Equivalence of Axiom Schemata for Groups/Warning

Theorem
Suppose we build an algebraic structure with the following axioms:

Then this does not (necessarily) define a group (although clearly a group fulfils those axioms).

Proof
Let $\left({S, \circ}\right)$ be the algebraic structure defined as:
 * $\forall x, y \in S: x \circ y = x$

That is, $\circ$ is the left operation.

From Left Operation All Elements Right Identities, every element serves as a right identity.

Then given any $a \in S$, we have that $x \circ a = x$ and as $x$ is an identity, axiom $(3)$ is fulfilled as well.

But from the complementary result of More than One Left Identity then No Right Identity, there is no right identity and therefore no identity element, so $\left({S, \circ}\right)$ is not a group.