Linearly Independent Set is Basis iff of Same Cardinality as Dimension
Let $H$ be a linearly independent subset of $E$.
Let $H$ be a basis for $E$.
By Bases of Finitely Generated Vector Space have Equal Cardinality, it follows that $H$ also contains exactly $n$ elements.
Let $H$ contain exactly $n$ elements.
By Sufficient Conditions for Basis of Finite Dimensional Vector Space $H$ is itself a basis for $E$.