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;
reserve p for Polynomial of F_Real;

theorem
   for p0 be Element of Polynom-Ring F_Real, p be Polynomial of F_Real
   st p0 = p holds poly_diff(p) = (Der1(F_Real)).p0
   proof
     let p0 be Element of Polynom-Ring F_Real, p be Polynomial of F_Real;
     assume
A1:  p0 = p;
     for n holds (poly_diff(p)).n = ((Der1(F_Real)).p0).n
     proof
       let n;
       (poly_diff(p)).n = p.(n+1) * (n+1) by Def9
       .= (n+1)*p.(n+1) by BINOM:18
       .= ((Der1(F_Real)).p0).n by A1,Def8;
       hence thesis;
     end;
     hence thesis;
   end;
