Group implies quasigroup

Verbal statement
Any group is a quasigroup.

Statement with symbols
Let $$G$$ be a group, and $$a,b \in G$$ be (not necessarily distinct) elements. Then, there exist unique $$x,y \in G$$ satisfying $$ax = b$$ and $$ya = b$$ respectively.

Quasigroup
A magma $$(S,*)$$ (a set $$S$$ with binary operation $$*$$) is termed a quasigroup if for any $$a,b \in S$$, there exist unique $$x,y \in S$$ such that $$a * x = y * a = b$$.

Proof
Given: A group $$G$$, elements $$a,b \in G$$

To prove: There exist unique solutions to $$ax = b$$ and $$ya = b$$

Proof: We have:

$$ax = b \implies a^{-1}(ax) = a^{-1}b \implies x = a^{-1}b$$

Conversely:

$$x = a^{-1}b \implies ax = a(a^{-1}b) \implies ax = b$$

Thus:

$$ax = b \iff x = a^{-1}b$$

So, $$ax = b$$ has a unique solution.

Similarly:

$$ya = b \implies (ya)a^{-1} = ba^{-1} \implies y = ba^{-1}$$

Conversely:

$$y = ba^{-1}\implies ya = (ba^{-1})a \implies ya = b$$

Thus:

$$ya = b \iff y = ba^{-1}$$

So, $$ya = b$$ as a unique solution.