theorem Th73:
  for n being Nat st rng R c= dom R holds dom iter(R,n) = dom R &
  rng iter(R,n) c= dom R
proof
  let n be Nat;
  defpred P[Nat] means dom iter(R,$1) = dom R & rng iter(R,$1) c= dom R;
A1: for k being Nat holds P[k] implies P[k+1]
  proof
    let k be Nat;
    assume
A2: dom iter(R,k) = dom R & rng iter(R,k) c= dom R;
    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;
    iter(R,k+1) = iter(R,k)*R by Th70;
    hence thesis by A2,A3,RELAT_1:27,XBOOLE_1:1;
  end;
  assume rng R c= dom R;
  then iter(R,0) = id dom R by Lm4;
  then
A4: P[ 0];
  for k being Nat holds P[k] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
