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;
