
theorem
for R being Ring
for a being Element of R holds (Deriv R).(X-a) = 1_.(R)
proof
let F be Ring, a be Element of F;
set p = 1_.(F), q = (Deriv F).(X-a);
now let o be object;
  assume o in NAT;
  then reconsider i = o as Element of NAT;
  per cases;
  suppose A: i = 0; then
    q.i = (0+1) * (X-a).(0+1) by RINGDER1:def 8
       .= (X-a).1 by BINOM:13
       .= rpoly(1,a).1 by FIELD_9:def 2
       .= 1_F by HURWITZ:25
       .= p.i by A,POLYNOM3:30;
    hence p.o = q.o;
    end;
  suppose A: i <> 0; then
    B: i+1 <> 1;
    q.i = (i+1) * (X-a).(i+1) by RINGDER1:def 8
       .= (i+1) * rpoly(1,a).(i+1) by FIELD_9:def 2
       .= (i+1) * 0.F by B,HURWITZ:26
       .= p.i by A,POLYNOM3:30;
    hence p.o = q.o;
    end;
  end;
hence thesis by FUNCT_2:12;
end;
