I have developed the way to count the faces of a $d-$dimensional polytope $P$ and show that the face numbers satisfy the Euler-Poincare relation, and we'll say $C(k,P)$ is the number of $k-$dimensional faces in $P.$ I'm also tasked with doing the same thing for its dual $P^\bigtriangleup.$
If $L(P)$ is the face lattice for $P$, then $L(P)^\text{op}$ is the face lattice for $P^\bigtriangleup.$ Does it suffice to make the argument that because of this relation, we know that $C(k,P) = C(d-k,P^\bigtriangleup),$ and therefore the sum of the face numbers of $P^\bigtriangleup$ is the same as the sum of the face numbers of $P,$ just reordered?