
theorem
for L being add-associative right_zeroed right_complementable
            right-distributive associative non empty doubleLoopStr
for p,q being Polynomial of L
for a being Element of L
holds (a * p) *' q = a * (p *' q)
proof
let L be add-associative right_zeroed right_complementable
         right-distributive associative non empty doubleLoopStr;
let p,q be Polynomial of L;
let a being Element of L;
set f = (a * p) *' q, g = a * (p *' q);
A1: dom f = NAT by FUNCT_2:def 1 .= dom g by FUNCT_2:def 1;
now let i be object;
  assume i in dom f;
  then reconsider n = i as Element of NAT;
  consider fr being FinSequence of the carrier of L such that
  A2: len fr = n+1 & f.i = Sum fr &
      for k being Element of NAT st k in dom fr
      holds fr.k = (a*p).(k-'1) * q.(n+1-'k) by POLYNOM3:def 9;
  consider fa being FinSequence of the carrier of L such that
  A3: len fa = n+1 & (p *' q).i = Sum fa &
      for k being Element of NAT st k in dom fa
      holds fa.k = p.(k-'1) * q.(n+1-'k) by POLYNOM3:def 9;
  Seg(len fa) = dom fr by A2,A3,FINSEQ_1:def 3;
  then A4: dom fa = dom fr by FINSEQ_1:def 3;
  A5: now let k be Element of NAT;
      assume A6: k in dom fa;
      then fa.k = fa/.k by PARTFUN1:def 6;
      then reconsider x = fa.k as Element of L;
      thus fr/.k = fr.k by A6,A4,PARTFUN1:def 6
        .= (a*p).(k-'1) * q.(n+1-'k) by A4,A6,A2
        .= (a * p.(k-'1)) * q.(n+1-'k) by POLYNOM5:def 4
        .= a* (p.(k-'1) * q.(n+1-'k)) by GROUP_1:def 3
        .= a * x by A6,A3
        .= a * (fa/.k) by A6,PARTFUN1:def 6;
      end;
  g.n = a * (Sum fa) by A3,POLYNOM5:def 4
     .= f.n by A5,A3,A2,Th1;
  hence f.i = g.i;
  end;
hence thesis by A1,FUNCT_1:2;
end;
