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

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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.

$\blacksquare$