Subfield of Subfield is Subfield

Theorem
Let $R$ be a ring with unity.

Let $K_1, K_2$ be fields, such that:


 * $K_1$ is a subfield of $R$
 * $K_2$ is a subfield of $K_1$

Then $K_2$ is a subfield of $R$.

Proof
Let $K_1$ be a subfield of $R$ and $K_2$ be a subfield of $K_1$.

Then by definition:
 * $K_1 \subseteq R$
 * $K_2 \subseteq K_1$

From Subsets Transitive it follows that $K_2 \subseteq R$

So by definition $K_2$ is a subfield of $R$.