# Trying to undertand the proof of Pelczynski decomposition

I am following Albiac and Kalton's book Topics in Banach Space Theory. Theorem 2.2.3 is Pelczynski Decomposition: let $$X$$ and $$Y$$ be Banach spaces. Suppose that $$X$$ is isomorphic to a complemented subspace of $$Y$$, and that $$Y$$ is isomorphic to a complemented subspace of $$X$$. Then, under certain conditions, $$X$$ and $$Y$$ are isomorphic.

The condition I am having trouble with is asking that $$X$$ is isomorphic to $$\ell_p(X)$$. If that is the case, $$X$$ is isomorphic to $$X^2$$, and after some manipulation you get that $$Y$$ is isomorphic to $$X \oplus Y$$, so far so good.

The problem is the penultimate line of the proof. Suppose $$X \cong Y \oplus E$$. The penultimate line reads: $$X \cong Y \oplus \ell_p(Y) \oplus \ell_p(E) \cong Y \oplus \ell_p(X) \cong Y \oplus X$$

Where has the first isomorphism come from?

Thank you very much!

$$X \cong \ell_p(X) \cong \ell_p(Y \oplus E) \cong \ell_p(Y) \oplus \ell_p(E) \cong Y \oplus \ell_p(Y) \oplus \ell_p(E) \cong Y \oplus \ell_p(X) \cong Y \oplus X$$.
Only the third isomorphism requires thought. That is where the infinite direct sum is in the sense of $$\ell_p$$ (rather than in some other sense) is used.