Conjugacy Class Equation/Proof 2

Proof
Let the distinct orbits of $G$ under the conjugacy action be:
 * $\Orb {x_1}, \Orb {x_2}, \ldots, \Orb {x_s}$

Then from the Partition Equation:
 * $\order G = \order {\Orb {x_1} } + \order {\Orb {x_2} } + \cdots + \order {\Orb {x_s} }$

From the Orbit-Stabilizer Theorem:
 * $\order {\Orb {x_i} } \divides \order G, i = 1, \ldots, s$

The result follows from the definition of the conjugacy action.