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Let $L,M$ two subspaces of the vector space, $V$ such that both $L,M \ne V$.
Prove: $L\cup M \ne V$.
I think this is a case of a proof by contradiction.
Lets assume $L \cup M = V$.
$$\dim(V) = \dim(L) + \dim(M) - \dim(L\cap M)$$
How to proceed?