So I've been reading Atiyah's book titled $\textit{K-theory}$ and I'm stuck at Corollary 2.7.15 (pg 113) which is basically trying to prove a Kuenneth formula for K-theory. Ordinarily, I would try and be more specific about what I don't understand, but here I have very little understanding of his proof outline. From his first paragraph, I think he's trying to use a five-lemma kind of argument, by first considering the exact sequence $\require{AMScd}$ \begin{CD} K^{*}(Z, f(X))\rightarrow K^{*}(Z)\rightarrow K^{*}(f(X))\rightarrow 0 \end{CD} and tensoring it with $K^{*}(Y)$, so that one gets an exact sequence $\require{AMScd}$ \begin{CD} 0\rightarrow Tor^1(K^{*}(Z, f(X))\rightarrow 0\rightarrow Tor^1(K^{*}(X), K^{*}(Y))\rightarrow K^{*}((Z/X)\times Y)\rightarrow K^{*}(Z\times Y)\rightarrow K^{*}(X)\otimes K^{*}(Y)\rightarrow 0 \end{CD}
(where the middle zero comes from torsion-freeness of $K^{*}(Z)$) and then making some comparison to the exact sequence obtained from the pair $(Z\times Y, X\times Y)$. But it's not entirely clear as to how he obtains the desired exact sequence, nor is it clear where the $K^{*}(Y/X)$ comes from (as Y/X appears in the exact sequence of $\tilde{K}$ and not $K^{*}$, if I understood correctly).
I'd be grateful to anyone to help me answer this!