# B-Algebra Identity: 0(0x)=x

## Theorem

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

Then:

$\forall x \in X: 0 \circ \paren {0 \circ x} = x$

## Proof

Let $x \in X$.

Then:

 $\displaystyle 0 \circ x$ $=$ $\displaystyle \paren {0 \circ x}\circ 0$ Axiom $\text A 2$ for $B$-algebra $\displaystyle$ $=$ $\displaystyle 0 \circ \paren {0 \circ \paren {0 \circ x} }$ Axiom $\text A 3$ for $B$-algebra

From 0 in B-Algebra is Left Cancellable Element, we conclude:

$x = 0 \circ \paren {0 \circ x}$

as desired.

$\blacksquare$