1. Let \(d\theta_{X,\CC}=\omega_{X,\CC}\) be primitives of each symplectic form. By assumption we have \((\theta_{X}\oplus\theta_\CC) |_{K}=df\) for some smooth function \(f:K\to \RR\). Then \(\theta_X|_{L_i}=d(\iota_{i}^*f)\), where \(\iota_{i}:L_i\into K\) includes \(L_i\) as a fiber over one of the ends. Indeed, for \(v\in TL_i\) we have \[ \theta_X (v)=(\theta_X\oplus\theta_{\CC})(v,0)=df(v,0)=(\iota_i^*(df))(v)=d(\iota_i^* f)(v) \] which proves that \(\theta_X|_{L_i}=d(\iota_i^*f)\).
2. If \(p\in (L\times\gamma^-)\cap K\), then \(\pi_\CC(p)\in \pi_\CC(L\times\gamma^-)\cap \pi_\CC(K)=\set{z}\), so \((L\times\gamma^-)\cap K \cong L\cap L_2\) and these are precisely the generators of both vector spaces.
3. By the choice of almost complex structure, the composition \(p:=\pi_\CC\circ u:\RR\times[0,1]\to\CC\) is holomorphic. First of all note that \(\text{Im}(p)\) must be bounded: the finite energy condition says that \(u\) extends to a map \(\tilde p:D\to \CC\) from the disc, which is compact and thus has compact --- in particular, bounded --- image. Now if \(\text{Im}(p)\not\subset\set{z}=\text{Im}(\gamma^-)\cap \RR_{<<0}\), then \(\text{Im}(p)\cap U\neq\emptyset\) for an open unbounded region \(U\subset \CC\setminus(\gamma^-\cup \pi_\CC(K))\). Furthermore:
   • \(\tilde p(D)\) is compact and thus closed, so \(\tilde p(D)\cap U=\text{Im}(p)\cap U\) is closed in \(U\).
   • \(p(\RR\times(0,1))\) is open by the open mapping theorem, thus \(p(\RR\times(0,1))\cap U=\text{Im}(p)\cap U\) is open in \(U\).
   We have a non-empty open and closed subset of \(U\); as \(U\) is connected, we conclude that \(\text{Im}(p)\cap U=U\), which implies that \(\text{Im}(p)\) contains the unbounded region \(U\). This is a contradiction, so \(\text{Im}(p)=\set{z}\). Now we know that all pseudoholomorphic strips with boundary on \(L\times\gamma^-\) and \(K\) --- which are precisely those contributing to the differential of \(CF^\bullet(L\times\gamma^-,K)\) --- are of the form \(u(s,t)=(v(s,t),z)\), where \(v\) is a pseudoholomorphic strip in \(X\) with boundary in \(L\) and \(L_2\) --- which are precisely those contributing to the differential of \(CF^\bullet(L,L_2)\). Thus we have a bijection betweeen generators of \(CF^\bullet(L\times \gamma^-,K)\) and \(CF^\bullet(L,L_2)\), as well as between their pseudoholomorhic strips.
4. As a vector space, \(CF^\bullet(L\times\gamma^+,K)=CF^\bullet(L,L_1)\oplus CF^\bullet(L,L_2)\) by the same argument as in part 2. We now have three types of pseudoholomorphic strips:
   1. Those connecting two points in \(CF^\bullet(L,L_1)\), which are encoded in the differential \(\partial_{L,L_1}\).
   2. Those connecting two points in \(CF^\bullet(L,L_2)\), which are encoded in the differential \(\partial_{L,L_2}\).
   3. Those connecting points in \(CF^\bullet(L,L_1)\) with one in \(CF^\bullet(L,L_2)\), which make use of the bounded region between \(\gamma^+\) and \(\pi_\CC(K)\). Call this map \(f:CF^\bullet(L,L_1)\to CF^\bullet(L,L_2)\).
   Recalling the definition of a {\it mapping cone} of chain complexes, this says precisely that (up to grading considerations) \[ CF^\bullet(L\times\gamma^+,K)=cone(CF^\bullet(L,L_0)\xrightarrow{f} CF^\bullet(L,L_1)). \]
5. By the previous part we have a long exact sequence \[ \dots HF^{i-1}(L\times\gamma^+,K)\to HF^i(L,L_0) \to HF^i(L,L_1) \to HF^i(L\times\gamma^-,K) \to HF^{i+1} (L,L_0)\to\dots \] The Hamiltonian isotopy between \(K\times\gamma^-\) and \(L\times\gamma^+\) together with the invariance of Lagrangian Floer homology under Hamiltonian isotopy gives \(HF^\bullet(L\times\gamma^+,K)\cong HF^\bullet(L\times\gamma^-,K)\), and by part 3 the latter is isomorphic to \(HF^\bullet(L_2,K)\). Putting all together we get a long exact sequence \[ \dots HF^{i-1}(L_2,K)\to HF^i(L,L_0) \to HF^i(L,L_1) \to HF^i(L_2,K) \to HF^{i+1} (L,L_0)\to\dots \] as required.