Group Product Identity therefore Inverses

Theorem
Let $$\left({G, \circ}\right)$$ be a Group.

Let $$x, y \in \left({G, \circ}\right)$$.

Then if either $$x \circ y = e$$ or $$y \circ x = e$$, it follows that $$x = y^{-1}$$ and $$y = x^{-1}$$.

Proof

 * By the Latin Square Property, $$x \circ y = e \Longrightarrow x = e \circ y^{-1} = y^{-1}$$.


 * Also by the Latin Square Property, $$x \circ y = e \Longrightarrow y = x^{-1} \circ e = x^{-1}$$.


 * The same results are obtained by exchanging $$x$$ and $$y$$ in the above.