| Theorem 4.9 If V has a basis, then any two bases have the same number of elements.
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| Proof Suppose that X and Y are both maximal. By the Steinitz exchange axiom, if |X| < |Y|, then we could obtain a linearly independent set containing X by adding an element of Y , which contradicts the maximality of X. Similarly, the assumption that |Y| < |X| leads to a contradiction. So |X| = |Y|.
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