Applications of subtraction techniques in many-body scattering theory

General subtraction procedures for many-body scattering equations are developed which allow one to separate different contributions to the full N-body dynamics and to treat them independently in different nested sets of equations. Compact matrix subtraction procedures for both two-body connected-kernel and highly-connected-kernel equations are presented. We explicitly show how the original complicated labeling structure of two-body connected equations drastically reduces itself upon the selection of few dominant clusters or channels