 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th22:
  for R be domRing
  for f be Element of the carrier of Polynom-Ring R, k be Nat holds
  for i be Nat holds (((Der1 R)|^k).f).i = (eta(i+k,k))*(f.(i+k))
    proof
      let R be domRing;
      let f be Element of the carrier of Polynom-Ring R, k be Nat;
      set D = Der1 R;
      set PR = Polynom-Ring R;
      defpred P[Nat] means for i be Nat holds
      for i be Nat holds ((D|^$1).f).i = (eta((i+$1),$1))*(f.(i+$1));
A1:   for k be Nat holds P[k] implies P[k+1]
      proof
        let k be Nat;
        assume
A2:     P[k];
        set fk = ((Der1 R)|^k).f;
A3:     D.fk = (D|^(k+1)).f by RINGDER1:9;
        for i be Nat holds (D.fk).i = (eta(i+(k+1),k+1))*(f.(i+(k+1)))
        proof
         let i be Nat;
reconsider m1 = i+1, m2 = eta((i+1)+k,k) as Element of NAT;
         reconsider fk1 = f.((i+1)+k) as Element of R;
         (D.fk).i = (i+1)*(fk.(i+1)) by RINGDER1:def 8
         .= m1*(m2 * fk1) by A2
         .= (i+1)*(eta((i+1)+k,k)) * fk1 by Lm11
         .= (eta(i+(1+k),k+1))*fk1 by Lm13;
         hence thesis;
       end;
       hence thesis by A3;
     end;
     for i be Nat holds ((D|^0).f).i = (eta((i+0),0))*(f.(i+0))
     proof
       let i be Nat;
A5:    eta((i+0),0) = 1 by XCMPLX_1:60;
       ((D|^0).f).i = ((id PR).f).i by VECTSP11:18
       .= eta((i+0),0)*(f.i) by A5,BINOM:13;
       hence thesis;
     end; then
A6:  P[0];
     for k be Nat holds P[k] from NAT_1:sch 2(A6,A1);
     hence thesis;
   end;
