A version of Bezout's Theorem I have read the following version of Bezout's Theorem, but I don't get to understand how it implies the classical version.

Let $F,G\in K[X_{0},X_{1},X_{2}]$ be non-constant homogeneous polynomials of respective degrees $d$ and $e$ with no common irreducible components. Then, there exists a ring $A$ such that 
  $$
\mathrm{Proj}\left(\frac{K[X_{0},X_{1},X_{2}]}{\langle F,G\rangle}\right)\simeq \mathrm{Spec}(A)
$$
  and $\dim_{Krull} (A)=0$. By Noether's Theorem $A$ is a finite dimensional $K$-vector space. Furthermore, $\dim_{K}(A)=d\cdot e$.

Now, how could we prove that this imply $\#\mathrm{Spec}(A)=d\cdot e$ (or less, depending on how the multiplicity works in this version)?
$\mathbf{Remark}.$ The reason why there exists a ring $A$ such that 
$$
\mathrm{Proj}\left(\frac{K[X_{0},X_{1},X_{2}]}{\langle F,G\rangle}\right)\simeq \mathrm{Spec}(A)
$$
and $\dim_{Krull} (A)=0$ is because $\mathrm{Proj}\left(\frac{K[X_{0},X_{1},X_{2}]}{\langle F,G\rangle}\right)$ has dimension $0$, and therefore it is affine.
 A: (I should say that my intent here is merely to explain what the left- and right-hand sides of the isomorphism mean intuitively so that you can see how this version of Bezout's Theorem implies the classical one, which you said you have trouble seeing.)
As you probably already know, $\mbox{Proj}\left(\frac{K[X_0,X_1,X_2]}{\langle F, G\rangle}\right)$ can be interpreted as the scheme-theoretic intersection of the subschemes of $\mathbb{P}^2 := \mbox{Proj}(K[X_0,X_1,X_2])$ cut out by the polynomials $F$ and $G$.
So now let's look at the other object, $\mbox{Spec}(A)$. Since you're applying Noether's theorem, it looks (as would be expected) like the assumption is that $A$ is a finitely-generated $K$-algebra. In this case, having Krull dimension $0$ is a characterization of Artin rings. It is a standard result that $\mbox{Spec}(A)$ is finite and discrete if and only if $A$ is an Artin ring (see the linked page, or e.g. Atiyah and MacDonald's Introduction to Commutative Algebra). Moreover, the best way to say what $\mbox{dim}_K(A)$ represents is that it represents the sum of all multiplicities of the points in the finite discrete set $\mbox{Spec}(A)$.
Hopefully from here it is clear how this version of Bezout is equivalent to the classical version.
