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

theorem Th83:
  dim(p) = 2 implies Sum alternating-proper-f-vector(p) =
  num-polytopes(p,0) - num-polytopes(p,1)
proof
  reconsider t = 2 as Nat;
  reconsider o = 1 as Nat;
  set apcs = alternating-proper-f-vector(p);
  reconsider apcso = apcs.o as Integer;
  reconsider apcst = apcs.t as Integer;
  assume
A1: dim(p) = 2;
  then
A2: apcs.o = ((-1)|^(o+1))*num-polytopes(p,o-1) & apcs.t = ((-1)|^(t+1))*
  num-polytopes(p,t-1) by Def27;
A3: (-1)|^(o+1) = 1 & (-1)|^(t+1) = -1 by Th4,Th7,Th8;
  len apcs = 2 by A1,Def27;
  then apcs = <*apcso,apcst*> by FINSEQ_1:44;
  then Sum apcs = apcso + apcst by RVSUM_1:77
    .= num-polytopes(p,0) - num-polytopes(p,1) by A2,A3;
  hence thesis;
end;
