reserve p,p1,p2,p3,q for Point of TOP-REAL 2;
reserve n for Nat;

theorem Th19:
  for P being non empty Subset of TOP-REAL 2, p1, p2, q1, q2 being
  Point of TOP-REAL 2, g being Function of I[01], TOP-REAL 2, s1, s2 being Real
st g is continuous one-to-one & rng g = P & g.0 = p1 & g.1 = p2 & g.s1 = q1 & 0
<= s1 & s1 <= 1 & g.s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds s1 <=
  s2
proof
  let P be non empty Subset of TOP-REAL 2, p1, p2, q1, q2 be Point of TOP-REAL
  2, g be Function of I[01], TOP-REAL 2, s1, s2 be Real;
  assume g is continuous one-to-one & rng g = P;
  then ex f being Function of I[01],(TOP-REAL 2)|P st f=g & f is
  being_homeomorphism by Th16;
  hence thesis by JORDAN5C:def 3;
end;
