 Note that these explanations are not in terms of proof conditions, but of assertion conditions. This may make a difference in particular for the explanation of implication, where, instead of only the information under what condition something counts as a proof of \(p\), we now can also take into consideration that, by hypothesis, a concrete construction for \(p\) has been effected. As we saw in  section 3.1.2,  the possibility to do so is crucial for Brouwer’s proof of the Bar Theorem.