Combinatorics of the $R$ operation

By using the functional language the new proof is given for the fundamental combinatorial statement in the renormalization theory [1], i. e. the application of $R$-operation to the diagrams of the initial theory is equivalent to the addition to the initial interaction $V(\varphi)$ the counterterms $\Delta V(\varphi)=-LH(\varphi)$, where $L$ defines $R=R(L)$ counter term operation on the diagrams such that the counter term $L\gamma$ corresponds with the graph $\gamma$, and $H(\varphi)$ is the $S$-matrix functional represented by the diagrams. (In the quantum field theory the operator of $S$-matrix is given by $T\exp V(\hat\varphi)=NH(\hat\varphi)$, where $T$ is a Wick chronological product, $N$ is a normal product, $\hat\varphi$ is a free field operator, $V(\hat\varphi) = iS_\mathrm{int}(\hat\varphi)$ is an interaction quantum operator.) The statement is proved for any $V$ and for an arbitrary operation $L$. The composite operators and the Wilson expansion are also considered.