previous set has the following definition: $$I_1(k_r)=\left\{ T_{E}=(p_1,\ldots,p_{r-1}):E=\left\{M_1,\ldots,M_r \right\}; T_{E_{r-1}}=(p_1,\ldots,p_{r-2}) \in \textrm{Cond}(\textbf{c}_{r-1})\right.$$ $$\left. M_r \in \textrm{Cond}^*(k_r) ;p_{r-1}=l(M_r)+c[r]-k_r\leq \min(^*p_{r-1},\textrm{Comp}(M_{r-1},M_r)) \right\}.$$\
$\bullet\ \textrm{Case}\  T_1=T_{E^{t-1}}=(p_t,\ldots,p_{r-1})\in \textrm{Cond}(\textbf{c}^{t-1}).$

We only have to adapt the considerations made in the previous case to this case. Thus we directly give the sets which arise without further justifications.

-   If $t\neq 2$, $$S_2^1(\textbf{c})=\left\{ T_{E}=(p_1,\ldots,p_{r-1}): E=\left\{M_1,\ldots,M_r \right\}; T_{E^{t-1}}=(p_t,\ldots,p_{r-1})\in \textrm{Cond}(\textbf{c}^{t-1});\right.$$ $$\left. T_{E_{t-1}}=(p_1,\ldots,p_{t-2}) \in \textrm{Cond}(\textbf{c}_{t-1}) \textrm{ con }1\leq p_{t-1} \leq \min(^*p_{t-1},p^*_{t-1},\textrm{Comp}(M_t,M_{t-1})) \right\}.$$

-   If $t=2$, $$S_2^1(\textbf{c})=\left\{ T_{E}=(p_1,\ldots,p_{r-1}): E=\left\{M_1,\ldots,M_r \right\}; \right.  T_{E^1}=(p_2,\ldots,p_{r-1}) \in \textrm{Cond}(\textbf{c}^{1});$$ $$\left.  M_1 \in \textrm{Cond}^*(c[1]) ;1\leq p_{1} \leq \min(l(M_1),p^*_{1},\textrm{Comp}(M_{1},M_2)) \right\}.$$

We have $S_2^1(\textbf{c}) \subseteq  \textrm{Cond}(\textbf{c}).$

For all $k_{t-1} \in \mathbb{N}$ such that $1\leq k_{t-1} < c[t-1]$ we consider:

-   If $t \neq 2,$ $$I_2(k_{t-1})=\left\{ T_{E}=(p_1,\ldots,p_{r-1}):E=\left\{M_1,\ldots,M_r \right\}; T_{E^{t-1}}=(p_t,\ldots,p_{r-1}) \in\textrm{Cond}(\textbf{c}^{t-1}); \right.$$ $$\left.  T_{E_{t-1}}=(p_{1},\ldots,p_{t-2}) \in \textrm{Cond}((c[1],\ldots,c[t-2],k_{t-1}));\right.$$ $$\left. p_{t-1}={}^*p_{t-1}+c[t-1]-k_{t-1}\leq  \min(p^*_{t-1},\textrm{Comp}(M_{t-1},M_{t}))\right\}.$$

-   If $t=2$, $$I_2(k_1)=\left\{ T_{E}=(p_1,\ldots,p_{r-1}):E=\left\{M_1,\ldots,M_r \right\}; T_{E^{1}}=(p_2,\ldots,p_{r-1}) \in \textrm{Cond}(\textbf{c}^{1})\right.$$ $$\left. M_1 \in \textrm{Cond}^*(k_1) ;p_{1}=l(M_1)+c[1]-k_1\leq \min(p^*_{1},\textrm{Comp}(M_{1},M_2)) \right\}.$$

We have that: $$S_2^2(\textbf{c})= \bigcup_{1\leq k_{t-1} < c[t-1]}{I_2(k_{t-1})} \subseteq   \textrm{Cond}(\textbf{c}).$$ The previous lemma ensures that we have considered all the possibilities. So we showed that $$\textrm{Cond}(\textbf{c}) \subseteq  S_1^1(\textbf{c}) \cup S_1^2(\textbf{c}) \cup S_2^1(\textbf{c}) \cup S_2^2(\textbf{c}),$$ hence $$S_1^1(\textbf{c}) \cup S_1^2(\textbf{c}) \cup S_2^1(\textbf{c}) \cup S_2^2(\textbf{c})=\textrm{Cond}(\textbf{c}).$$ Due to our induction hypothesis all the previous sets can be computed so we developed an 