reserve n for Nat,
  k for Integer;
reserve p for polyhedron,
  k for Integer,
  n for Nat;

theorem Th49:
  1 <= n & n <= len (alternating-proper-f-vector(p)) implies
  (alternating-proper-f-vector(p)).n = ((-1)|^(n+1))*(dim ((n-2)
  -bounding-chain-space(p))) + ((-1)|^(n+1))*(dim ((n-1)-circuit-space(p)))
proof
  set apcs = alternating-proper-f-vector(p);
  assume
A1: 1 <= n;
  set a = (-1)|^(n+1);
  assume n <= len apcs;
  then n <= dim(p) by Def27;
  then apcs.n = a*num-polytopes(p,n-1) by A1,Def27
    .= a*(dim ((n-1)-chain-space(p))) by Th34
    .= a*(rank ((n-1)-boundary p) + nullity ((n-1)-boundary p)) by RANKNULL:44
    .= (a*dim ((n-2)-bounding-chain-space(p))) + (a*dim ((n-1)-circuit-space
  (p)));
 hence thesis;
end;
