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https://en.m.wikipedia.org/wiki/Great_Circle#Derivation_of_shortest_paths

This proof require the path to be differentiable, which is not a part of the statement.

https://math.stackexchange.com/questions/1180923/shortest-path-on-a-sphere

There are several proofs in the page, they are rely on the uniqueness of the shortest path. They thought the uniqueness is intuitive, but I cannot persuade myself on this assumption.

Can anyone provide me a strict proof without other assumption of the statement?