
theorem
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL 2 st
  p in L~f & p <> f.len f & f is being_S-Seq holds Index (p, L_Cut(f,p)) = 1
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: p <> f.len f and
A3: f is being_S-Seq;
  L_Cut(f,p) is being_S-Seq by A1,A2,A3,JORDAN3:34;
  then
A4: 2 <= len L_Cut(f,p) by TOPREAL1:def 8;
  then 1 <= len L_Cut(f,p) by XXREAL_0:2;
  then 1 in dom L_Cut(f,p) by FINSEQ_3:25;
  then (L_Cut(f,p))/.1 = (L_Cut(f,p)).1 by PARTFUN1:def 6
    .= p by A1,JORDAN3:23;
  hence thesis by A4,JORDAN3:11;
end;
