The proof

We want to show that Mn = Mn-2 + Mn-3 + Mn-4
We know that Mn = An + Bn + Cn, therefore we have

Mn-2 + Mn-3 + Mn-4

1. Substitute Mn = An + Bn + Cn for all Mn
= [An-2 + Bn-2 + Cn-2] + [An-3 + Bn-3 + Cn-3] + [An-4 + Bn-4 + Cn-4]

2. Substitute An = Bn+1 for all An
=Bn-1 +Bn-2 + Cn-2 + Bn-2 + Bn-3 + Cn-3 + Bn-3 + Bn-4 + Cn-4

3. Substitute Cn = Bn-1 + Cn-1 for all Bn + Cn
=Bn-1 + Cn-1 + Bn-2 + Cn-2 + Bn-3 + Cn-3

4. Substitute Cn = Bn-1 + Cn-1 for all Bn + Cn
= Cn + Cn-1 + Cn-2

5. Substitute An = Cn-1 for Cn-1 and Cn-2
=Cn + An + An-1

6. Substitute Bn = An-1
=Cn + An + Bn

7. Substitute Mn = An + Bn + Cn
=Mn

Therefore, Mn = Mn-2 + Mn-3 + Mn-4