Generalized Schanuel Lemma This is on page 128, ex 3.15, of Rotman's AIHA, 

(Schanuel) Let $B$ be a left $R$-module over some ring $R$ consider two exact sequences, 
      $$ 0 \rightarrow K \rightarrow P_n \rightarrow \cdots \rightarrow B \rightarrow 0 $$
$$ 0 \rightarrow K' \rightarrow P'_n \rightarrow \cdots \rightarrow B \rightarrow 0 $$ 
      \noindent where $P_*, P'_*$ are projectives, prove that 
      $$ K \oplus P'_n \oplus P_{n-1} \oplus \cdots \cong K' \oplus P_n \oplus P'_{n-1} \oplus \cdots $$ 

I could not really apply the usuall Schanuel's lemma, any hint? 

I found the following proof in Lectures on Modules and Rings by T. Y. Lam.
We do an induction on $n$. Assume the claim is true for $n-1$. Write $f$ and $g$ for the arrows $P_{0}\to B$ and $Q_{0}\to B$.
Applying the usual version of Schanuel's lemma to the sequences
\begin{gather*}
0\to\ker f\to P_{0}\to B\to0,\\
0\to\ker g\to Q_{0}\to B\to0,
\end{gather*}
we deduce that $\ker g\oplus P_{0}\cong\ker f\oplus Q_{0}$. Now the induction hypothesis applies to the sequences
\begin{gather*}
0\to K\to P_{n}\to\dots\to P_{2}\to P_{1}\oplus Q_{0}\to\ker f\oplus Q_{0}\to0,\\
0\to K'\to Q_{n}\to\dots\to Q_{2}\to Q_{1}\oplus P_{0}\to\ker g\oplus P_{0}\to0.
\end{gather*}
