EDIT: So my question is distinct from the question asked here because I am asking an easier question. Why should we have to invoke something as powerful as the "Annulus Theorem" to show that the connected sum is well-defined here?
Define the connected sum of two surfaces $\Sigma_1$, $\Sigma_2$ to be the surface we get by taking away a small disk from each of $\Sigma_1$, $\Sigma_2$ and sewing the two boundary components together. The surface we get is denoted by $\Sigma_1 \# \Sigma_2$.
Is it possible someone could provide a concise but complete proof that for orientable $\Sigma_1$, $\Sigma_2$ this operation is well-defined, i.e. the resulting surface doesn't depend on which disks are removed from $\Sigma_1$, $\Sigma_2$ or how the boundaries are glued?
I had searched a bit, but out of the two sources I looked at, one simply asserted that the operation was well-defined, which does not seem to be obvious at all, and the other one simply gave some vague "intuition" as to why the statement should be true.