Monotonically normal implies hereditarily collectionwise normal

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

Related facts

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

Facts used

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

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

To prove: Every subspace of $$X$$ is a collectionwise normal space.

Proof: By fact (1), every subspace of $$X$$ is monotonically normal, and by fact (2), every subspace is thus collectionwise normal.