I'm studying the book "Algebraic Topology" by Tammo Tom Dieck: https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf
I don't understand the proof of proposition 8.5.1:
$\textbf{(8.5.1) Proposition.}$ Suppose $X$ is obtained from $A$ by attaching $(n+1)$-cells. Then $(X,A)$ is $n$-connected.
And that's the proof he gives:
$\textit{Proof.}$ We know that $(D^{n+1},S^n)$ is $n$-connected. Now apply (6.4.2).
And this is (6.4.2):
$\mathbf{(6.4.2) Proposition.}$ Let $Y$ be the union of open subspaces $Y_1$ and $Y_2$ with nonempty intersection $Y_0$. Suppose $(Y_2,Y_0)=0$ is $q$-connected. Then $(Y, Y_1)$ is $q$-connected.
However he does not give any detail about it. How can you prove proposition 8.5.1?
Suppose given a pushout diagram \begin{array}{ccc}\bigsqcup_{e\in\mathcal{E}} S^n_e &\rightarrow & \bigsqcup_{e\in\mathcal{E}} D^{n+1}_e\\ \varphi\downarrow & &\ \downarrow \\ A &\rightarrow& X. \end{array}
Write $D^{n+1}=S^n\times I/(S^n\times\{0\})$ and consider its subspaces $D^{n+1}(+)=(S^n\times[0,3/4))/(S^n\times\{0\})$ and $D^n(-)=S^{n-1}\times(1/4,1]$. Clearly $D^{n+1}(+)$ is contractible and the inclusion $S^n=S^n\times \{0\}\subseteq D^{n+1}(-)$ is a strong deformation retraction.
Now, find copies of these subspaces in each $(n+1)$-cell of $X$ and put $$X_1=A\cup\bigcup_{e\in\mathcal{E}} D^{n+1}_e(-),\qquad\qquad X_2=\bigcup_{e\in\mathcal{E}} D^{n+1}_e(+).$$
Then both $X_1,X_2$ are open in $X$ and $X=X_1\cup X_2$. The intersection $X_0=X_1\cap X_2$ is homeomorphic to a disjoint union $\bigsqcup_{e\in\mathcal{E}} S^n\times(1/4,3/4)$. The inclusion $$X_0=\bigsqcup_{e\in\mathcal{E}} S^n\times(1/4,3/4)\subseteq X_2=\bigsqcup_{e\in\mathcal{E}} D^{n+1}_e(+)$$ induces an isomorphism on path components, and otherwise neither side has nontrivial homotopy groups until dimension $n$. We see that $X_0\subseteq X_2$ is $n$-connected.
According to Proposition 6.4.2, therefore, the inclusion $X_1\subseteq X$ is $n$-connected. But the deformation retract of $S^n$ in $D^{n+1}(-)$ induces on of $A$ in $X_1$. Thus the inclusion $A\subseteq X$ factors as a deformation retraction $A\subseteq X_1$ and an $n$-connected map $X_1\subseteq X$. In particular $(X,A)$ is $n$-connected.