How could I prove the folowing using the tower property of conditional expectations?
$$E\left(E\left[\frac{D(t,T)D(T,S)H}{P(T,S)}|F_T\right]|F_t\right)=E\left(\frac{D(t,T)H}{P(T,S)}E[D(T,S)|F_T]|F_t\right)=$$
given : $$ t\le T$$ $$ r(s) \,\, is\,a\,stochastic\,process\,(interest\,rate) $$ $$D(t,T)=e^{-\int_t^Tr(s)ds}$$ $$D(t,S)=D(t,T)D(T,S)$$ $$H \,\, is \,\, F_T-measurable$$ $$ P(t,T) =E\left[e^{-\int_t^Tr(s)ds} \right] $$
To be more accurate, I don't particularly understand how one could take $$P(T,S)$$ out from the $$E[*|F_T]$$ expectation. Thank you
Note that $P(t,T)$ is constant; $D(t,T)$ is $T$ measurable cause $t<T$ and $H$ is $T$ measurable by hypotesis.