Distinct addable nodes of a partition have distinct contents

By [F1] every addable node has the form $(i,{\lambda }_i+1)$ for a unique index $i\in I:=\{1\le i\le k+1:{\lambda }_{i-1}>{\lambda }_i\}$, where we use ${\lambda }_0=+\mathrm{\infty }$ and ${\lambda }_{k+1}=0$; indeed for $i\le k$ the condition is exactly the addability criterion, and $i=k+1$ is the new-row node $(k+1,1)=(k+1,{\lambda }_{k+1}+1)$ with ${\lambda }_k>{\lambda }_{k+1}=0$.

For such an index $i$ the content of the addable node is $c(i,{\lambda }_i+1)={\lambda }_i+1-i$ by [F2], and the partition is weakly decreasing, so ${\lambda }_i\ge {\lambda }_{i+1}\ge \cdots \ge 0$.

Let $i<i^{\mathrm{\prime }}$ be two indices in $I$. By weak monotonicity ${\lambda }_i\ge {\lambda }_{i^{\mathrm{\prime }}}$, and since $i<i^{\mathrm{\prime }}$ we get ${\lambda }_i-i>{\lambda }_{i^{\mathrm{\prime }}}-i^{\mathrm{\prime }}$, that is ${\lambda }_i+1-i>{\lambda }_{i^{\mathrm{\prime }}}+1-i^{\mathrm{\prime }}$.

Combined with step 1.1, distinct addable nodes $(i,{\lambda }_i+1)$ and $(i^{\mathrm{\prime }},{\lambda }_{i^{\mathrm{\prime }}}+1)$ with $i\mathrm{\ne }{i}^{\mathrm{\prime }}$ have contents differing by the strict inequality of step 2.1; hence $c$ is injective on $\mathrm{Add}(\lambda )$.