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

theorem
  p is simply-connected & dim(p) = 3 implies
  num-vertices(p) - num-edges(p) + num-faces(p) = 2
proof
  set s = num-polytopes(p,0) - num-polytopes(p,1) + num-polytopes(p,2);
  set c = alternating-f-vector(p);
  assume p is simply-connected;
  then
A1: p is eulerian by Th86;
  assume
A2: dim(p) = 3;
  then
A3: s = Sum(alternating-proper-f-vector(p)) by Th84;
  0 = Sum c by A1
    .= s - 2 by A2,A3,Th5,Th80;
  hence thesis;
end;
