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Using this notation, we see that
if there is a decomposition $J_r\rightarrow m_1,m_2,\ldots,m_t,k^*$ and a
decomposition $J^+_s\rightarrow m'_1,m'_2,\ldots,m'_{t'},+l$, then there is a
decomposition 
$J_{r+s}\rightarrow m_1,m_2,\ldots,m_t,m'_1,m'_2,\ldots,m'_{t'},k+l$.
Moreover, if $l=s$ and the edge $\{n+3,n+6\}$ is in the path of the 
decomposition $J^+_s\rightarrow m'_1,m'_2,\ldots,m'_{t'},+l$, then the 
resulting $(r+s)$-cycle has property $(*)$, and the resulting decomposition is 
$J_{r+s}\rightarrow m_1,m_2,\ldots,m_t,m'_1,m'_2,\ldots,m'_{t'},(k+l)^*$.
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