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

theorem Th81:
  dim(p) is even implies
  Sum (alternating-f-vector(p)) = Sum (alternating-proper-f-vector(p))
proof
  reconsider minusone = -1 as Integer;
  set acs = alternating-f-vector(p);
  set apcs = alternating-proper-f-vector(p);
  reconsider lastterm = (-1)|^(dim(p)) as Integer;
  assume dim(p) is even;
  then
A1: (-1)|^(dim(p)) = 1 by Th7;
  acs = <*-1*> ^ apcs ^ <*(-1)|^(dim(p))*> by Th79;
  then Sum acs = (Sum <*minusone*>) + (Sum apcs) + (Sum <*lastterm*>) by Th19
    .= (Sum <*minusone*>) + (Sum apcs) + 1 by A1,RVSUM_1:73
    .= (-1) + (Sum apcs) + 1 by RVSUM_1:73
    .= Sum apcs;
  hence thesis;
end;
