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;

theorem
   for f be Element of the carrier of Polynom-Ring R, f1 be Polynomial of R
   st f = f1 & f1 is constant holds (Der1(R)).f = 0_.R
   proof
     let f be Element of the carrier of Polynom-Ring R, f1 be Polynomial of R;
     assume
A1:  f = f1 & f1 is constant;
     for i be Element of NAT holds ((Der1(R)).f).i = (0_.R).i
     proof
       let i be Element of NAT;
       ((Der1(R)).f).i = (i+1)*(f1.(i+1)) by A1,Def8
       .= (i+1)*0.R by ALGSEQ_1:8,A1,XREAL_1:20
       .= (0_.R).i by Th3;
       hence thesis;
     end;
     hence (Der1(R)).f = 0_.R;
   end;
