Claim: Let $V: [0, T] (\subset \mathbb{R}) \to \mathbb{R}$ be a Lipschitz continuous function, i.e. $$|V(x,y)|:=|V(x)-V(y)| \leq K_1 | x - y |$$ for all $x,y$ in $[0, T] $, where $V(.,.)$ denotes the difference between values of $V$. Then, it follows that the difference satisfies a Lipschitz-like condition, i.e that there exists $K_2$ such that, for all $x,y,x’,y’$ in $\mathbb{R}$,
$$ | V(x,y) - V(x’,y’)| \\ \leq K_2 | ( x - y) - (x’- y’) |$$
I know there is a chance this may be false from academic vox populi, can you find a counterexample?
This post here is a follow-up to this question, so more background can be found there. I believe this specific case is more tractable: How to prove Lipschitz functions are not Lipschitz in the $p$-variation metric?
Take $y'=-x, x'=-y$. Then the inequality you want is $|V(x)-V(y)-V(-y)+V(-x)| \leq 0$. This is false for $V(x)=|x|$, for example.