Composite of Monomorphisms is Monomorphism

Theorem
Let:
 * $\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
 * $\struct {S_2, *_1, *_2, \ldots, *_n}$
 * $\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be algebraic structures.

Let:
 * $\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
 * $\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be monomorphisms.

Then the composite of $\phi$ and $\psi$ is also a monomorphism.

Proof
From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.

From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.

A monomorphism is an injective homorphism.

Hence $\psi \circ \phi$ is a monomorphism.