So I'm trying to follow a proof from some online notes for Bézout's Theorem over affine space. In the proof we're predominantly using the following lemma which was already proven:
Say F, G $\in k[x,y]$, and say the degree of $F$ is $m$ and the degree of $G$ is $n.$ Then either $Res_x(F,G)$ = 0 or $Res_x(F,G) \leq$ $mn.$
So, Bezout's Theorem supposes that the two affine algebraic curves have no common components, so we know there's a finite number of intersection points. The rest is the write up of the proof. Since we know there are no common components, we know that there is a finite number of intersecting points. Now, we select some $\lambda \in k$ $\textit{such that we can write a coordinate $u = y + \lambda x$, such that each u has a distinct}$ $\textit{value for every two intersecting points of our curves.}$ We now turn to these curves, which as we know give the equations $C_F(x,y)$ = 0, and $C_G(x,y)$ = 0. Given our construction of u, we can express y = $u - \lambda(x)$. Thus, when determining the resultants of our two polynomials $F$ and $G$, we substitute y with this alternate denotation. Based on the lemma above, we know that $Res_x(F(x, u -\lambda(x)),G(x, y - \lambda(x)))$ results in a polynomial, where the degree of this polynomial is less than or equal to $mn$ in $u$. Thus, there are no more than $mn$ zeros in $u$, $\textit{where for our $\lambda$ we can match each individual zero in u to exactly one intersecting point.}$ Thus, the upper bound of intersecting points of these two curves is exactly $mn.$
I understand most of the proof with the exception of the italicized portions. I've tried looking at other proofs of the theorem online, but due to time restrictions this is the particular proof that I need to figure out. I'm just confused about the usage of the coordinate of u and its relation to $\lambda$, both in how there is a distinct $u$ values for every $\textit{two}$ intersecting points, and how our choice of $\lambda$ allows us to match one $u$ to one intersecting point. I apologize for the length of this question; I just wanted to ensure my confusion and what tools I had to work with.
We have a finite number of intersection points $a_i=(x_i,y_i)$. Suppose that the $u$-coordinate of $a_i$ and $a_j$ is the same. It means that $y_i+\lambda x_i=y_j+\lambda x_j$, i.e. $\lambda=\lambda_{ij}=\frac{y_j-y_i}{x_i-x_j}$ and $x_i\neq x_j$. It suffice to choose $\lambda$ not equal to any of $\lambda_{ij}$.