I'm studying $CW$ complex from Massey "A basic course in algebraic topology" and I don't understand the following senteces in order to prove Lemma 5.2 (page 241).
The setup is the following, we have
$$ H_n(\overline{e}_{\lambda}^n,\partial \overline{e}_{\lambda}^n )\overset{l_{\lambda*}}{\longrightarrow} H_n(K^n,K^{n-1})\overset{m_{\lambda*}}{\longrightarrow} H_n(K^n, K^n-e_{\lambda}^n)$$
With $$m_{\lambda*} \circ l_{\lambda*} = l'_{\lambda*} : H_n(\overline{e}_{\lambda}^n,\partial \overline{e}_{\lambda}^n ) \longrightarrow H_n(K^n, K^n-e_{\lambda}^n)$$
Here $l'_{\lambda*}$ and $m_{\lambda*}$ are induced by incluson which is why the diargram commute.
In order to prove Lemma 5.2 the book says "To prove the assertion about $m_{\lambda *}$, one must prove that if $e_{\lambda}^n \ne e_{\mu}^n$ then $m_{\mu*} \circ l_{\lambda*} =0$; this is an easy consequence on Lemma 5.3 below".
The cited lemma is the following :
Lemma 5.3 : Let $f: (X,A) \longrightarrow (Y,B)$ be a map of pair which is homotopic to a map $g : (X,A) \longrightarrow (Y,B)$ sucht that $g(X) \subset B$. Then the induced homomorphism $$f_* : H_n(X,A) \longrightarrow H_n(Y,B)$$ is zero for all $n$.
I do understand the proof of Lemma 5.3 but I don't see how I should use in this case with $m_{\mu*} \circ l_{\lambda*}$, if this is $f$, who's $g$?
Any help would be appreciated.
Consider $f = m_{\mu} \circ l_{\lambda} : (\overline{e}_{\lambda}^n,\partial \overline{e}_{\lambda}^n )\overset{l_{\lambda}}{\longrightarrow} (K^n,K^{n-1})\overset{m_{\mu}}{\longrightarrow} (K^n, K^n \setminus e_{\mu}^n)$. Clearly $f(\overline{e}_{\lambda}^n) \cap e_{\mu}^n = \emptyset$, i.e. Lemma 5.3 applies (with $g = f$).