Module isomorphism and length of tensor product.

See image: Why is the isomorphism of  $\Psi_A$ modules true and how do we show the length equality? Notation: $A$ is a commutative, Noetherian, local ring together with a surjective homomorphism to a discrete valuation ring: $\lambda: A \rightarrow \mathcal{O} , p_A=ker(\lambda),  I_A= Ann[p_A]$ (the annihilator of the kernel of the map), $\Psi_A=\mathcal{O}/\lambda(I_A), M[I]=\{m \in M: mi=0, \forall i \in I\}$ (for every ideal $I$ of $A$), $\hat{\Psi}_A= \frac{M}{M[I_A]+I_AM}$. Please let me know if you need to know any more facts.

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