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

theorem
  for p being Function st dom p c= NAT for k being Nat holds
  rng Shift(p,k) = rng p
proof
  let p be Function such that
A1: dom p c= NAT;
  let k be Nat;
  thus rng Shift(p,k) c= rng p by Th25;
  let y be object;
  assume y in rng p;
  then consider x being object such that
A2: x in dom p and
A3: y = p.x by FUNCT_1:def 3;
  reconsider x as Element of NAT by A1,A2;
  x+k in dom Shift(p,k) & Shift(p,k).(x+k) = y by A2,A3,Def12,Th24;
  hence thesis by FUNCT_1:def 3;
end;
