reserve n for Nat;

theorem Th56:
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2
  st f is being_S-Seq & p in rng f & p <> f.1 holds Index(p,f)+1 = p..f
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume that
A1: f is being_S-Seq and
A2: p in rng f and
A3: p <> f.1;
A4: 1 <= p..f by A2,FINSEQ_4:21;
  p..f <> 1 by A2,A3,FINSEQ_4:19;
  then
A5: 1 < p..f by A4,XXREAL_0:1;
A6: f.(p..f) = p by A2,FINSEQ_4:19;
  p..f <= len f by A2,FINSEQ_4:21;
  hence thesis by A1,A5,A6,JORDAN3:12;
end;
