reserve n for Nat;

theorem :: "Nie oba"
  for C be Simple_closed_curve
  for p be Point of TOP-REAL 2 st p`1 = (W-bound C + E-bound C)/2 holds
  not (p in North_Arc C & p in South_Arc C)
proof
  let C be Simple_closed_curve;
  let p be Point of TOP-REAL 2;
A1: W-bound C < E-bound C by SPRECT_1:31;
  assume
A2: p`1 = (W-bound C + E-bound C)/2;
  then
A3: W-bound C < p`1 by A1,XREAL_1:226;
  p`1 < E-bound C by A1,A2,XREAL_1:226;
  hence thesis by A3,Th26;
end;
