
theorem poly2:
for L being add-associative right_zeroed right_complementable right_unital
            distributive associative commutative non empty doubleLoopStr,
    p,q being sequence of L,
    a being Element of L holds a * (p *' q) = p *' (a * q)
proof
let R be add-associative right_zeroed right_complementable right_unital
         distributive associative commutative non empty doubleLoopStr,
    p,q be sequence of R, a be Element of R;
set t = a * (p *' q);
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
  H1: len F = i+1 & (p*'q).i = Sum F &
      for k being Element of NAT st k in dom F
      holds F.k = p.(k-'1) * q.(i+1-'k) by POLYNOM3:def 9;
  consider G being FinSequence of the carrier of R such that
  H2: len G = i+1 & (p*'(a*q)).i = Sum G &
      for k being Element of NAT st k in dom G
      holds G.k = p.(k-'1) * (a*q).(i+1-'k) by POLYNOM3:def 9;
  H3: dom F = Seg(i+1) by H1,FINSEQ_1:def 3 .= dom G by H2,FINSEQ_1:def 3;
  now let x be object;
    assume H4: x in dom F;
    then reconsider j = x as Element of NAT;
    H5: F.j = p.(j-'1) * q.(i+1-'j) by H4,H1;
    G/.j = G.j by H3,H4,PARTFUN1:def 6
        .= p.(j-'1) * (a*q).(i+1-'j) by H2,H4,H3
        .= p.(j-'1) * (a * q.(i+1-'j)) by POLYNOM5:def 4
        .= (p.(j-'1) * q.(i+1-'j)) * a by GROUP_1:def 3
        .= a*(F/.j) by PARTFUN1:def 6,H5,H4;
    hence G/.x = a * (F/.x);
    end;
  then G = a * F by H3,POLYNOM1:def 1;
  then Sum G = a * (Sum F) by POLYNOM1:12;
  hence t.x = (p *' (a * q)).x by H1,POLYNOM5:def 4,H2;
  end;
hence thesis by FUNCT_2:12;
end;
