reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th33:
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
2 st f is being_S-Seq & p in L~f & p<>f.len f holds L_Cut(f,p) is_S-Seq_joining
  p,f/.len f
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume that
A1: f is being_S-Seq and
A2: p in L~f and
A3: p<>f.len f;
A4: f <> {} by A2,CARD_1:27,TOPREAL1:22;
A5: Rev f is being_S-Seq by A1;
A6: p in L~Rev f by A2,SPPOL_2:22;
A7: p <> (Rev f).1 by A3,FINSEQ_5:62;
  L_Cut(f,p) = L_Cut(Rev Rev f,p)
    .= Rev R_Cut(Rev f,p) by A1,A6,Th22;
  then L_Cut(f,p) is_S-Seq_joining p,(Rev f)/.1 by A5,A6,A7,Th15,Th32;
  hence thesis by A4,FINSEQ_5:65;
end;
