reserve n for Nat;

theorem Th43:
  for f be S-Sequence_in_R2 for p be Point of TOP-REAL 2 st p in
  L~f & f.len f in L~R_Cut(f,p) holds f.len f = p
proof
  let f be S-Sequence_in_R2;
  let p be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: f.len f in L~R_Cut(f,p);
A3: L~f = L~(Rev f) by SPPOL_2:22;
A4: (Rev f).1 = f.len f by FINSEQ_5:62;
  L_Cut(Rev f,p) = Rev R_Cut(f,p) by A1,JORDAN3:22;
  then (Rev f).1 in L~L_Cut(Rev f,p) by A2,A4,SPPOL_2:22;
  hence thesis by A1,A3,A4,JORDAN1E:7;
end;
