Hurewicz theorem for extension dimension

We prove a new selection theorem for multivalued mappings of C-space. Using this theorem we prove extension dimensional version of Hurewicz theorem for a closed mapping f :X→Y of k-space X onto paracompact C-space Y: if for finite CW-complex M we have e-dimY⩽[M] and for every point y∈Y and every compactum Z with e-dimZ⩽[M] we have e-dim(f−1(y)×Z)⩽[L] for some CW-complex L, then e-dimX⩽[L].