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

theorem
  for j be Nat st j > k holds
  for f be Element of the carrier of Polynom-Ring INT.Ring
    st ((Der1(INT.Ring))|^1).(f|^1) = 1.Polynom-Ring INT.Ring holds
     ((Der1(INT.Ring))|^j).(f|^k) = 0.Polynom-Ring INT.Ring
     proof
       set L = Polynom-Ring INT.Ring;
       set D = Der1(INT.Ring);
       let j be Nat;
       assume
A1:    j > k;
       for f be Element of the carrier of L
       st (D|^1).(f|^1) = 1.Polynom-Ring INT.Ring holds (D|^j).(f|^k) = 0.L
       proof
         let f be Element of the carrier of L;
         assume
A2:      (D|^1).(f|^1) = 1.L;
A3:      j - k > k -k by A1,XREAL_1:8; then
         j-k in NAT by INT_1:3; then
         reconsider l = j - k as Nat;
         reconsider l1 = l-1 as Nat by A3,NAT_1:14;
         l + k = j; then
         (D|^j).(f|^k) = ((D|^l)*(D|^k)).(f|^k) by VECTSP11:20
         .= (D|^l).((D|^k).(f|^k)) by FUNCT_2:15
         .= (D|^(l1+1)).((k!)*(1.L)) by A2,Th20
         .= ((D|^l1)*(D|^1)).((k!)*(1.L)) by VECTSP11:20
         .= (D|^l1).((D|^1).((k!)*(1.L))) by FUNCT_2:15
         .= (D|^l1).(D.((k!)*(1.L))) by Lm12
         .= (D|^l1).((k!)*(D.(1.L))) by RINGDER1:6
         .= (D|^l1).((k!)*0.L) by RINGDER1:5
         .= (D|^l1).(0*1.L) by BINOM:12
         .= 0*((D|^l1).1.L) by Th18
         .= 0.L by BINOM:12;
         hence thesis;
       end;
       hence thesis;
     end;
