= Local continuation proof of the monodromy theorem
Along one <path>, <analytic continuation> is a chain of <function elements> on overlapping <open discs>, each agreeing with the preceding <germ of a holomorphic function> near the crossing point. The <identity theorem> makes this chain unique along that fixed <path>. Equality at a crossing means equality on a <neighbourhood>, not just equality of a single value.
For the deformation step, fix one intermediate <path> $t\mapsto H(s_0,t)$ and its continuation. By <compactness> of $[0,1]$, choose a finite chain of representative <open discs> and subdivide the parameter so that each subarc fits in its assigned <open disc>, with consecutive junctions in overlap <neighbourhoods> where the representatives agree. Joint <continuity> of $H$ ensures that every sufficiently nearby intermediate <path> uses that same chain and those same overlaps. The endpoints stay fixed, so the terminal <germ of a holomorphic function> stays unchanged. Existence along every intermediate <path> supplies this argument at each $s_0$; local constancy and <connectedness> of $[0,1]$ give equality at $s=0$ and $s=1$. This proves the <monodromy theorem> without assuming that every <analytic germ projection> is a <covering map>.
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