theorem
  for n being Nat for f being Function of X,X holds iter(f,n) is
  Function of X,X
proof
  let n be Nat;
  let f be Function of X,X;
A1: X = {} implies X = {};
  then
A2: dom f = X by FUNCT_2:def 1;
A3: rng f c= X;
  then dom iter(f,n) = X & rng iter(f,n) c= X by A2,Th73;
  then reconsider R = iter(f,n) as Relation of X,X by RELSET_1:4;
  dom R = X by A2,A3,Th73;
  hence thesis by A1,FUNCT_2:def 1;
end;
