Here's an excerpt from M. Ziegler's "Lectures on Polytopes":
Remarks:
- polytope here means what in other sources may be referred to as a convex polytope - a convex hull of a finite set of points (vertices);
- $F_1,\dots,F_s$ being a shelling order seems to be irrelevant to the claim (but I'm not sure) , one can probably treat them just as all the facets of $P$.
The highlited claim is given without a proof (as far as I was able to find) also in some other sources on the topic. I see that this is pretty obvious for three or less dimensions. Maybe someone can formalize this claim?