Conjugacy Class Equation/Proof 2

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

Then from the Partition Equation:
 * $\left|{G}\right| = \left|{\operatorname{Orb} \left({x_1}\right)}\right| + \left|{\operatorname{Orb} \left({x_2}\right)}\right| + \cdots + \left|{\operatorname{Orb} \left({x_s}\right)}\right|$

From the Orbit-Stabilizer Theorem:
 * $\left|{\operatorname{Orb} \left({x_i}\right)}\right| \backslash \left|{G}\right|, i = 1, \ldots, s$

The result follows from the definition of the Conjugacy Action.