Subfield of Subfield is Subfield
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Theorem
Let $R$ be a ring with unity.
Let $K_1, K_2$ be fields, such that:
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 Subset Relation is Transitive it follows that $K_2 \subseteq R$
So by definition $K_2$ is a subfield of $R$.
$\blacksquare$