
theorem
  for f being FinSequence of TOP-REAL 2, i be Nat st 1 <= i &
i+1 <= len f & f is being_S-Seq & Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i)
) in LSeg (f,i) holds Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.(i+1)
proof
  let f be FinSequence of TOP-REAL 2, i be Nat;
  assume that
A1: 1 <= i & i+1 <= len f and
A2: f is being_S-Seq and
A3: Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i);
  reconsider Q = LSeg (f,i) as non empty Subset of TOP-REAL 2 by A3;
  Q = LSeg (f/.i, f/.(i+1)) by A1,TOPREAL1:def 3;
  then Q c= L~f by A1,SPPOL_2:16;
  then L~f meets Q by A3,XBOOLE_0:3;
  then
A4: Last_Point (L~f, f/.1, f/.len f, Q) = Last_Point (Q, f/.i, f/.(i+1), Q)
  by A1,A2,A3,Th20;
  Q is closed & Q is_an_arc_of f/.i, f/.(i+1) by A1,A2,JORDAN5B:15;
  hence thesis by A4,Th7;
end;
