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
    for x be Element of Polynom-Ring R
    st x = anpoly(1.R,1) holds ex y be Element of Polynom-Ring R
    st y = anpoly(1.R,n) & (Der1(R)).(x|^(n+1)) = (n+1)*y
    proof
      let x be Element of Polynom-Ring R;
      assume
A1:   x = anpoly(1.R,1);
      reconsider x1 = anpoly(1.R,1) as Polynomial of R;
A2:   <%0.R,1.R%> = (<%0.R,1.R%>)`^1 by POLYNOM5:16 .= x1 by FIELD_1:12;
      reconsider D = (Der1(R)) as Derivation of Polynom-Ring R;
      D.x = anpoly(1.R,0) by A1,Th30 .= 1_.R; then
      (x|^n) = x1`^n & D.x = 1_.R by A1,Th37; then
A3:   (x|^n)*D.x = (x1`^n)*'1_.R by POLYNOM3:def 10
      .= anpoly(1.R,n) by A2,FIELD_1:12;
      reconsider y = anpoly(1.R,n)
        as Element of Polynom-Ring R by POLYNOM3:def 10;
      D.(x|^(n+1)) = (n+1)*y by A3,Th7;
      hence thesis;
    end;
