Fibers and Push-forward

Let $$f\colon X\longrightarrow Y$$ be a surjective morphism of projective varieties. Consider a non-singular irreducible projective curve $$T$$ and surjective morphism $$p_Y\colon Y\longrightarrow T$$. Then we get the following commutative diagram $$\require{AMScd}$$ $$\begin{CD} X_t @>f_t>> Y_t\\ @V V V @VV V\\ X @>f>> Y\\ @Vp_X V V @Vp_YV V\\ T @>>\mathsf{id}_T> T \end{CD}$$ so in particular an induced map between fibers $$f_t\colon X_t\longrightarrow Y_t$$. I would to check the following result:

Result: Let $$V\subseteq X$$ be an irreducible projective variety such that $$\dim V=\dim W$$, where $$W:=f(V)$$. Denote by $$[V_t]$$ the cycle associated to the projective variety $$p_X^{-1}(t)\cap V$$ and by $$f_\ast$$ the push-forward induced by $$f$$. Then $$(\#) \: \: \: \:(f_t)_\ast([V_t])=\big[f_\ast[V]\big]_t=\deg(f_{|V})[W_t].$$ This is a well know formula (eg. Prop. 10.1 of [Ful]), but in the reference I cited the Author proves it in a different way. Indeed, he consider intersection fibers instead of cycle associated to scheme theoretic fibers as I am doing.

My Attempt: We can assume that $$V$$ and $$Y$$ are affine and $$f$$ is dominant. Denote the coordinate rings of the fibers $$V_t,W_t$$ by $$A:=k[V_t]$$ and $$R=k[W_t]$$. Then we have $$(f_t)_\ast[V_t]=\sum_S\ell(A_{\mathfrak q_S})\deg({f_t}_{|S})f_t(S),$$ where $$S$$ runs over the irreducible components of $$V_t$$ such that $$\dim f(S)=dim W-1$$. On the other hand, we have $$\deg(f_{|V})\cdot[W_t]=[k(V):k(W)]\cdot\sum_Z\ell(R_{\mathfrak p_Z})Z,$$ where $$Z$$ runs over the irreducible components of $$W_t$$. Now fix an irreducible component $$Z$$ (corresponding to the minimal prime $$\mathfrak p\subset R$$) and consider the irreducible components $$S_1,\ldots,S_n$$ of $$V_t$$ (corresponding to the minimal primes $$\mathfrak q_1,\ldots \mathfrak q_n\subset A$$) whose image via $$f_t$$ is just $$Z$$. Proving $$(\#)$$ is then equivalent to provving the following formula $$[k(V):k(W)]\cdot\ell(R_\mathfrak p)=\sum_{i=1}^n[k(\mathfrak q_i):k(\mathfrak p)]\cdot\ell(A_{\mathfrak q_i}).$$

A similar aproach can be found here. More precisely, this lemma should be the key-point of the proof.

Add: The proof of this result would be important because one could check that the push-forward preserves algebraic equivalence of cycles by considering the following commutative diagram:

$$\begin{CD} \lbrace t\rbrace\times X @>f>> \lbrace t \rbrace\times Y\\ @V V V @VV V\\ T\times X @>\mathsf{id}\times f>> T\times Y\\ @Vp_X V V @Vp_YV V\\ T @>>\mathsf{id}_T> T \end{CD}$$

Thanks in advance for any help.

[Ful] Fulton, W. (1988) Intersection Theory. New York: Springer-Verlag.

By mean of standard geometric arguments, He deduces that it is enough to check the claim in the affine case and consider $$V$$ in place of $$X$$ and $$f(V)$$ in place of $$Y$$. Thus, the proposition is proved by showing the formula