Two-quantum annihilation of positrons - the simplest antimatter particles - with electrons inside the matter is of fundamental interest. Annihilation with inner-shell electrons of heavy systems, in turn, provides a unique opportunity to perform antimatter researches in the presence of a strong nucleus field. This process requires a rigorous QED description with a nonperturbative account of the positron- and electron-nucleus interactions. We have worked out an approach aimed at performing such a description. We have applied the developed formalism, utilizing the exact Dirac-Coulomb Green function, for the description of the two-quantum annihilation of positrons with K-shell electrons of uranium ion. The infrared divergences which occur in this process are extracted and subtracted from the final expressions.