Embedding Theorem/Corollary

Corollary to Embedding Theorem
Let:
 * $(1): \quad \struct {T_2, \oplus_2, \otimes_2}$ be a submagma of $\struct {S_2, *_2, \star_2}$
 * $(2): \quad f: \struct {T_1, \oplus_1, \otimes_1} \to \struct {T_2, \oplus_2, \otimes_2}$ be an isomorphism

then there exists:
 * $(1): \quad$ a magma $\struct {S_1, *_1, \star_1}$ which algebraically contains $\struct {T_1, \oplus_1, \otimes_1}$
 * $(2): \quad g: \struct {S_1, *_1, \star_1} \to \struct {S_2, *_2, \star_2}$ where $g$ is an isomorphism which extends $f$.

Proof
By the Embedding Theorem, there exists:
 * a magma $\struct {S_1, *_1}$ which algebraically contains $\struct {T_1, \oplus_1}$
 * an isomorphism $g: \struct {S_1, *_1} \to \struct {S_2, *_2}$ which extends $f$.

Let $\star_1$ be the transplant of $\star_2$ under $g^{-1}$.

We have that:
 * $g^{-1}$ is an isomorphism from $\struct {S_2, \star_2}$ to $\struct {S_1, \star_1}$.

Hence:
 * $g$ is an isomorphism from $\struct {S_1, \star_1}$ to $\struct {S_2, \star_2}$.

It follows that:
 * $g$ is an isomorphism from $\struct {S_1, *_1, \star_1}$ to $\struct {S_2, *_2, \star_2}$.

It remains to be shown that:
 * $\struct {T_1, \otimes_1}$ is closed

and:
 * the operation induced on $T_1$ by $\star_1$ is $\otimes_1$.

Let $x, y \in T_1$.

Then:
 * $\map g x = \map f x$

and:
 * $\map g y = \map f y$

such that:
 * $\map g x, \map g y \in T_2$

So by definition of $\star_1$:

Hence $\struct {T_1, \oplus_1, \otimes_1}$ is embedded in $\struct {S_1, *_1, \star_1}$.