Vtyjkyuk

I came across this presentation on the Grassmannian, Plücker coordinates and Plücker relations (https://sites.math.washington.edu/~morrow/papers/matt-masters.pdf). It states that

where $X$ denotes the image of the Grassmannian $Gr(k,n)$ under the Plücker embedding and for some vectorspace $S_k in Gr(k,n)$ and corresponding matrix $A$, $P_L$ denotes the determinant of the submatrix of $A$ obtained by only considering the columns indexed by a set $L$ (gives a $k times k$ matrix). Moreover, the following definition is given:

So, I wanted to prove this Theorem, but I'm having some trouble. I know that for some $p in X$, $F_I,J(p)=0$. Then I tried to argue along the lines of showing $V(G)=V(P_I,J)$ by showing both inclusions, but having this result does not necessarily mean that $G$ is a multiple of $F_I,J$.

So, I have no idea how to show this. I'm sure this has to hold even for a more general setting (i.e. not necessarily Plücker coordinates etc.). Thank you for any suggestions.