  The basic theorems of intuitionistic analysis — the bar theorem, fan theorem, and continuity theorem — are in “On the Domains of Definition of Functions” of 1927. The first two are structural theorems on spreads; the third (not to be confused with the continuity principle for choice sequences) states that every total function \([0,1] \rightarrow \mathbb{R}\) is continuous and even uniformly continuous. The fan theorem is, in fact, a corollary of the bar theorem; combined with the continuity principle, which is not classically valid, it yields the continuity theorem, which is not classically valid either. The bar and fan theorems on the other hand are classically valid, although the classical and intuitionistic proofs for them are not exchangeable. The classical proofs are intuitionistically not acceptable because of the way they depend on PEM; the intuitionistic proofs are classically not acceptable because they depend on reflection on the structure of mental proofs. In this reflection, Brouwer introduced the notion of the “fully analysed” or “canonical” form of a proof, which would be adopted much later by Martin-Löf and by Dummett. In a footnote, Brouwer mentions that such proofs, which he identifies with mental objects in the subject’s mind, are often infinite.