This is a basic question,  I am however not able to reduce my solution to the simple form proposed in a book and thus seeking help.

A patient can go via two stages in a disease. first stage is followed up by second stage and that is followed up by death. 

Notation: 

$p({t^{a}_{1})}$ be the pdf of time spent in stage A/stage 1 

$p({t^{b}_{2})}$ be the pdf of time spent in stage B / stage 2 before death and this is independent of time spent in stage A


let $p(t^{d})$ be the *convolution* of  $p(t^{a})$ and $p(t^{b})$ that gives probability distribution of months from start of disease till death i.e $p(t^{a}_{1}+t^{b}_{2})$. 

From here on notation *P* refers to probability (& not pdf):

I further calculated probability of atleast total 'T' months (a constant) till death given stage 1 already happened should be $P(\frac{t>T}{t_{1}<T})= \frac{P(t^{d}>T) {\cap}P(t^{a}_{1} <= T)}{P(t^{a}_{1} <= T)}$. 

I am however not able to tie this out to solution that's given out for 

$P(\frac{t>T}{t_{1}<T}) = P(t^{a}_{1}<=T)-P(t^{d}<=T)$

Please can you suggest what I am doing incorrect here and help me derive?