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 Th71:
  dom iter(R,n) c= field R & rng iter(R,n) c= field R
proof
  defpred P[Nat] means dom iter(R,$1) c= field R & rng iter(
  R,$1) c= field R;
A1: P[k] implies P[k+1]
  proof
    iter(R,k+1) = iter(R,k)*R by Th70;
    then
A2: dom iter(R,k+1) c= dom iter(R,k) by RELAT_1:25;
    iter(R,k+1) = R*iter(R,k) by Th68;
    then
A3: rng iter(R,k+1) c= rng iter(R,k) by RELAT_1:26;
    assume dom iter(R,k) c= field R & rng iter(R,k) c= field R;
    hence thesis by A2,A3,XBOOLE_1:1;
  end;
  iter(R,0) = id(field R) by Th67;
  then
A4: P[ 0];
  P[k] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
