Identity Property in Semigroup

Theorem
Let $$\left({S, \circ}\right)$$ be a semigroup.

Let $$s \in S$$ be such that:

$$\forall a \in S: \exists x, y \in S: s \circ x = a = y \circ s$$

Then $$\left({S, \circ}\right)$$ has an identity.

Proof
Suppose $$\forall a \in S: \exists x, y \in S: s \circ x = a = y \circ s$$.

Since $$s \in S$$, it follows that $$\exists v, w \in S: s \circ v = s = w \circ s$$.

Let $$a \in S$$. Then $$\exists x, y \in S: s \circ x = a = y \circ s$$.

Thus:

$$a = s \circ x \Longrightarrow w \circ a = w \circ s \circ x = s \circ x = a$$

$$a = y \circ s \Longrightarrow a \circ v = y \circ s \circ v = y \circ s = a$$

Hence $$w \circ a = a$$ and $$a \circ v = a$$ for any $$a \in S$$.

In particular, letting $$a = v$$ in the first of these gives $$w \circ v = v$$ and letting $$a = w$$ in the second gives $$w \circ v = w$$.

Thus $$v = w \circ v = w$$ is the identity element in $$S$$.