Associative and Anticommutative

Theorem
Let $\circ$ be a binary operation on a set $S$.

Let $\circ$ be both associative and anticommutative.

Then:
 * $\forall x, y, z \in S: x \circ y \circ z = x \circ z$

Proof
Let $\circ$ be both associative and anticommutative.

Then from Associative Idempotent Anticommutative:
 * $\forall x, z \in S: x \circ z \circ x = x$

and $\circ$ is idempotent.

Consider $x \circ y \circ z \circ x \circ z$.

We have:

Also:

Hence the result.