 reserve n for Nat;

theorem
  for R being non degenerated Ring, n being non zero Nat,
    a,x being Element of R holds eval(anpoly(a,n),x) = a * x|^n
    proof
      let R be non degenerated Ring, n be non zero Nat, a,x be Element of R;
      set q = anpoly(a,n);
      per cases;
        suppose
A1:     a = 0.R; then
        q = 0_.R by UPROOTS:def 5;
        hence eval(q,x) = a * x|^n by A1,POLYNOM4:17;
        end;
        suppose a <> 0.R; then
A2:      a is non zero;
        consider F be FinSequence of R such that
A3:     eval(q,x) = Sum F and
A4:     len F = len q and
A5:     for j be Element of NAT st j in dom F holds
        F.j = q.(j-'1) * (power R).(x,j-'1) by POLYNOM4:def 2;
        n = deg q by A2,Lm1 .= len q - 1; then
A7:      dom F = Seg(n+1) by A4,FINSEQ_1:def 3;
A8:      1 <= n + 1 by NAT_1:11;
A9:      n+1-'1 = n+1-1 by NAT_1:11,XREAL_1:233;
A10:     F/.(n+1) = F.(n+1) by A7,FINSEQ_1:3,PARTFUN1:def 6
                .= q.n * (power R).(x,n+1-'1) by A9,A5,A8,A7,FINSEQ_1:1
                .= a * (power R).(x,(n+1)-'1) by POLYDIFF:24
                .= a * x|^ n by A9,BINOM:def 2;
        now let j be Element of NAT;
        assume
A11:    j in dom F & j <> n+1; then
A12:    1 <= j <= n + 1 by A7,FINSEQ_1:1; then
        j < n + 1 by A11,XXREAL_0:1; then
A13:    1 <= j <= n by A11,NAT_1:13,A7,FINSEQ_1:1;
A14:    j -' 1 = j - 1 by A12,XREAL_0:def 2;
A15:    j - 1 < j - 0 by XREAL_1:15;
        thus F/.j = F.j by A11,PARTFUN1:def 6
           .= q.(j-'1) * (power R).(x,j-'1) by A5,A11
           .= 0.R * (power R).(x,j-'1) by A15,A13,A14,POLYDIFF:25
           .= 0.R;
        end;
        hence thesis by A3,A10,A7,FINSEQ_1:3,POLYNOM2:3;
      end;
    end;
