
theorem poly1:
for L being add-associative right_zeroed right_complementable
            left-distributive left_unital non empty doubleLoopStr
for p being sequence of L holds (1_.(L)) *' p = p
proof
let R be add-associative right_zeroed right_complementable
         left-distributive left_unital non empty doubleLoopStr,
    p be sequence of R;
set q = 1_.(R);
now let x be object;
  assume x in NAT;
  then reconsider i = x as Element of NAT;
  consider F being FinSequence of the carrier of R such that
  H: len F = i+1 & (q*'p).i = Sum F &
     for k being Element of NAT st k in dom F
     holds F.k = q.(k-'1) * p.(i+1-'k) by POLYNOM3:def 9;
  D: now let j be Element of NAT;
     assume B: j in dom F & j <> 1;
     then j in Seg(len F) by FINSEQ_1:def 3;
     then 1 <= j & j <= i+1 by H,FINSEQ_1:1;
     then j > 1 by B,XXREAL_0:1;
     then q.(j-'1) = 0.R by POLYNOM3:30,NAT_D:36;
     hence 0.R = q.(j-'1) * p.(i+1-'j)
              .= F.j by B,H
              .= F/.j by B,PARTFUN1:def 6;
     end;
  G: dom F = Seg(i+1) by H,FINSEQ_1:def 3;
  E: 1 <= 1 & 1 <= i + 1 by NAT_1:11; then
  F: Sum F = F/.1 by D,POLYNOM2:3,G,FINSEQ_1:1;
  F.1 = q.(1-'1) * p.(i+1-'1) by H,E,G,FINSEQ_1:1
     .= q.0 * p.(i+1-'1) by XREAL_1:232
     .= q.0 * p.i by NAT_D:34
     .= 1.R * p.i by POLYNOM3:30
     .= p.i;
  hence (q*'p).x = p.x by E,H,F,PARTFUN1:def 6,G,FINSEQ_1:1;
  end;
hence thesis by FUNCT_2:12;
end;
