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
From Associative Idempotent Anticommutative, for $$\circ$$ to be associative and anticommutative, then $$\forall x, z \in S: x \circ z \circ x = x$$, and $$\circ$$ is idempotent.

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

We have:

$$ $$ $$

Also:

$$ $$

The proof follows.

Reference
Seth Warner: Modern Algebra, 1965, Exercise 2:17 (b).