Intersection product of line bundles with $\mathcal{F}$. I am having trouble with computing the intersection number using the following definition [Ref Vakil Chapter 20]:

Suppose $\mathcal{F}$ is a coherent sheaf on $X$ with proper support
  of dimension at most $n$, and $\mathcal{L}_{1}$,..., $\mathcal{L}_{n}$
  are invertible sheaves on $X$. The intersection of the line bundles
  with the coherent sheaf is defined to be 
  $$(\mathcal{L}_1\cdot...\mathcal{L}_n\cdot\mathcal{F})=\sum_{\{i_{1},...,i_{m}\}\subseteq
 \{1,...,n\}}(-1)^m\chi(X,\mathcal{L}_{i_1}^\vee\otimes...\otimes\mathcal{L}_{i_m}^\vee\otimes\mathcal{F}).$$

So trying to attempt both questions 20.1.B(b) and 20.1.C, I run unto certain problems understanding the definition while doing the calculations:

Exercise 20.1.B(b) [Proving Bézout's theorem] Suppose $k$ is an infinite field, $X=\mathbb{P}^N$ and $Y$ is a dimension $n$ subvariety
  of $X$. If $H_1$, ..., $H_{n}$ are generally chosen hypersurfaces of
  degree $d_1$, ..., $d_n$ respectively, show that 
$$(\mathcal{O}_X(H_1)\cdot...\cdot\mathcal{O}_X(H_n)\cdot
Y)=d_{1}...d_{n}\deg Y$$

Suppose exercise 20.1.C is true that is 

Exercise 20.1.C Suppose $D$ is an effective Cartier divisor on $X$ that restricts to an effective Cartier divisor $D|_Y$ on $Y$, show
  that $$(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n-1}\cdot
 \mathcal{O}(D)\cdot Y)=(\mathcal{L}_1\cdot...\cdot
 \mathcal{L}_{n-1}\cdot D|_Y).$$ More generally if $D$ is an effective
  Cartier divisor on $X$ that does not contain any associated point of
  $\mathcal{F}$, show that  $$(\mathcal{L}_1\cdot...\cdot
 \mathcal{L}_n-1\cdot \mathcal{O}(D)\cdot
 \mathcal{F})=(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n-1}\cdot
 \mathcal{F}|_Y).$$

Suppose exercise C is true, then we may prove by induction I think, on the dimension $n$ of $Y$. Suppose it is true for $n-1$, then the formula in C gives 
$$(\mathcal{O}_X(H_1)\cdot...\cdot\mathcal{O}_X(H_n)\cdot
Y)=(\mathcal{O}_Y(H_1)\cdot...\cdot\mathcal{O}_Y(H_{n-1})\cdot\mathcal{O}_Y(H_n))\\
=\deg_{Y}(H_1)\cdot...\cdot\deg_{Y}(H_n) (\text{Or } =\deg_{Y}(H_{1}\cap...H_{n})???.)$$
Then we conclude by $\deg_{Y}K=\deg K\cdot \deg Y$ (?????). Then I will have a problem, either I get $d_{1}...d_{n}(\deg Y)^n$ or $d_{1}...d_{n}(\deg Y)$. Which one is correct?
Assume for the sake of consistency that the second one is correct...
Which I think remains to solve C. I am stuck at this stage...
So using the following exact sequence $$0\rightarrow
 \mathcal{O}_X(-Y)\rightarrow\mathcal{O}_X\rightarrow
 \mathcal{O}_X|_{Y}\rightarrow 0,$$
by tensoring with the any line bundle $\mathcal{L}$ we have
$$\chi(X,\mathcal{L}^\vee\otimes\mathcal{O}_{X}(-Y))=\chi(X,\mathcal{L}^\vee)-\chi(Y,\mathcal{L}^\vee|_Y).$$
Hence doing this for several line bundles and taking signed sum, we get
$$(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n}\cdot Y)=\sum_{\{i_{1},...,i_{m}\}\subseteq
 \{1,...,n\}}(-1)^m\chi(X,\mathcal{L}_{i_1}^\vee\otimes...\otimes\mathcal{L}_{i_m}^\vee)+\sum_{\{i_{1},...,i_{m}\}\subseteq
 \{1,...,n\}}(-1)^{m-1}\chi(Y,\mathcal{L}_{i_1}|_{Y}^\vee\otimes...\otimes\mathcal{L}_{i_m}|_{Y}^\vee),$$
where it is suggested that the first term in the RHS should vanish and the second term should be something like $(\mathcal{L}_1\cdot...\cdot
 \mathcal{L}_{n-1}\cdot D|_Y)$, but I don't see why is this evident. I am quite lost in all these manipulations.
 A: I realise this might be stupid, the reason why I was stuck because I have been looking at the wrong exact sequence. I start by answering question C. We consider the following exact sequence
$$0\rightarrow \mathcal{O}_{Y}(-D)\rightarrow\mathcal{O}_{Y}\rightarrow\mathcal{O}_{Y\cap D}\rightarrow 0$$
For any line bundle $L$ (or invertible sheaf whichever applies) we have
$$\chi(Y\cap,L^{\vee}\otimes\mathcal{O}_{Y\cap D})=\chi(Y,L^{\vee}\otimes\mathcal{O}_{Y})-\chi(X,L^{\vee}\otimes\mathcal{O}_{Y}(-D)).$$
Therefore, we have
$$\sum_{\{i_{1},...,i_{m}\}\subseteq \{1,...,n-1\}}(-1)^{m}\chi(Y\cap D,L_{i_{1}}^{\vee}\otimes...\otimes L_{i_{m}}^{\vee}\otimes\mathcal{O}_{Y\cap D})\\
=\sum_{\{i_{1},...,i_{m}\}\subseteq \{1,...,n-1\}}(-1)^{m}\chi(Y,L_{i_{1}}^{\vee}\otimes ...\otimes L_{i_{m}}^{\vee}\otimes\mathcal{O}_{Y})+(-1)^{m+1}\chi(Y,L_{i_{1}}^{\vee}\otimes ...\otimes L_{i_{m}}^{\vee}\otimes\mathcal{O}_{Y}(-D))\\
=\sum_{\{i_{1},...,i_{m}\}\subseteq \{1,...,n-1\}}(-1)^{m}\chi(Y,L_{i_{1}}^{\vee}\otimes ...\otimes L_{i_{m}}^{\vee}\otimes\mathcal{O}_{Y})+(-1)^{m+1}\chi(Y,L_{i_{1}}^{\vee}\otimes ...\otimes L_{i_{m}}^{\vee}\otimes\mathcal{O}(D)^{\vee}\otimes\mathcal{O}_{Y}).
$$
Hence we have proven 
$$(L_{1}\cdot...\cdot L_{n-1}\cdot\mathcal{O}(D)\cdot Y)=(L_{1}\cdot...\cdot L_{n-1}\cdot D\cap Y).$$
Now to prove B, we use induction
$$(H_{1}\cdot H_{n}\cdot Y)=...=(H_{1}\cdot H_{2}\cap...\cap Y)=\deg_{H_{2}\cap...\cap Y} H_{1}=d_{1}\cdot d_{n}\deg Y.$$
