 Lemma. Every formula \(F\) of \(L\) is equivalent to (i.e. is satisfied by exactly the same assignments as) some boolean combination of sentences of the form ‘There are exactly \(k\) elements in \(A\)’ and formulas of the form ‘There are exactly \(k\) elements that are in \(v_1\), not in \(v_2\), not in \(v_3\) and in \(v_4\)’ (or any other combination of this type, using only variables free in \(F)\). 