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

theorem
  p is simply-connected & dim(p) = 1 implies num-vertices(p) = 2
proof
  set acs = alternating-f-vector(p);
  set apcs = alternating-proper-f-vector(p);
  assume p is simply-connected;
  then
A1: p is eulerian by Th86;
  assume
A2: dim(p) = 1;
  0 = Sum acs by A1
    .= Sum apcs - 2 by A2,Th3,Th80
    .= num-polytopes(p,0) - 2 by A2,Th82;
  hence thesis;
end;
