B-Algebra Identity: x (y z) = (x (0 z)) y

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

Then:


 * $\forall x,y,z \in X: x \circ \paren {y \circ z} = \paren {x \circ \paren {0 \circ z} } \circ y$

Proof
Let $x, y, z \in X$.

Then:

Hence the result.

Also see
This identity is comparable to Axiom $(A3)$ for $B$-algebras.