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

theorem Th19:
  for k be Nat, f be Element of the carrier of Polynom-Ring INT.Ring
    st ((Der1(INT.Ring))|^1).(f|^1) = 1.Polynom-Ring INT.Ring holds
    for j be Nat st 1 <= j <= k holds
    ((Der1(INT.Ring))|^j).(f|^k) = (eta(k,j))*(f|^(k-'j))
    proof
      set L = Polynom-Ring INT.Ring;
      set D = Der1(INT.Ring);
      let k be Nat, f be Element of the carrier of L;
      assume
A1:   (D|^1).(f|^1) = 1.L;
A2:   D.f = (D|^1).f by VECTSP11:19 .= 1.L by A1,BINOM:8;
      defpred P[Nat] means
      for j be Nat st 1 <= j <= $1 holds
      (D|^j).(f|^$1) = (eta($1,j))*(f|^($1-'j));
A3:   for k be Nat st for n be Nat st n < k holds P[n] holds P[k]
      proof
        let k be Nat;
        assume
A4:     for n be Nat st n < k holds P[n];
        for l be Nat st 1<= l <= k holds (D|^l).(f|^k)=(eta(k,l))*(f|^(k-'l))
        proof
         let l be Nat;
         assume
A5:      1 <= l <= k; then
         reconsider k1 = k-1 as Nat;
A6:      k -' 1 = k1 by A5,XXREAL_0:2,XREAL_1:233;
A7:      (D|^1).(f|^k) = D.(f|^(k1+1)) by VECTSP11:19
         .= (k1+1)*((f|^k1)*1.L) by A2,RINGDER1:7
         .= k*(f|^(k-'1));
A8:      eta(k,1) = ((k1 +1)*(k1!))/(k1!) by NEWTON:15 .= k by XCMPLX_1:89;
         reconsider l1 = l-1 as Nat by A5;
A9:      (D|^l).(f|^k) = (D|^(l1+1)).(f|^k)
         .= ((D|^l1)*(D|^1)).(f|^k) by VECTSP11:20
         .= (D|^l1).(k*(f|^k1)) by A7,A6,FUNCT_2:15
         .= k*(D|^l1).(f|^k1) by Th18;
A10:     k1 < k - 0 by XREAL_1:15;
A11:     k1 -' l1 = (k-1) - (l - 1) by A5,XREAL_1:9,XREAL_1:233 .= k - l
         .= k-'l by A5,XREAL_1:233; then
A12:     k*(eta(k1,l1)) = ((k1+1)*(k1!))/((k-'l)!)
         .= eta(k,l) by NEWTON:15;
reconsider s = (eta(k1,l1)), k as Element of NAT by ORDINAL1:def 12;
         reconsider t = f|^(k1-'l1) as Element of L;
         per cases;
           suppose
             1 <= l1; then
             1 <= l1 <= k1 by A5,XREAL_1:9; then
             k*(D|^l1).(f|^k1) = k*(s*t) by A10,A4
             .= (eta(k,l))*(f|^(k-'l)) by A11,A12,Lm11;
             hence thesis by A9;
           end;
           suppose
             1 > l1; then
             l1 = 0 by NAT_1:14;
             hence thesis by A7,A8;
           end;
         end;
         hence thesis;
       end;
       for k be Nat holds P[k] from NAT_1:sch 4(A3);
       hence thesis;
     end;
