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:
 * $x * \iota_{_R}$ is a bijection and $x^{-1} * \iota_{_R}$ is the inverse of $x * \iota_{_R}$
 * $\iota_{_R} * x$ is a bijection and $\iota_{_R} * x^{-1}$ is the inverse of $\iota_{_R} * x$

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

From Equality of Mappings, $\paren{x * \iota_{_R} } \circ \paren{x^{-1} * \iota_{_R} } = \iota_{_R}$.

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

From Equality of Mappings, $\paren{x^{-1} * \iota_{_R} } \circ \paren{x * \iota_{_R} } = \iota_{_R}$.

Hence $x * \iota_{_R}$ has a left inverse and a right inverse.

From definition 2 of bijection, $x * \iota_{_R}$ is a bijection and the inverse is $x^{-1} * \iota_{_R}$.

Similarly, $\iota_{_R} * x$ is a bijection and the inverse is $\iota_{_R} * x^{-1}$.