We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis ∧, ∨) can be turned into a proof containing only monotone formulas of size m O(log m) and with the number of proof lines polynomial in m. Also we show that some interesting special cases, namely the functional and the onto versions of PHP and a version of the Matching Principle, have polynomial size monotone proofs