Partition Equation

Theorem
Let group $$G$$ act on a finite set $$X$$.

Let the distinct orbits of $$X$$ under the action of $$G$$ be $$\operatorname{Orb} \left({x_1}\right), \operatorname{Orb} \left({x_2}\right), \ldots, \operatorname{Orb} \left({x_s}\right)$$.

Then $$\left|{X}\right| = \left|{\operatorname{Orb} \left({x_1}\right)}\right| + \left|{\operatorname{Orb} \left({x_2}\right)}\right| + \cdots + \left|{\operatorname{Orb} \left({x_s}\right)}\right|$$.

Proof
Follows trivially from the fact that the orbit of an element is an equivalence class.