
theorem Th16:
  for f being FinSequence of TOP-REAL 2, Q being Subset of
  TOP-REAL 2, q being Point of TOP-REAL 2 st f is being_S-Seq & L~f /\ Q is
closed & q in L~f & q in Q holds LE q, Last_Point(L~f,f/.1,f/.len f,Q), L~f, f
  /.1, f/.len f
proof
  let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of
  TOP-REAL 2;
  set P = L~f;
  assume that
A1: f is being_S-Seq and
A2: L~f /\ Q is closed and
A3: q in L~f and
A4: q in Q;
  set q1 = Last_Point(P,f/.1,f/.len f,Q);
A5: L~f /\ Q c= L~f by XBOOLE_1:17;
  L~f meets Q & P is_an_arc_of f/.1,f/.len f by A1,A3,A4,TOPREAL1:25,XBOOLE_0:3
;
  then
  q1 in L~f /\ Q & for g being Function of I[01], (TOP-REAL 2)| P, s1, s2
being Real
 st g is being_homeomorphism & g.0=f/.1 & g.1=f/.len f & g.s1=q & 0<=
  s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1 holds s1<=s2 by A2,A4,Def2;
  hence thesis by A3,A5;
end;
