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I have a question concerning following proof, Lemma 1.

http://planetmath.org/allbasesforavectorspacehavethesamecardinality

So, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to ##S_1##. How do we know for sure that ##a_1## is not yet in B? Can we say this because we suppose that m < n, thus there is certainly such an element?(to derive a contradiction)

Thanks!

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# I Proof that every basis has the same cardinality

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