Lemma: Strongly separated witness class

Suppose $f,g$ have time supports in $[0,T]$ and spatial supports in

respectively. If

then both $B_f\gtrsim W_g$ and $W_g\gtrsim B_f$ hold.

Proof. A point in $P_F\cup F$ has time in $[0,T]$ and spatial distance at most $T+r$ from $x_0$. A point in $G$ has time in $[0,T]$ and spatial distance at most $s_0$ from $y_0$. Their spatial distance is greater than $T$, whereas their time difference is at most $T$. The upper support sets are causally disjoint. The actual supports are subsets of them by (eq:support-upper).