Congruence Relation on Naturals without Zero for Addition Distinct from Equality is Restricted Dipper Relation

Theorem
Let $\RR^*$ be a congruence relation for addition on the non-zero natural numbers $\N_{>0}$.

Let $\RR^*$ be distinct from the equality relation on $\N$.

Then there exist $m, n \in \N_{>0}$ such that:
 * $\RR^* = \RR^*_{m, n}$

where $\RR^*_{m, n}$ denotes the restricted dipper relation.