So if $T_x$ is the random variable for future lifetime of age $x$ how can I show that "The distribution of the future lifetime, of a life aged $x$, less $n$ years given the future life time is greater then $n$ year is the same as the distribution of a future life time aged $x+n$"
I use this "fact" a lot in an actuarial studies course im studying at the moment but I can't seem to prove it.
The problem here is to parse your claim, stated in english. When you do that, and state your claim with algebra, you will see there is nothing to prove!
Let $T$ denota a lifetime, so $T$ is a positive random variable, that is, it takes real values in $(0,\infty)$. The future lifetime at age $x$, when we then know that $T>x$, $T-x$. So "the future lifetime, of a life aged x, less n years given the future life time is greater then n year" means the distribution of $T-x-n$ given that $T-x>n$.
Then the next part is "he distribution of a future life time aged $x+n$" is the distribution of $T-(x+n)$ given that $T>x+n$. So, in both cases you ask for $$ P(T-x-n \le t \mid T>x+n) $$ so you simply ask for the same probability stated in two different ways! There is nothing to prove.