
theorem thE2:
for R being Ring
for p being Polynomial of R holds (<%0.R,1.R%> *' p).0 = 0.R
proof
let R be Ring, p be Polynomial of R;
set q = <%0.R,1.R%>;
consider r being FinSequence of the carrier of R such that
A: len r = 0+1 & (<%0.R,1.R%> *' p).0 = Sum r & for k being Element of NAT
   st k in dom r holds r.k = q.(k-'1) * p.(0+1-'k) by POLYNOM3:def 9;
dom r = { 1 } by FINSEQ_1:2,A,FINSEQ_1:def 3; then
1 in dom r by TARSKI:def 1; then
C: r.1 = q.(1-'1) * p.(0+1-'1) by A
      .= q.0 * p.(1-'1) by NAT_2:8
      .= q.0 * p.0 by NAT_2:8
      .= 0.R * p.0 by POLYNOM5:38;
r = <*r.1*> by A,FINSEQ_1:40;
hence thesis by A,C,RLVECT_1:44;
end;
