B-Algebra is Right Cancellable

Theorem
Let $\left({X, \circ}\right)$ be a $B$-algebra.

Then $\circ$ is right-cancellable for $X$. That is:


 * $\forall x,y,z \in X: x \circ z = y \circ z \implies x = y$

Proof
Let $x, y \in X$.

Then:

Now if for some $z \in X$ we have $x \circ z = y \circ z$, then also:


 * $\left({x \circ z}\right) \circ \left({0 \circ z}\right) = \left({y \circ z}\right) \circ \left({0 \circ z}\right)$

which by the above implies $x = y$.

Hence the result.