Group/Examples/Self-Inverse and Cancellable Elements

Theorem
Let $S$ be a set with an operation which assigns to each $\left({a, b}\right) \in S \times S$ an element $a \ast b \in S$ such that:


 * 1) $\exists e \in S: a \ast b = e \iff a = b$;
 * 2) $\forall a, b, c \in S: \left({a \ast c}\right) \ast \left({b \ast c}\right) = a \ast b$.

Then $\left({S, \circ}\right)$ is a group in which $\circ$ is defined as $a \circ b = a \ast \left({e \ast b}\right)$.