Let W, S, T be subspaces of finite-dimensional vector space such that S∩T=S∩W S+T=S+W, and W is subset of T. Prove that W=T.

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Definition of Composition?

I'm not sure of a formal proof but if you think about it the only way that S+T=S+W is if they are the same. If they were not the same that would mean there is an element of T not in W or an element of W not in T. If not you'd have something like {1,2}+{3,4,5}={1,2}+{3,4,6} implies {1,2,3,4,5}={1,2,3,4,6} which is not true obviously. It is still true that their intersections can be equal. {1,2} intersect {3,4,5} is the same as {1,2} intersect {3,4,6} (in this case the empty set). I think that the fact of their intersections with S are equal is superfluous information but I didn't care for proofs too much.

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