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 Th12:
  for A1,A2 being Subset of TOP-REAL 2, p1,p2,q1,q2 being Point of
  TOP-REAL 2 st A1 is_an_arc_of p1,p2 & A1 /\ A2 = {q1,q2} holds A1 <> A2
proof
  let A1,A2 be Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2 such
  that
A1: A1 is_an_arc_of p1,p2 and
A2: A1 /\ A2 = {q1,q2} & A1 = A2;
  p1 in A1 by A1,TOPREAL1:1;
  then
A3: p1= q1 or p1 = q2 by A2,TARSKI:def 2;
  p2 in A1 by A1,TOPREAL1:1;
  then
A4: p2= q1 or p2 = q2 by A2,TARSKI:def 2;
  ex p3 being Point of TOP-REAL 2 st p3 in A1 & p3<>p1 & p3<>p2 by A1,
JORDAN6:42;
  hence contradiction by A1,A2,A3,A4,JORDAN6:37,TARSKI:def 2;
end;
