Gyrogroup implies left-inverse property loop

Statement
 Suppose $$(G,*)$$ is a gyrogroup with neutral element $$e$$ and inverse map $${}^{-1}$$. Then, $$(G,*)$$ is a left inverse property loop with the same neutral element $$e$$ and where the left inverse map is the same as the inverse map.

Proof
Given: A gyrogroup $$(G,*)$$, elements $$a,b \in G$$.

To prove: There exists unique $$u \in G$$ such that $$a * u = b$$ and there exists unique $$y \in G$$ such that $$v * a = b$$ (together, these prove it's a loop). Further, $$u = a^{-1} * b$$ (this shows the left inverse property).

Proof:

Steps (4) and (6) complete the proof.