Prove that centre of mass is a unique point
I started with O as origin.G as CM and G' as assumed 2nd CM in the same sysyem of particles.
OG=R=(1/M) sum(i) [m_i*r_i]
OG'=R'=(1/M) sum(k) [m_k*r'_k]
The problem is theere is no unique relation between r_i and r'_k
centre of mass
r_i and r'_k locate the particle in the coordinate system. Since you're using the same coordinate system to describe supposedly 2 different centers of mass, it must be that r_i = r'_k. This is because OG and OG' are the same coordinate system (remember, you're describing two centers of mass in the same system. The result you seek follows immediately.