Product Inverse Operation Properties/Lemma 5

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Let $\struct {G, \oplus}$ be a closed algebraic structure on which the following properties hold:

\((\text {PI} 1)\)   $:$   Self-Inverse Property      \(\ds \forall x \in G:\) \(\ds x \oplus x = e \)      
\((\text {PI} 2)\)   $:$   Right Identity      \(\ds \exists e \in G: \forall x \in G:\) \(\ds x \oplus e = x \)      
\((\text {PI} 3)\)   $:$   Product Inverse with Right Identity      \(\ds \forall x, y \in G:\) \(\ds e \oplus \paren {x \oplus y} = y \oplus x \)      
\((\text {PI} 4)\)   $:$   Cancellation Property      \(\ds \forall x, y, z \in G:\) \(\ds \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y \)      

These four stipulations are known as the product inverse operation axioms.

Let $\circ$ be the operation on $G$ defined as:

$\forall x, y \in G: x \circ y = x \oplus \paren {e \oplus y}$


$\forall x, y \in G: \paren {x \circ y} \oplus y = x$


\(\ds \forall x, y \in G: \, \) \(\ds \paren {x \circ y} \oplus y\) \(=\) \(\ds \paren {x \oplus \paren {e \oplus y} } \oplus y\) Definition of $\circ$
\(\ds \) \(=\) \(\ds \paren {x \oplus \paren {e \oplus y} } \oplus \paren {y \oplus e}\) $\text {PI} 2$: Right Identity
\(\ds \) \(=\) \(\ds \paren {x \oplus \paren {e \oplus y} } \oplus \paren {e \oplus \paren {e \oplus y} }\) $\text {PI} 3$: Product Inverse with Right Identity
\(\ds \) \(=\) \(\ds x \oplus e\) $\text {PI} 4$: Cancellation Property
\(\ds \) \(=\) \(\ds x\)