reserve n for Element of NAT,
  V for Subset of TOP-REAL n,
  s,s1,s2,t,t1,t2 for Point of TOP-REAL n,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  a,p ,p1,p2,q,q1,q2 for Point of TOP-REAL 2;

theorem Th15:
  for C being Simple_closed_curve, P being Subset of TOP-REAL 2,
p1,p2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & P c= C ex R being non
empty Subset of TOP-REAL 2 st R is_an_arc_of p1,p2 & P \/ R = C & P /\ R = {p1,
  p2}
proof
  let C be Simple_closed_curve, P be Subset of TOP-REAL 2, p1,p2 be Point of
  TOP-REAL 2 such that
A1: P is_an_arc_of p1,p2 and
A2: P c= C;
A3: p1 <> p2 by A1,JORDAN6:37;
  p1 in P & p2 in P by A1,TOPREAL1:1;
  then consider P1,P2 being non empty Subset of TOP-REAL 2 such that
A4: P1 is_an_arc_of p1,p2 and
A5: P2 is_an_arc_of p1,p2 and
A6: C = P1 \/ P2 and
A7: P1 /\ P2 = {p1,p2} by A2,A3,TOPREAL2:5;
  reconsider P1,P2 as non empty Subset of TOP-REAL 2;
A8: P1 c= C & P2 c= C by A6,XBOOLE_1:7;
A9: now
    assume
A10: P1 = P2;
    ex p3 being Point of TOP-REAL 2 st p3 in P1 & p3 <> p1 & p3 <> p2 by A4,
JORDAN6:42;
    hence contradiction by A7,A10,TARSKI:def 2;
  end;
  per cases by A1,A2,A4,A5,A8,A9,JORDAN16:14;
  suppose
A11: P = P1;
    take P2;
    thus thesis by A5,A6,A7,A11;
  end;
  suppose
A12: P = P2;
    take P1;
    thus thesis by A4,A6,A7,A12;
  end;
end;
