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 Th15:
  for f be FinSequence of REAL holds
    k in dom (f/^n) & n in dom f implies n+k in dom f
  proof
    let f be FinSequence of REAL;
    assume
    A1: k in dom (f/^n) & n in dom f; then
    len (f|n) + k in dom ((f|n)^(f/^n)) by FINSEQ_1:28; then
    n+k in dom ((f|n)^(f/^n)) by A1,Th10;
    hence thesis by RFINSEQ:8;
  end;
