reserve m,j,p,q,n,l for Element of NAT;
reserve e1,e2 for ExtReal;

theorem
  for F being Function, k being Nat st k > 0
  holds not 0 in dom Shift(F,k)
proof
  let F be Function, k be Nat such that
A1: k > 0 and
A2: 0 in dom Shift(F,k);
  dom Shift(F,k) = { m+k where m is Nat: m in dom F } by Def12;
  then ex m being Nat st 0 = m+k &  m in dom F by A2;
  hence contradiction by A1;
end;
