
theorem der4:
for R being non degenerated comRing
for p being constant Element of the carrier of Polynom-Ring R
holds (Deriv R).p = 0_.R
proof
let F be non degenerated comRing,
    p be constant Element of the carrier of Polynom-Ring F;
consider a being Element of F such that A: p = a|F by RING_4:20;
B: 1.Polynom-Ring F = 1_.F & 0.Polynom-Ring F = 0_.F by POLYNOM3:def 10;
thus (Deriv F).p
   = (Deriv F).(a * 1_.(F)) by A,RING_4:16
  .= a * ((Deriv F).(1.Polynom-Ring F)) by B,der3
  .= a * 0_.(F) by B,der1
  .= 0_.F by POLYNOM5:28;
end;
