The existence of an indecomposable summand of a projective module such that it is isomorphic to a given simple module

Question

I am reading the book “ Elements of the Representation Theory of Associative Algebras:Volume 1” written by Ibrahim Assem, Daniel Simson, Andrzej Skowronski.

I have a question about the proof of Chapter $2$ lemma $2.12$.

Let $A$ be a basic, connected and finite dimensional $K-$algebra.

Let $P$ be a finitely generated right $A-$module and a projective cover of a simple module $S$.

Let $S^\prime$ be another simple right $A$-module. If we choose $0\ne f\in \mathrm{Hom}_A(P,S^\prime)$, then $f$ is surjective by Schur lemma.

Then in this book, author says there exists an indecomposable summand $P’$ of $P$ such that $f$ equals the composition of the canonical projection $P\rightarrow P’$ and the canonical homomorphism $P^\prime\rightarrow P ^\prime /\mathrm{rad}P’$, and an isomorphism $P^\prime/\mathrm{rad}P^\prime\cong S^\prime$.

My question is that how can I get the existence of $P’$ and the isomorphism?

Here are the details.

Any help and references are greatly appreciated.

Thanks!