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 $$a \circ b = a \ast \left({e \ast b}\right)$$.