This paper continues the exposition of a demonstrably consistent foundation for mathematics begun in An extension of basic logic (hereafter referred to as EBL) and in The Heine-Borel theorem in extended basic logic (hereafter referred to as HB). Still earlier papers related to these are A basic logic (referred to as BL) and Representations of calculi (referred to as RC). The main conventions and results of all the above papers will be presupposed in what follows. The numbering of sections and paragraphs will be a continuation of that of EBL and HB.