Let $\ f_1,\cdots, f_s \in K[x_0, x_1, \cdots, x_n]$ and $F_1 = f_1, \cdots, F_s = f_s \in K[x_0, x_1, \cdots, x_n, x_{n+1}]$, also let $X = V(f_1,\cdots, f_s) \subset \mathbb{P}^n$ and $X' = V(F_1,\cdots,F_s) \subset \mathbb{P}^{n+1}$, then apparently, $$ X' = \{ (a_1 : \cdots : a_n : a_{n+1})\ |\ (a_1 : \cdots : a_n) \in X \mathrm{\ and\ } a_{n+1} \in K\}$$ and (I think??) $\dim(X') = \dim(X) + 1$. What about the degree of $X'$? I think it should be $$\deg(X') = \deg(X).$$ My idea of proving this is by the geometric definition of degree of varieties, that is, the number of intersection points between $X$ and its complementary general linear subspace (i.e. generic plane). Let $\dim(X) = d$ and $L$ be a general subspace of dimension $n-d$, then the number of intersection points between $X$ and $L$ is $\deg(X)$, say $l$, and they are $q_1, \cdots, q_l$.
Now in $\mathbb{P}^{n+1}$, $\dim(X') = d+1$ and thus $\deg(X')$ is the number of intersection points between $X'$ and a general linear subspace $L$ (this can be the same as that above I believe) of dimension again $n-d$. But I am stuck in here about how to reason that the projections of the intersection points between $X'$ and $L$ from $\mathbb{P}^{n+1}$ to $\mathbb{P}^n$ are actually $q_1, \cdots, q_l$. Any advices on this will be greatly appreciated.
Another approach I am thinking is using Hilbert polynomials. However, I am still not confident on the familiarity with the formula of corresponding Hilbert polynomials. Is there any useful materials I can turn to on how to explicit write out (at least in combinations form) the Hilbert polynomial given an arbitrary variety?
Another question would be: is this true in affine space?
Thank you very much.