reserve n for Nat;

theorem Th5:
  for f be FinSequence of TOP-REAL 2 st f is being_S-Seq for p be
  Point of TOP-REAL 2 st p in L~f holds not f.1 in L~mid(f,Index(p,f)+1,len f)
proof
  let f be FinSequence of TOP-REAL 2 such that
A1: f is being_S-Seq;
  let p be Point of TOP-REAL 2 such that
A2: p in L~f and
A3: f.1 in L~mid(f,Index(p,f)+1,len f);
  len f <> 0 by A2,TOPREAL1:22;
  then f <> {};
  then 1 in dom f by FINSEQ_5:6;
  then
A4: f/.1 in L~mid(f,Index(p,f)+1,len f) by A3,PARTFUN1:def 6;
  Index(p,f) < len f by A2,JORDAN3:8;
  then Index(p,f)+1 <= len f by NAT_1:13;
  then Index(p,f)+1 = 0+1 by A1,A4,JORDAN5B:16,NAT_1:11;
  hence contradiction by A2,JORDAN3:8;
end;
