
theorem der1:
for R being non degenerated comRing
holds (Deriv R).(1_.R) = 0_.R & (Deriv R).(0_.R) = 0_.R
proof
let F be non degenerated comRing;
reconsider r = 1_.(F) as Element of the carrier of Polynom-Ring F
  by POLYNOM3:def 10;
H: (NAT --> 0.F) = 0_.(F) by POLYNOM3:def 7;
now let o be object;
  assume o in NAT; then
  reconsider i = o as Element of NAT;
  ((Deriv F).r).i
     = (i+1) * ((1_.(F)).(i+1)) by RINGDER1:def 8
    .= (i+1) * 0.F by POLYNOM3:30
    .= (0_.(F)).i by H;
  hence ((Deriv F).r).o = (0_.(F)).o;
  end;
hence 0_.(F) = (Deriv F).(1_.F) by FUNCT_2:12;
thus (Deriv F).(0_.F)
       = (Deriv F).(0.(Polynom-Ring F)) by POLYNOM3:def 10
      .= 0_.F by POLYNOM3:def 10;
end;
