 reserve n for Nat;

theorem
  for R being non degenerated Ring,
    a being Element of R holds LC anpoly(a,n) = a
  proof
    let R be non degenerated Ring, a being Element of R;
    set q = anpoly(a,n);
    per cases;
      suppose
A1:     a = 0.R; then
        q = 0_.R by UPROOTS:def 5; then
        q.(len q -' 1) = 0.R by FUNCOP_1:7;
        hence thesis by A1,RATFUNC1:def 6;
      end;
      suppose a <> 0.R; then
        a is non zero; then
A2:     n = deg q by Lm1 .= len q - 1; then
        n + 1 = len q; then
        len q -' 1 = n by A2,XREAL_1:233,NAT_1:11;
        hence a = q.(len q -' 1) by POLYDIFF:24 .= LC q by RATFUNC1:def 6;
      end;
    end;
