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

theorem Th82:
  dim(p) = 1 implies Sum alternating-proper-f-vector(p) = num-polytopes(p,0)
proof
  reconsider egy = 1 as Nat;
  set apcs = alternating-proper-f-vector(p);
  assume dim(p) = 1;
  then
A1: len apcs = 1 & apcs.egy = (-1)|^(egy+1)*num-polytopes(p,egy-1) by Def27;
  (-1)|^(egy+1) = 1 by Th4,Th7;
  then apcs = <*num-polytopes(p,0)*> by A1,FINSEQ_1:40;
  hence thesis by RVSUM_1:73;
end;
