Monotonically normal implies hereditarily normal

Statement
Any monotonically normal space is a hereditarily normal space: every subspace of the space is a normal space.

Related facts

 * Monotone normality is hereditary
 * Monotonically normal implies collectionwise normal
 * Monotonically normal implies hereditarily collectionwise normal

Facts used

 * 1) uses::Monotone normality is hereditary
 * 2) uses::Monotonically normal implies normal

Proof
Given: A monotonically normal space $$X$$.

To prove: Every subspace of $$X$$ is normal.

Proof: By fact (1), every subspace of $$X$$ is monotonically normal. Using fact (2), we obtain that every subspace of $$X$$ is normal.