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
  for C being Simple_closed_curve, A1,A2 being Subset of TOP-REAL 2, p1,
p2 being Point of TOP-REAL 2 st A1 is_an_arc_of p1,p2 & A2 is_an_arc_of p1,p2 &
  A1 c= C & A2 c= C & A1 /\ A2 = {p1,p2} holds A1 \/ A2 = C
proof
  let C be Simple_closed_curve, A1,A2 be Subset of TOP-REAL 2, p1,p2 be Point
  of TOP-REAL 2 such that
A1: A1 is_an_arc_of p1,p2 and
A2: A2 is_an_arc_of p1,p2 and
A3: A1 c= C & A2 c= C and
A4: A1 /\ A2 = {p1,p2};
  A1 <> A2 by A2,A4,Th12;
  hence thesis by A1,A2,A3,Th11;
end;
