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 P being non empty Subset of TOP-REAL 2, p1,p2,q1 being Point of
TOP-REAL 2 st P is_an_arc_of p1,p2 & q1 in P & p1<>q1 holds Segment(P,p1,p2,p1,
  q1) is_an_arc_of p1,q1
proof
  let P be non empty Subset of TOP-REAL 2, p1,p2,q1 be Point of TOP-REAL 2;
  assume that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P and
A3: p1<>q1;
  LE p1,q1,P,p1,p2 by A1,A2,JORDAN5C:10;
  hence thesis by A1,A3,Th21;
end;
