Doubt about a state transition matrix property

Question

I was going through this paper [DOI] and the step that yields equation (21) it is not very clear to me. More specifically, the left hand term of (21). Shouldn't be

$${\bf \Phi} (t_s,t_f-t_s) \, {\bf \Phi} (t_f-t_s,t) = {\bf \Phi} (t_s,t)$$

instead of ${\bf \Phi} (0,t)$ since the state transition matrix property says the following?

$${\bf \Phi} (t_2,t_1) \, {\bf \Phi} (t_1,t_0) = {\bf \Phi} (t_2,t_0)$$

Where am I wrong?

Answer 1

It looks like this is a typo. They meant to multiply by $\Phi(0,t_f-t_s)$ from the left. After that they substituted (19) for $\hat{X}(t_s)$.