reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;

theorem Th11:
  for P being Subset of the carrier of TOP-REAL 2, p1,p2,q1,q2
  being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & p1 <> q1 & p2 <> q2 holds
  not p1 in Segment(P,p1,p2,q1,q2) & not p2 in Segment(P,p1,p2,q1,q2)
proof
  let P be Subset of the carrier of TOP-REAL 2, p1,p2,q1,q2 be Point of
  TOP-REAL 2;
  assume P is_an_arc_of p1,p2 & p1 <> q1 & p2 <> q2;
  then
A1: ( not p2 in L_Segment(P,p1,p2,q2))& not p1 in R_Segment(P,p1,p2,q1) by
JORDAN6:59,60;
  Segment(P,p1,p2,q1,q2) = R_Segment(P,p1,p2,q1) /\ L_Segment(P,p1,p2,q2)
  by JORDAN6:def 5;
  hence thesis by A1,XBOOLE_0:def 4;
end;
