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 Th11:
  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 holds A1 \/ A2 = C & A1 /\ A2 = {p1,p2}
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 and
A4: A2 c= C & A1 <> A2;
A5: p2 in A1 by A1,TOPREAL1:1;
  p1 <> p2 & p1 in A1 by A1,JORDAN6:37,TOPREAL1:1;
  then consider P1,P2 being non empty Subset of TOP-REAL 2 such that
A6: P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & C = P1 \/ P2 & P1 /\
  P2 = {p1,p2} by A3,A5,TOPREAL2:5;
  reconsider P1,P2 as non empty Subset of TOP-REAL 2;
  A1=P1 or A1=P2 by A1,A3,A6,Th10;
  hence thesis by A2,A4,A6,Th10;
end;
