The stochastic discount factor $M_t$ in a simple RBC model In these class notes, page 3, the author defines the stochastic discount factor as $M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$. 

I'm trying to find the rationale behind it.

A cash flow at period t, is usually discounted by $\prod^t_{i=0} \frac{1}{1+r_i}$ (if I'm not mistaken). I was thinking of using equation (5) - in the notes - where we have that $\frac{1}{1+r_t}=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$. 

However, since $\frac{E_{t-1}(u'(C_{t}))}{u'(C_{t})}\neq 1$, I'm not exactly sure how to proceed.  

Also, just below equation (10), the author states that $E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$. How is this possible? Isn't the tower property exactly the inverse: $I_0\subseteq I_t\implies E(E(M_{t+1}|I_t)|I_0)=E(M_{t+1}|I_0)$ ?

Any help would be appreciated.
Somewhat inconsistent/careless use of indices. In 

$$M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$$

it is assumed that we stand at period $0$ for any $t$ ahead we want to look at.

In 

$$E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$$

it is assumed that we stand at any $t$ and we look one period ahead.

The two expressions don't give the same formulae for any given $t$, and there lies the problem/confusion.