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

theorem Th18:
  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 P is_an_arc_of p1,p2 & 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 & s1 <=
  s2 holds LE q1,q2,P,p1,p2
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 such that
A1: P is_an_arc_of p1,p2 and
A2: g is continuous one-to-one & rng g = P;
  ex f being Function of I[01],(TOP-REAL 2)|P st f=g & f is
  being_homeomorphism by A2,Th16;
  hence thesis by A1,JORDAN5C:8;
end;
