Let $f : X \to Z$ be a continuous mapping

from a compact $C-$space $X$ onto a compact

space $Z$. Then there are a compact  $C-$space $Y$ and

continuous mappings $g : X \to Y$ and $h : Y \to Z$

such that

$ {\dimC}Y \leq {\dimC}X, wY \leq wZ$ and $f = hg$.



Here ${\dimC}$ is Borst's transfinite extension of the covering

dimension $\dim$ in the class of compact $C$-spaces.