Weird result about seminorm in $\operatorname{BV}[a,b]$

Let $f\in \operatorname{BV}[a,b]$, where $\operatorname{BV}[a,b]$ stay for the set of functions $f:[a,b]\to \mathbb{R}$ of bounded variation. It follows that $f$ can be written as the difference of two increasing functions, specifically $f=f_1-f_2$ where $f_1(t):=V_a^t(f)$ (where $V_a^t(f)$ is the variation of $f$ in $[a,t]$) and $f_2=f-f_1$. Its well-known that $V_a^b$ is a seminorm in $\operatorname{BV}[a,b]$.

Then, if I'm not wrong, we have that $V_a^b (f_1)=V_a^b(f)$. By the other hand it follows that $V_a^b(f_1+f_2)=V_a^b (f_1)+V_a^b (f_2)$ as both $f_1$ and $f_2$ are increasing, therefore $V_a^b (f+2f_2)=V_a^b(f)+V_a^b(f_2)$. As these results seem to me extremely non‑intuitive I opened this question to confirm if they are right or if there is something that I'm overlooking. Thank you in advance.