  A lot of work has been devoted in the past to understand the role of vertical
branch of the time path in thermal field theories, and in particular to see how
to deal with it in the real-time formalism. Unlike what is commonly believed, I
emphasize on the fact that the vertical part of the path contributes to
real-time Green's functions, and I prove that this contribution is taken into
account simply by the substitution $n(\omega_{\boldsymbol k})\to n(|k_o|)$ in
the real time Feynman rules. This new proof is based on very simple algebraic
properties of the contour integration.
