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

theorem
  p is simply-connected & dim(p) = 2 implies num-vertices(p) = num-edges (p)
proof
  set s = num-polytopes(p,0) - num-polytopes(p,1);
  set c = alternating-f-vector(p);
  assume p is simply-connected;
  then
A1: p is eulerian by Th86;
  assume
A2: dim(p) = 2;
  then
A3: s = Sum(alternating-proper-f-vector(p)) by Th83;
  0 = Sum c by A1
    .= s by A2,A3,Th4,Th81;
  hence thesis;
end;
