Questions in Proposition $8.4$ of Textbook Clader and Ross Beginning in Algebraic Geometry

I am self studying Algebraic Geometry from the textbook of Clader and Ross and I have question in Propositions $8.4$ on page $217-218$.

Proposition $8.4$: Let $X\subseteq \mathbb{A}^m $ and $Y\subseteq \mathbb{A}^n$ be affine varieties. Then we have $I(X\times Y)= \langle I(X)\rangle+\langle I(Y)\rangle$.

Proof: The inclusion $ \langle I(X)\rangle + \langle I(Y) \rangle \subseteq I(X\times Y)$ is clear to me.

For the converse part,let $h\in I(X\times Y)$. It is always possible to find an expression of the form

$(8.5)$ $ h=f_1g_1+....+f_k g_k$ with $f_1,...,f_k \in K[x_1,...,x_m]$ and $g_1,...,g_k \in K[y_1,...,y_n]$. In any expression of the form $(8.5)$ , we may assume after rearranging that $g_1,...,g_j \notin I(Y)$ and $g_{j+1} ,....,g_k\in I(Y)$

I am not able to see why the last line must be true.Please help.

Fix an expression of this form for which $j$ is minimal. For this particular expression with $j$ minimal, we claim that $f_1,...,f_j\in I(X)$ from which it follows that $h=f_1g_1+...+f_j g_j+f_{j+1}g_{j+1} +...+ f_k g_k \in \langle I(X)\rangle +\langle I(Y) \rangle$ and the result follows.

To prove the claim, suppose towards a contradiction that one of $f_1,....,f_j$ does not lies in $I(X)$. WLOG, suppose that $f_1(a)\neq 0$ for some $a\in X$. Define $g=h(a,y)= f_1(a)g_1+ f_2(a)g_2+...+f_k(a) g_k \in K[y_1,...,y_n]$ Since $h$ valishes on $X\times Y$ it follows that $g\in I(Y)$ . Solving for $g_1$ , we have $g_1= {f_1(a)}^{-1}(g-f_2(a) g_2 -...-f_k(a) g_k).$ ----------- $(8.6)$

Replacing $g_1$ by this expression in $(8.5)$ , we arrive at an expression for $h$ of the form $h=(f_2-{f_1(a)}^{-1}f_2(a) f_1) g_2+....+ (f_k-{f_1(a)}^{-1} f_k(a) f_1) g_k + {f_1(a)}^{-1} f_1 g.$ ---------------$(8.7)$

Unfortunately, I am not able to deduce $(8.7)$ using $(8.6)$ and $(8.5)$. Can you please help me with the same?

In this new expression $g_2,...,g_j \notin I(Y)$ while $g_{j+1},...,g_k,g \in I(Y)$,contradicting the minimality of $j$ and completing the proof.

How to deduce that $g_2,...,g_j \notin I(Y)$ while $g_{j+1},...,g_k,g \in I(Y)$ . I am unable to see this.

Please help me with these questions. I shall be very grateful.