 Bolzano also presents two important criteria for the correctness of proofs: according to the first, “If the subject (or the hypothesis) of a proposition is as wide as it can be so that the predicate (or the thesis) can be applied to it, then in any correct proof of this proposition all characteristics of the subject must be used, i.e., they must be applied in the derivation of the predicate.” The second is Bolzano’s version of Aristotle’s ban on crossing from one genus to another: one cannot deduce a general proposition from one of its special cases (Bolzano 1810, II, §28, 29; Bolzano 2004b: 122–126). In a proof of the intermediate value theorem, for instance, one of the necessary conditions is that the functions be real-valued. An attempted proof that makes no use of any attribute separating the reals from the rationals can accordingly be diagnosed as incorrect according to the first criterion, while an attempted proof that appeals to geometry or kinematics will run afoul of the second. These insights immediately bore fruit in Bolzano’s mathematical work (1816, 1817).