One conservative extension of formal mathematic analysis with a scheme of dependent choice

This paper studies an extension of classical analysis, the language of which is obtained by adding to the language of analysis a two-place predicate symbol $\rho$. To the axioms of this extension, in addition to all the axioms of analysis (a convolution scheme is selected for all the formulas of the new language), there also belong a series of axioms asserting that relationship $\rho$ completely orders the class of all sets of natural numbers. It is proven that the theory described herein is a conservative extension of analysis with a scheme of dependent choice.