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 Th17:
  A is_an_arc_of p1,p2 & LE q1, q2, A, p1, p2 & q1 <> q2 implies
  ex g being Function of I[01], (TOP-REAL 2)|A, s1, s2 being Real st g is
  being_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1 & g.s2 = q2 & 0 <= s1 &
  s1 < s2 & s2 <= 1
proof
  assume that
A1: A is_an_arc_of p1,p2 & LE q1, q2, A, p1, p2 and
A2: q1 <> q2;
  consider g being Function of I[01], (TOP-REAL 2)|A, s1, s2 being Real such
  that
A3: g is being_homeomorphism & g.0 = p1 & g.1 = p2 and
A4: g.s1 = q1 & g.s2 = q2 and
A5: 0 <= s1 and
A6: s1 <= s2 and
A7: s2 <= 1 by A1,Th16;
  take g,s1,s2;
  thus g is being_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1 & g.s2 = q2
  & 0 <= s1 by A3,A4,A5;
  thus s1 < s2 by A2,A4,A6,XXREAL_0:1;
  thus thesis by A7;
end;
