Let {X-i(t), t >= 0}, 1 <= i <= n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of
P (there exists(t epsilon[0,T])for all(i=1),..,X-n(i)(t) > u) as u -> infinity
for both locally stationary X-i 's and X-i 's with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.