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

theorem Th18:
  for L be domRing, D be Derivation of L holds
  for f be Element of the carrier of L, j,n be Nat
    holds (D|^n).(j*f) = j*((D|^n).f)
    proof
      let L be domRing, D be Derivation of L;
      for f be Element of the carrier of L, j,n be Nat
      holds (D|^n).(j*f) = j*((D|^n).f)
      proof
        let f be Element of the carrier of L, j,n be Nat;
        defpred P[Nat] means (D|^$1).(j*f) = j*((D|^$1).f);
A1:     P[0]
        proof
A2:       (D|^0).(j*f) = (id L).(j*f) by VECTSP11:18 .= j*f;
          j*((D|^0).f) = j*((id L).f) by VECTSP11:18 .= j*f;
          hence thesis by A2;
        end;
A3:     for n be Nat st P[n] holds P[n+1]
        proof
          let n be Nat;
          assume P[n]; then
          (D|^(n+1)).(j*f) = D.(j*((D|^n).f)) by RINGDER1:9
          .= j*D.((D|^n).f) by RINGDER1:6
          .= j*(D|^(n+1)).f by RINGDER1:9;
          hence thesis;
        end;
        for n be Nat holds P[n] from NAT_1:sch 2(A1,A3);
        hence thesis;
      end;
      hence thesis;
    end;
