reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th16:
  for k be positive Nat holds n+k in dom f implies k in dom (f/^n)
  proof
    let k be positive Nat;
    A0: k >= 1 by NAT_1:14;
    assume n+k in dom f; then
    A2: n+k <= len f by FINSEQ_3:25; then
    A3: n+k-n <= len f - n by XREAL_1:9;
    n + k > n +0 by XREAL_1:6; then
    len f > n by A2,XXREAL_0:2; then
    k <= len (f/^n) by A3,RFINSEQ:def 1;
    hence thesis by A0,FINSEQ_3:25;
  end;
