 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
  for R be domRing
  for f be Element of the carrier of Polynom-Ring R st len f > 1 & Char R = 0
    holds len ((Der1 R).f) = (len f)-1
    proof
      let R be domRing;
      let f be Element of the carrier of Polynom-Ring R;
      reconsider n = 0 as Nat;
      assume
A1:   len f > 1 & Char R = 0; then
A2:   (len f) - 1 > 1 - 1 by XREAL_1:14;
      reconsider lf1 = (len f) - 1 as Nat by A1;
      lf1 +0 > 0 by A2; then
A4:   lf1 >= 1 by NAT_1:19;
      reconsider lf2 = lf1 - 1 as Nat by A2;
A5:   lf1 + 1 = len f;
      for i be Nat st i>= lf1 holds ((Der1 R).f).i = 0.R
      proof
        let i be Nat;
        assume i>= lf1; then
        i + 1 >= lf1 +1 by XREAL_1:6; then
A8:     f.(i+1) = 0.R by ALGSEQ_1:8;
        ((Der1 R).f).i = (i+1)*(f.(i+1)) by RINGDER1:def 8 .= 0.R by A8;
        hence thesis;
      end; then
A9:   lf1 is_at_least_length_of ((Der1 R).f);
A10:  ((Der1 R).f).lf2 = (lf2+1)*(f.(lf2+1)) by RINGDER1:def 8 .= lf1*f.lf1;
      reconsider flf1 = f.lf1 as Element of R;
      flf1 is non zero by A5,ALGSEQ_1:10;
      hence thesis by Th15,A4,A9,A10,A1,Th12;
    end;
