$ \blacktriangleright \textrm{Subcase}\ d_2(t+1)=d(t+1).$

In this case we have $T_2 \in \textrm{Cond}(\textbf{c}^t)$ and $p_t \leq p^*_t$ (if $t=r-1$ we have $T_2=M_r \in \textrm{Cond}^*(c[r])$). We also recall that we must have the compatibility condition $p_t \leq \textrm{Comp}(M_t,M_{t+1})$.

Thus we have discovered that $T$ belongs to the following set: $$S_1^1(\textbf{c})=\left\{ T_{E}=(p_1,\ldots,p_{r-1}): E=\left\{M_1,\ldots,M_r \right\}; T_{E_t}=(p_1,\ldots,p_{t-1})\in \textrm{Cond}(\textbf{c}_t);\right.$$ $$\left. T_{E^t}=(p_{t+1},\ldots,p_{r-1}) \in \textrm{Cond}(\textbf{c}^t) \textrm{ with }1\leq p_t \leq \min({}^*p_{t},p^*_{t},\textrm{Comp}(M_t,M_{t+1})) \right\}.$$ It is very easy to check that we also have $S_1^1(\textbf{c}) \subseteq \textrm{Cond}(\textbf{c}).$ If $t=r-1$ the previous set has the following definition: $$S_1^1(\textbf{c})=\left\{ T_{E}=(p_1,\ldots,p_{r-1}): E=\left\{M_1,\ldots,M_r \right\}; \right.  T_{E_{r-1}}=(p_1,\ldots,p_{r-2}) \in \textrm{Cond}(\textbf{c}_{r-1});$$ $$\left.  M_r \in \textrm{Cond}^*(c[r]) ;1\leq p_{r-1} \leq \min(l(M_r),^*p_{r-1},\textrm{Comp}(M_{r-1},M_r)) \right\}.$$\
$ \blacktriangleright \textrm{Subcase}\ d_2(t+1) \neq d(t+1).$

In this case we have $$p_t=d(t+1)-1 >d_2(t+1)-1=\max(l(M_{t+1}),p_{t+1})=p^*_t.$$ Hence $$c[t+1]=\sum_{k=1}^{d(t+1)-1} M_{t+1}[k]=\sum_{k=1}^{p_t} M_{t+1}[k] =\sum_{k=1}^{p^*_t} M_{t+1}[k]+p_t-p^*_t$$ and from this it follows that $T_2 \in \textrm{Cond}((k[t+1],c[t+2],\ldots,c[r]))$ where $$\sum_{k=1}^{p^*_t} M_{t+1}[k]=\sum_{k=1}^{d_2(t+1)-1} M_{t+1}[k]=k[t+1]<c[t+1],$$ and we have $T_2 \in \textrm{Cond}^*(k[t+1])$ in the case $t=r-1$. Thus we have $k[t+1]=c[t+1]-(p_t-p^*_t)$. For all the $k_{t+1} \in \mathbb{N}$ such that $k_{t+1}<c[t+1]$ we define the set: $$I_1(k_{t+1})=\left\{ T_{E}=(p_1,\ldots,p_{r-1}):E=\left\{M_1,\ldots,M_r \right\}; T_{E_t}=(p_1,\ldots,p_{t-1}) \in\textrm{Cond}(\textbf{c}_t); \right.$$ $$\left.  T_{E^t}=(p_{t+1},\ldots,p_{r-1}) \in \textrm{Cond}((k_{t+1},c[t+2],\ldots,c[r])); \right.$$ $$\left.  p_t=p_t^* +c[t+1]-k_{t+1}\leq\min(^*p_{t},\textrm{Comp}(M_t,M_{t+1}))\right\}.$$ Hence $T$ belongs to the following set: $$S_1^2(\textbf{c})= \bigcup_{1\leq k_{t+1} < c[t+1]}{I_1(k_{t+1})},$$ and it is clear that $ S_1^2(\textbf{c}) \subseteq   \textrm{Cond}(\textbf{c})$.

If $t=r-1$ the 