 Compatibility bound for complementary generating families

Let $n>2r-t$ and let $G,H$ generate respectively a t-intersecting family of $r$-sets and its complement family. If their union had size at most $n-r+t-1$, contain it in a set $T$ of that size. Since $|T|\ge r,n-r$, extend $G,H$ within $T$ to an $r$-set $A$ and an $(n-r)$-set $B$. Both $A$ and $B^c$ belong to the original family, but $|A\cap B^c|=|A\cup B|-|B|\le t-1$. This contradiction proves the bound.