Every discrete subgroup of Euclidean space is free Abelian on a linearly independent set

Statement
Let $$H$$ be a discrete subgroup of $$\R^n$$. Then, $$H$$ is a free Abelian group of rank at most $$n$$, with a freely generating set of linearly independent vectors in $$\R^n$$.

Facts used

 * Every nontrivial discrete subgroup of reals is infinite cyclic
 * Every nontrivial subgroup of the group of integers is cyclic on its smallest element