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

theorem Th25:
  for p being Function, k being Nat holds rng Shift(p,k) c= rng p
proof
  let p be Function, k being Nat;
  let y be object;
  assume y in rng Shift(p,k);
  then consider x being object such that
A1: x in dom Shift(p,k) and
A2: y = Shift(p,k).x by FUNCT_1:def 3;
  x in { m+k where m is Nat:m in dom p } by A1,Def12;
  then consider m being Nat such that
A3: x = m+k and
A4: m in dom p;
  p.m = Shift(p,k).x by A3,A4,Def12;
  hence thesis by A2,A4,FUNCT_1:def 3;
end;
