How does the Heaviside step function rewrite time-ordered integrals?

[aaaa202]

I really asked this question in another thread but it seems the original respondent gave up explaining me. My question is about the rewriting of the integrals from first to second line on the attached picture. The θ denotes the heaviside step function such that:
θ(t1-t2) = {1 t1>t2 , 0 t1<t2}
I think the idea is to extend the integration domain from respectively t1 or t2 to t but doing so we have to multiply by the step function in such a way that the integration over [t1,t] or [t2,t] yields zero. But how is this exactly related to when t1>t2 or t2>t1?

 

[vanhees71]

Have a look at p. 16 of my QFT manuscript:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

I hope that helps to explain the appearance of the time-ordering operator in the solution of the time-evolution operator with a time-dependent Hamiltonian (or interaction part in the interaction picture).