 This is a strengthening of the corresponding weak counterexample from 1923, but the argument is entirely different. The strong counterexample depends on the theorem in intuitionistic analysis, obtained in 1924 and improved in 1927, that all total functions \([0,1] \rightarrow \mathbf{R}\) are uniformly continuous. The non-classical elements in that theorem are the conception of the continuum as a spread of choice sequences, and the bar theorem based on it (for further explanations of this conception, see Heyting 1956 (Ch.3) and van Atten 2004b (Chs.3 and  4)).[10]