# Is it generally true that $T_x - n \mid T_x \gt n$ has the same distribution of $T_{(x+n)}$

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.

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.