How does C(n) = P(down)[C( n - 1 ) + C(n) + k (n)] + P(same)[(C(n) + k(n)] + P(break)[C(n) + R + k(n)] + P(up)k(n)
become
C(n)[1-P(down)-P(same)-P(break) =
P(down)C(n-1) + P(break)R + k(n) ?

Please explain to me the factorisation process of the C(n) terms as thoroughly as possible and the summation of the k(n) terms as thoroughly as possible.

Either I have forgotten how factorisation works or I dont understand the maths of Markov chain & recurrence relations, or the poster skipped a lot of steps in his mathematical presentation.

Please help.