Reading the paper "Floer Cohomology of Lagrangian intersections" the authors construct a map $f: \mathbb{R}^n \times [0,2^N]\rightarrow \mathbb{C}\mathbb{P}^n$ such that $f(\tau,0)=f(\tau,2^N)$ with finite energy, by using $J-$holomorphic strips $u$ which glue smoothly. Then they claim that $C_{2^N}=\mathbb{R}\times [0,2^N]$ is confirmally equivalent to $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ by a map $\chi$. Then if we consider the map $f\circ \chi$ they claim that we can extend this map to a map from $\mathbb{C}\mathbb{P}^1$ by the removal of singularity theorem, since it will have finite energy .
Now my questions are doesn't the removal of singularity theorem just apply for a single singularity ? Say to extend the map from $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ to one from $\mathbb{C}\mathbb{P}^1-\{\infty\}$? Or can we apply it iteratively?
Also why do we to construct a map with the property that $f(\tau,0)=f(\tau,2^N)$ , and from $C_{2^N}$?
If we just considered a holomorphic strip $u:\mathbb{R}\times [0,1]\rightarrow \mathbb{C}\mathbb{P}^n$ couldn't we apply these results ? I'm guessing not otherwise the author would just do it, but I am not sure how we can use these conditions and how they are relevant.
Any help is appreciated. Thanks in advance.