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 Th70:
  for n being Nat holds iter(R,n+1) = iter(R,n)*R
proof
  let n be Nat;
  defpred P[Nat] means iter(R,$1+1) = iter(R,$1)*R;
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A2: iter(R,k+1) = iter(R,k)*R;
    thus iter(R,k+1)*R = R*iter(R,k)*R by Th68
      .= R*(iter(R,k)*R) by RELAT_1:36
      .= iter(R,k+1+1) by A2,Th68;
  end;
  iter(R,0+1) = R by Th69
    .= id(field R)*R by Lm3
    .= iter(R,0)*R by Th67;
  then
A3: P[ 0];
  for k be Nat holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
