Property. Transitivity [MACB]

Let $S \mathbin{⪧} X$ be a sieve on $X$, and $T \mathbin{⪧}_J X$ be a sieve in $J(X)$. If for all $f \in T$ we have $f^*(S) \mathbin{⪧}_J \text{dom}(f)$, then $S \mathbin{⪧}_J X$.