
theorem
for R being non degenerated commutative Ring
for n being non trivial Nat
for a being Element of R holds (Deriv R).(X^(n,a)) = n * X^(n-1,0.R)
proof
let R be non degenerated commutative Ring, n be non trivial Nat;
let a be Element of R;
set q = X^(n,a), D = (Deriv R).(X^(n,a));
reconsider p = n * X^(n-1,0.R) as Polynomial of R by POLYNOM3:def 10;
A: n <> 0 & n <> 1 by NAT_2:def 1;
now let o be object;
  assume o in NAT; then
  reconsider j = o as Element of NAT;
  per cases;
  suppose C: j = 0;
    hence D.o
        = (0+1) * q.(0+1) by RINGDER1:def 8
       .= 1 * (0.R) by A,Lm11
       .= n * (0.R)
       .= n * ((X^(n-1,0.R)).j) by C,Lm10
       .= p.o by BBB;
    end;
  suppose C: j = n - 1;
    hence D.o
        = ((n-1)+1) * q.((n-1)+1) by RINGDER1:def 8
       .= n * (1.R) by Lm10
       .= n * (X^(n-1,0.R).j) by C,Lm10
       .= p.o by BBB;
    end;
  suppose C: j <> 0 & j <> n - 1; then
    D: j + 1 <> n;
    thus D.o
       = (j+1) * q.(j+1) by RINGDER1:def 8
      .= (j+1) * 0.R by D,Lm11
      .= n * 0.R
      .= n * (X^(n-1,0.R).j) by C,Lm11
       .= p.o by BBB;
    end;
  end;
hence D = n * (X^(n-1,0.R)) by FUNCT_2:12;
end;
