  We develop the method of averaging in Clifford (geometric) algebras suggested
by the author in previous papers. We consider operators constructed using two
different sets of anticommuting elements of real or complexified Clifford
algebras. These operators generalize Reynolds operators from the representation
theory of finite groups. We prove a number of new properties of these
operators. Using the generalized Reynolds operators, we give a complete proof
of the generalization of Pauli's theorem to the case of Clifford algebras of
arbitrary dimension. The results can be used in geometry, physics, engineering,
computer science, and other applications.
