# Product Inverse Operation Properties/Lemma 5

## Theorem

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:

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

Then:

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

## Proof

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

$\blacksquare$