:
 * $\forall z \in R: \map {\paren{\iota_{_R} * y}} z = \map {\iota_{_R}} z * y$

Let $x \in R$ be a unit of $R$ with product inverse $x^{-1}$.

Then:
 * $x^{-1} * \iota_{_R}$ is the inverse mapping of $x * \iota_{_R}$
 * $\iota_{_R} * x^{-1}$ is the inverse mapping of $\iota_{_R} * x$

Proof
Consider the composite of $x * \iota_{_R}$ with $x^{-1} * \iota_{_R}$.