$k[X,Y]/(F,G)$ is a finite dimensional $k$-vector space

Let $k$ be an algebraically closed field and $V$ be an affine variety.

From Page 19 of Daniel Perrin’s Algebraic geometry.

Lemma: Let $F,G\in k [X,Y]$ be non-zero polynomials without common factors, there is a non-zero polynomial $d\in k[X]$ and polynomials $A,B\in k[X,Y]$ such that $d= AF+BG$

Proof: If $d$ is the GCD of the polynomials $F,G \in k[X,Y]$ then by Bezout’s elemenary theorem we can get that there exists polynomials $A,B \in k[X,Y]$ such that $d=AF+BG$.

I hope I am correct.

Use the lemma to prove the following theorem:

Theorem 1: Let $F,G\in k [X,Y]$ be non-zero polynomials without common factors. Then $V(F)\cap V(G)$ is finite.

The proof is clear to me using the hints given in the text.

Theorem 2: Under the hypothesis of Theorem 1, the ring $k[X,Y]$ is a finite dimensional $k$-vector space.

Proof hint given: We use similar argument as given in 5.1 applied to the images of the monomials $X^i Y^j$ in the quotient ring, we see by Lemma that a finite number of these monomials generate $k[X,Y]/(F,G)$.

But I am not able to prove Theorem 2 using the hints and would appreciate a complete argument.

Can you please help me with this?