I am writing a paper on K-analytic spaces and need the following known characterization.

Theorem.For a regular topological space $X$ the following conditions are equivalent:(1) $X$ is a continuous image of a Lindelof Cech-complete space;

(2) $X$ is the image of a Polish space under an upper semicontinuous compact-valued map.

The condition (2) is usually taken as the definition of K-analytic spaces.

On the other hand, the equivalent condition (1) is shorter and intuitively is a better generalization of the analytic spaces (which are defiend as continuous images of Polish spaces).

The only book in which (1) is taken as the definition of K-analycity is the book "General Topology, III" of Arhangelski (see page 37). But this book does not prove that these two conditions are indeed equivalent.

So I asking the MO-community for help with finding a proper reference to the above theorem (in order to avoid writing a proof which is rather standard).