composition $$\begin{aligned}
\label{decom1}
{\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}\left({\bf K}\right)
 \,=\,
 \sum_{I\in {\frak I}}{\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \,+\,\sum_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset \left\{I,I-1,\dots\right\}}}
  {\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,i-1,\dots\right\}}
    \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
     \left({\bf K}\right)\end{aligned}$$ which is exact and as always ${\left\{i,i-1,\dots\right\}}$ denotes the subset of all needed product subspaces for constructing the correlation blocks. We like to stress that the decomposition (\[decom1\]) in sums of system blocks and correlation blocks is not unique. Later we will examine another decomposition to which in contrast we will refer to as the product decomposition.\
Let us apply the RG transformation (\[transf4\]) on the sum decomposition (\[decom1\]) $$\begin{aligned}
\label{decom2}
{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)
 \,&=\,
 G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}\,\circ\,\left[
 \sum_{I\in {\frak I}}{\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)\right]
  \,\circ\, G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\nonumber\\[0.3cm]
  &\;+\,
 G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}\,\circ\,\left[
   \sum_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset
       \left\{I,I-1,\dots\right\}}}
  \!{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,i-1,\dots\right\}}
   \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
    \left({\bf K}\right)
  \right]\,\circ\, G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}
  \nonumber\\[0.3cm]\end{aligned}$$ and all quantities are used in the context of additional auxiliary spaces. Let us first consider the system block summand in (\[decom2\]), which can be rewritten as\
$$\begin{aligned}
\label{decom3}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
 \sum_{I\in {\frak I}}&\left[ G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}
  \,\circ\, {\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\!\left({\bf K}\right)
  \,\circ\, G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\right] =\nonumber\\[0.2cm]
 &\qquad\qquad\sum_{I\in {\frak I}}
  \left[G^+_{{\cal H}_{I}\,\otimes\,
   {\left({\cal H}_{\text{aux}}\right)}_{I}}\,\circ\,
     {{\cal O}^{\text{system}}_{{\cal H}_{I}\otimes
         {\left({\cal H}_{\text{aux}}\right)}_{I}}
      \!\left({\bf K}\right)}\,\circ\, 