Flexible and existence of neutral element implies equality of left and right inverses for cancellative element

Statement
Suppose $$(S,*)$$ is a magma satisfying the following conditions:


 * 1) $$S$$ is a fact about::flexible magma, i.e., $$x * (y * x) = (x * y) * x$$ for all $$x,y \in S$$.
 * 2) $$S$$ has a fact about::neutral element, i.e., there exists $$e \in S$$ such that $$x * e = e * x = x$$ for all $$x \in S$$.

Then, if $$x \in S$$ is a fact about::cancellative element, the following are equivalent:


 * 1) $$x$$ has a left inverse, i.e., there exists $$y$$ such that $$y * x = e$$.
 * 2) $$x$$ has a right inverse, i.e., there exists $$y$$ such that $$x * y = e$$.
 * 3) $$x$$ has a two-sided inverse, i.e., there exists $$y$$ such that $$x * y = y * x = e$$.
 * 4) $$x$$ has a unique left inverse.
 * 5) $$x$$ has a unique right inverse.
 * 6) $$x$$ has a unique two-sided inverse.