= Tower proof of the loop theorem
Start with a singular <disk> whose <boundary> represents a prescribed nontrivial class in a <boundary> subsurface. Repeatedly lift it through double <covering spaces> of regular neighbourhoods of its image. The number of distinct image <simplexes> strictly increases and is bounded by the number of domain <simplexes>, so the tower ends. At the top, vanishing first mod-two <homology> forces every <boundary> component to be a <sphere>. An embedded <disk> can then be constructed and projected down the tower. Resolving double curves preserves a <boundary> class outside a chosen <normal subgroup>, giving an embedded compression <disk> at the bottom.
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