Understanding Time Ordering Operator in Imaginary Time Formalism

[aaaa202]

I am studying how to express Greens functions in imaginary time formalism. I have however big problems understanding the attached derivation. In equation 10.15 and 10.16 in http://www.phys.lsu.edu/~jarrell/CO...hysics Henrik Bruus and Karsten Flensberg.pdf
how do they go from one equation to the other?

 

[qbert]

short answer: under the time ordering you can collect all of the U(τ) terms together and they cancel.

 

[aaaa202]

How does that happen? The time ordering operator for τ>τ' has no effect on A(τ)B(τ') and reverses them in the case τ'>τ