Isomorphic finite dimensional vector spaces

Question

Theorem: Two finite-dimensional vector spaces over $F$ are isomorphic if and only if
they have the same dimension.

$F$ denotes either $\mathbb{R}$ or $\mathbb{C}$

Below is proof of the theorem. Where in the proof do we assume that the vector spaces are over $F$? It seems to me like that assumption is not used. Can the theorem be restated as: "Two finite-dimensional vector spaces over the same field are isomorphic if and only if
they have the same dimension"?

Answer 1

On page $10$ of the book the author states that you can think of $F$ denoting arbitrary fields throughout chapter $1,2$ and $3$.