0 in B-Algebra is Left Cancellable Element

Theorem
Let $\struct {X, \circ}$ be a $B$-Algebra.

Then:
 * $\forall x, y \in X: 0 \circ x = 0 \circ y \implies x = y$

Proof
Let $x, y \in X$ and let $0 \circ x = 0 \circ y$.

Then:

So we have shown:


 * $\forall x, y \in X: 0 \circ x = 0 \circ y \implies x \circ y = 0$

From $B$-Algebra Identity: $x \circ y = 0 \iff x = y$:


 * $\forall x, y \in X: 0 \circ x = 0 \circ y \implies x = y$

Hence the result.