My attempt

Let V be a finite- dimensional vector space. Let B = and B'= be 2 bases for B. By the (one half of) exchange lemma, a spanning set has at least as many elements as a lin. independent set. Hence m is greater than or equal to n and n is greater than or equal to m. So m=n

When I set out to do this proof I thought I would have to use all the exchange lemma so I'm a bit unsure of my 'proof'. Thanks