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

theorem Th34:
  num-polytopes(p,k) = dim (k-chain-space(p))
proof
  set n = dim (k-chain-space(p));
  singletons(k-polytopes(p)) is Basis of k-chain-space(p) by BSPACE:40;
  then n = card (singletons(k-polytopes(p))) by VECTSP_9:def 1;
  hence thesis by BSPACE:41;
end;
