reserve L for Abelian left_zeroed add-associative associative right_zeroed
              right_complementable distributive non empty doubleLoopStr;
reserve a,b,c for Element of L;
reserve R for non degenerated comRing;
reserve n,m,i,j,k for Nat;
 reserve D for Function of R, R;
 reserve x,y,z for Element of R;
reserve D for Derivation of R;
reserve s for FinSequence of the carrier of R;
reserve h for Function of R,R;
 reserve R for domRing;
 reserve f,g for Element of the carrier of Polynom-Ring R;
reserve a for Element of R;

theorem Th30:
    for f being Element of the carrier of Polynom-Ring R, a be Element of R
    st f = anpoly(a,1) holds (Der1(R)).f = anpoly(a,0)
    proof
      let f be Element of the carrier of Polynom-Ring R, a be Element of R;
      reconsider f1 = f as Polynomial of R;
      assume
A1:   f = anpoly(a,1);
      for n be Element of NAT holds ((Der1(R)).f).n = anpoly(a,0).n
      proof
        let n be Element of NAT;
        per cases;
          suppose
A2:        n = 0; then
           ((Der1(R)).f).n = (0+1)*(f1.(0+1)) by Def8
           .= 1 * a by A1,POLYDIFF:24
           .= a by BINOM:13
           .= anpoly(a,0).n by A2,POLYDIFF:24;
           hence thesis;
         end;
         suppose
A3:        n <> 0; then
A4:        n + 1 <> 1;
           ((Der1(R)).f).n = (n+1)*(anpoly(a,1).(n+1)) by A1,Def8
           .= (n+1) * 0.R by A4,POLYDIFF:25
           .= 0.R by Th3
           .= anpoly(a,0).n by A3,POLYDIFF:25;
           hence thesis;
         end;
       end;
       hence (Der1(R)).f = anpoly(a,0);
     end;
