Consider a construction sequence $<\epsilon_{1},\ldots,\epsilon_{n}>$ for formula $\phi$ , i.e. $\epsilon_{n}=\phi$ . Consider the last expression $\epsilon_{k}$ , that uses sentence symbol $A_{4}$ . Since, $\phi$ does not use it, $k<n$ . Moreover, for all $j$ , $\epsilon_{j}$ does not use $\epsilon_{k}$ , as $\epsilon_{j}$ cannot use it for $j\le k$ , and for $j>k$ , $\epsilon_{j}$ does not use $A_{4}$ used in $\epsilon_{k}$ . Hence, if we remove step $\epsilon_{k}$ from the sequence, the sequence will remain well-formed, as nothing uses the expression obtained at this step. The resulting sequence is still a construction sequence for $\phi$ , since $\epsilon_{n}=\phi$ , but now it has one less expression using $A_{4}$ . Continuing this way, we can ensure that no expression in the sequence uses $A_{4}$ , while still the sequence is a construction sequence for the wff $\phi$ . Alternatively, we could use induction by the number of expressions in the sequence that use $A_{4}$ .