reserve n for Nat;

theorem lem1e:
for R being non degenerated Ring,
    x being Element of R holds eval(npoly(R,0),x) = 1.R
proof
let L be non degenerated Ring, x be Element of L;
set q = npoly(L,0);
consider F be FinSequence of L such that
A3: eval(q,x) = Sum F and
A4: len F = len q and
A5: for n be Element of NAT st n in dom F holds
      F.n = q.(n-'1) * (power L).(x,n-'1) by POLYNOM4:def 2;
0 = deg q by lem6 .= len q - 1 by HURWITZ:def 2;
then C: F = <*F.1*> by A4,FINSEQ_1:40;
then Seg 1 = dom F by FINSEQ_1:38;
then F.1 = q.(1-'1) * (power L).(x,1-'1) by A5,FINSEQ_1:3
        .= q.0 * (power L).(x,1-'1) by NAT_2:8
        .= q.0 * (power L).(x,0) by NAT_2:8
        .= 1.L * (power L).(x,0) by Lm10
        .= 1.L * 1_L by GROUP_1:def 7
        .= 1.L;
hence thesis by A3,C,RLVECT_1:44;
end;
