Note that is a vector space over the field of rational numbers, with ordinary real multiplication as the scalar multiplication. Let H be the basis of that space (a so-called Hamel basis). The dimension over of the defined space is .
Note that is also a vector space over the field ; it has a basis so its dimension is because is infinite, so this dimension is too.
Both and are vector spaces over of the same dimension; therefore they are isomorphic (as vector spaces). However, every isomorphism of linear spaces is additive and for that reason it is also an isomorphism of their additive groups. We have proven that .
The isomorphism between and is a trivial consequence of the isomorphism proven above.