X is a vector space such that dim(X)=n. $M \subset X $ is a subspace. let $B=\{b_1,b_2,...,b_m\} $ is a hamel basis for M.

Question

if X is a vector space such that dim(X)=n and $M \subset X $ is a subspace. let $B=\{b_1,b_2,...,b_m\} $ is a hamel basis for M. then

1. $dim(M)\le dim(X) $
2. there is a set $ D=\{d_1,d_2,...,d_{m-n} \}$ such that $B \cup D$ is a hamel basis for X.

I tried to prove the first sentence like this:

Since B is a basis the B is linear independent then if Span B=X then dim(X)=dim(M). let $span B\neq X $ so B, is not a basis for X. since B is linear independent then $dim(M)=m \le dim(X) $

for the second sentence:

B isn't a basis for X then $ \exists d_1\in X : x\notin Span B$ then $ B\cup \{d_1\} $ is linear independent . if $ span ( B\cup \{d_1\}) $ is a basis for X then the proof is finished. if $ span ( B\cup \{d_1\}) $ is not a basis for X then we continue in the same way until we obtain the basis.

is this correct?