Lemma: Weak monotonicity on independent factors

For every state ${\rho }_{ABE}$ on the stated tensor product,

Proof. Purify ${\rho }_{ABE}$ by $F$. Strong subadditivity gives $S(B|A)\ge S(B|AF)$. Purity of $ABEF$ gives $S(B|AF)=-S(B|E)$. Combining the two identities proves the claim.