  This work develops a rigorous setting allowing one to prove several features
related to the behaviour of the Heisenberg-Ising (or XXZ) spin-$1/2$ chain at
finite temperature $T$. Within the quantum inverse scattering method the
physically pertinent observables at finite $T$, such as the \textit{per}-site
free energy or the correlation length, have been argued to admit integral
representations whose integrands are expressed in terms of solutions to
auxiliary non-linear integral equations. The derivation of such representations
was based on numerous conjectures: the possibility to exchange the infinite
volume and the infinite Trotter number limits, the existence of a real,
non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix,
the existence and uniqueness of solutions to the auxiliary non-linear integral
equations, as well as the possibility to take the infinite Trotter number limit
on their level. We rigorously prove all these conjectures for temperatures
large enough. As a by product of our analysis, we obtain the large-$T$
asymptotic expansion for a subset of sub-dominant Eigenvalues of the quantum
transfer matrix and thus of the associated correlation lengths. This result was
never obtained previously, not even on heuristic grounds.
