# Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $$\tilde{\Phi_*}$$ in the diagram it's written that:

Let $$\pi:X\to B$$ be a fibration with $$B$$ being a CW complex. Considering $$\Phi_{\alpha}:D_{\alpha}^p\to B^p$$ being the characteristic map for the $$p$$-th cell of the CW complex $$B$$ and defining $$\tilde{D_{\alpha}^p}=\Phi_{\alpha}^*(X_p)$$ we have a map $$\tilde{\Phi}:\sqcup_{\alpha}(\tilde{D_{\alpha}^p},\tilde{S_{\alpha}^{p-1}})\to (X_p,X_{p-1})$$, where $$\tilde{S_{\alpha}^{p-1}}$$ is the part of $$\tilde{D_{\alpha}^p}$$ over $$S_{\alpha}^{p-1}$$. Which will induce the map on homology.

It's written that as there is a neighbourhood $$N$$ of $$B^{p-1}$$ in $$B^p$$, which deformation retracts onto $$B^{p-1}$$, by the homotopy lifting property of $$\pi$$, we have $$X_{p-1}\hookrightarrow \pi^{-1}(N)$$ is a homotopy equivalence. Now it's written that using excision,this implies $$\tilde{\Phi}$$ induces the isomorphism $$\tilde{\Phi_*}$$.

I couldn't understand the last line, how excision is used. Please give some hint.