reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem
  A1 c= C & A2 c= C & A1 <> A2 & A1 is_an_arc_of p1,p2 & A2 is_an_arc_of
  p1,p2 implies for A st A is_an_arc_of p1,p2 & A c= C holds A = A1 or A = A2
proof
  assume that
A1: A1 c= C & A2 c= C & A1 <> A2 and
A2: A1 is_an_arc_of p1,p2 & A2 is_an_arc_of p1,p2;
A3: A1 \/ A2 = C & A1 /\ A2 = {p1,p2} by A1,A2,Th11;
  let A;
  assume A is_an_arc_of p1,p2 & A c= C;
  hence thesis by A2,A3,Th10;
end;
