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 Function, n being Nat holds iter(f,n) = compose
  (n|->f,field f)
proof
  let f be Function;
  defpred P[Nat] means iter(f,$1) = compose($1|->f,field f);
A1: now
    let n be Nat;
    assume P[n];
    then iter(f,n+1) = f*compose(n|->f,field f) by Th70
      .= compose((n|->f)^<*f*>,field f) by Th40
      .= compose((n+1)|->f,field f) by FINSEQ_2:60;
    hence P[n+1];
  end;
  iter(f,0) = id (field f) by Th67
    .= compose({},field f) by Th38
    .= compose(0|->f,field f);
  then
A2: P[ 0];
  thus for n being Nat holds P[n] from NAT_1:sch 2(A2, A1);
end;
