
theorem
  for p being FinSequence, n being non zero Nat st
  p is one-to-one & n <= len p holds (p.n)..p = n
proof
  let S be FinSequence;
  let n be non zero Nat such that
A1: S is one-to-one and
A2: n <= len S;
  set m = (S.n)..S;
  0+1 <= n by NAT_1:14;
  then
A3: n in dom S by A2,FINSEQ_3:25;
  then
A4: S.n in rng S by FUNCT_1:3;
  then
A5: S.m = S.n by FINSEQ_4:19;
  m in dom S by A4,FINSEQ_4:20;
  hence thesis by A1,A3,A5,FUNCT_1:def 4;
end;
