reserve j for Nat;

theorem Th21:
  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 & p2<>q1 holds Segment(P,p1,p2,
  q1,p2) is_an_arc_of q1,p2
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: p2<>q1;
  LE q1,p2,P,p1,p2 by A1,A2,JORDAN5C:10;
  hence thesis by A1,A3,JORDAN16:21;
end;
