Let $m,n,o \in \mathbb{N} $, when $o$ is $m+n$?!

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let $ \mathbb{N}:=\{ \emptyset, (\emptyset)^+, ((\emptyset)^+)^+,...,(...((\emptyset)^+)^+...)^+,...\} $, $n \in \mathbb{N} $ with

$ I_n:=\{t \in \mathbb{N}|t \leq n\} $

$ I_n^*:=I_n\backslash \{0\} $

in this case $ 0:=\emptyset $, $ t \leq n $ if $ t \subseteq n $, and $ (A)^+ := A \cup \{A\} $

let $ m,p,o \in \mathbb{N} $, $ o $ is sum of $ m $ and $ p $ if $o \sim (I_m^*\oplus I_p^*) $

in this case $\oplus$ is http://en.wikipedia.org/wiki/Disjoint_union

Is correct? Thanks in advance!