reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th16:
  for D being non empty set, sq being FinSequence of D st 1<=k & k
  <len sq holds (sq/^1).k=sq.(k+1)
proof
  let D be non empty set, sq be FinSequence of D;
  assume that
A1: 1<=k and
A2: k<len sq;
A3: len sq=(len sq-1)+1;
  k+1<=len sq by A2,NAT_1:13;
  then
A4: k<=len sq-1 by A3,XREAL_1:6;
A5: len(sq)>=1 by A1,A2,XXREAL_0:2;
  then len(sq/^1)=len sq-1 by RFINSEQ:def 1;
  then k in dom(sq/^1) by A1,A4,FINSEQ_3:25;
  hence thesis by A5,RFINSEQ:def 1;
end;
