reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;

theorem
  for f being PartFunc of X,X holds iter(f,n) is PartFunc of X,X
proof
  let f be PartFunc of X,X;
A1: field f = dom f \/ rng f;
  rng iter(f,n) c= field f by Th71;
  then
A2: rng iter(f,n) c= X by A1,XBOOLE_1:1;
  dom iter(f,n) c= field f by Th71;
  then dom iter(f,n) c= X by A1,XBOOLE_1:1;
  hence thesis by A2,RELSET_1:4;
end;
