reserve j for Nat;

theorem Th19:
  for P being non empty Subset of TOP-REAL 2, p1,p2,q1,q2 being
Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & q1 in P & q2 in P holds LE q1,q2,
  P,p1,p2 or LE q2,q1,P,p1,p2
proof
  let P be non empty Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2;
  assume that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P and
A3: q2 in P;
  per cases;
  suppose
    q1<>q2;
    hence thesis by A1,A2,A3,JORDAN5C:14;
  end;
  suppose
    q1=q2;
    hence thesis by A2,JORDAN5C:9;
  end;
end;
