reserve n for Nat;

theorem Th40:
  for f be non empty FinSequence of TOP-REAL 2 for p be Point of
TOP-REAL 2 st f is almost-one-to-one special unfolded s.n.c. & p in L~f & p<>f.
  len f & p <> f.1 holds L_Cut(f,p) is_S-Seq_joining p,f/.len f
proof
  let f be non empty FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume that
A1: f is almost-one-to-one special unfolded s.n.c. and
A2: p in L~f and
A3: p<>f.len f and
A4: p <> f.1;
A5: Rev f is special by A1,SPPOL_2:40;
A6: p in L~Rev f by A2,SPPOL_2:22;
  p <> Rev f.len f by A4,FINSEQ_5:62;
  then
A7: p <> Rev f.len Rev f by FINSEQ_5:def 3;
A8: Rev f is s.n.c. by A1,SPPOL_2:35;
A9: p <> (Rev f).1 by A3,FINSEQ_5:62;
A10: Rev f is unfolded by A1,SPPOL_2:28;
  L_Cut(f,p) = L_Cut(Rev Rev f,p)
    .= Rev R_Cut(Rev f,p) by A1,A7,A10,A8,A6,Th38;
  then L_Cut(f,p) is_S-Seq_joining p,(Rev f)/.1 by A1,A5,A10,A8,A6,A9,Th39,
JORDAN3:15;
  hence thesis by FINSEQ_5:65;
end;
