the elements of , and therefore is a linear combination of elements of ). As the cardinality of is greater than or equal to the cardinality of , one may replace with ; that is, one may suppose, without loss of generality, that is a basis.

Thus, every can be written as a finite sum

where is a finite subset of Let . Since is spanned by , which is itself spanned by the , the latter set spans . Since this set is a subset of a basis,

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