reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem Th27:
  (for f being FinSequence st i in dom(f/^n) holds (f/^n).i = f.(n+i)) &
  (i in dom(f/^n) implies (f/^n)/.i = f/.(n+i))
proof
  thus AA: for f being FinSequence st i in dom(f/^n) holds (f/^n).i = f.(n+i)
  proof let f be FinSequence;
  assume
A1: i in dom(f/^n);
  per cases;
  suppose
    n <= len f;
    hence (f/^n).i = f.(n+i) by A1,RFINSEQ:def 1;
  end;
  suppose
    n > len f;
    then f/^n = {} by RFINSEQ:def 1;
    hence (f/^n).i = f.(n+i) by A1;
  end;
  end;
  assume B1: i in dom (f/^n); then
B2:  n+i in dom f by Th26;
  (f/^n).i = f.(n+i) by AA,B1; then
  (f/^n)/.i = f.(n+i) by B1,PARTFUN1:def 6 .= f/.(n+i) by PARTFUN1:def 6,B2;
  hence thesis;
end;
