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 Th29:
    for f being Element of the carrier of Polynom-Ring R, a be Element of R
    st f = anpoly(a,0) holds (Der1(R)).f = 0_.R
    proof
      let f be Element of the carrier of Polynom-Ring R, a be Element of R;
      assume
A1:   f = anpoly(a,0);
      for n be Element of NAT holds ((Der1(R)).f).n = (0_.R).n
      proof
        let n be Element of NAT;
        ((Der1(R)).f).n = (n+1)*(anpoly(a,0).(n+1)) by A1,Def8
           .= (n+1) * 0.R by POLYDIFF:25
           .= (0_.R).n by Th3;
         hence thesis;
       end;
       hence (Der1(R)).f = 0_.R;
     end;
