Proving two elements of a set are equal based on a two-sided identity

Question

Say I have a set S w/ an associative binary operation *: S x S -> S and a two-sided identity e, and let . Let _L and _R be elements of S such that _L * s = e = s * _R

How can I prove that _L = _R ?

Since the left and right identity are equal to e which is a two-sided identity, that proves that the left and right are equal just by the definition of a two-sided identity right? Is there some more in-depth proof I'm not seeing here?

Answer 1

$$\tilde s_R=e*\tilde s_R=(\tilde s_L*s)*\tilde s_R=\tilde s_L*(s*\tilde s_R)=\tilde s_L*e=\tilde s_L\;.$$