# 2nd uniqueness Theorem in primary decomposition

By Theorem 4.10 of Atiyah's book which states that

"Let $$a$$ be a decomposable ideal, let $$a=\cap_{i=1}^{n}q_i$$ be a minimal primary decomposition of $$a$$, let $$\{p_{i_1},...,p_{i_m}\}$$ be an isolated set of prime ideals of $$a$$. Then $$q_{i_1}\cap...\cap q_{i_m}$$ is independent of the decomposition."

we know that intersection of isolated primary ideals are uniquely determined. Corollary 4.11 states that each of the isolated primary ideal is uniquely determined. I cannot conclude this corollary.

A singleton set containing a minimal prime ideal belonging to $$\mathfrak a$$ is an isolated set.