B-Algebra is Right Cancellable

Theorem
Let $\struct {X, \circ}$ 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:


 * $\paren {x \circ z} \circ \paren {0 \circ z} = \paren {y \circ z} \circ \paren {0 \circ z}$

which by the above implies $x = y$.

Hence the result.