reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;
reserve
  v,v1,v2 for FinSequence of REAL,
  n,m,k for Nat,
  x for set;

theorem
  for D being non empty set, f1,f2 being FinSequence of D, n st 1
  <= n & n <= len f2 holds (f1^f2)/.(n + len f1) = f2/.n
proof
  let D be non empty set, f1,f2 be FinSequence of D, n such that
A1: 1 <= n and
A2: n <= len f2;
A3: len f1 < n + len f1 by A1,NAT_1:19;
  len(f1^f2) = len f1 + len f2 by FINSEQ_1:22;
  then
A4: n + len f1 <= len(f1^f2) by A2,XREAL_1:6;
  n + len f1 >= n by NAT_1:11;
  then n + len f1 >= 1 by A1,XXREAL_0:2;
  hence (f1^f2)/.(n + len f1) = (f1^f2).(n + len f1) by A4,FINSEQ_4:15
    .= f2.(n + len f1 - len f1) by A3,A4,FINSEQ_1:24
    .= f2/.n by A1,A2,FINSEQ_4:15;
end;
