 The Reduction Theorem makes proving completeness of the axiomatic theory with respect to the appropriate class of pointed Kripke models easy: since every \(\LPAL\)-formula can all be expressed using a provably equivalent announcement-free \eqref{ML}-formula, completeness of the theory \(\PAL\) follows by the Reduction Theorem, the soundness of \(\PAL\), and the known completeness of the underlying modal theory \(\L\). 