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!