
theorem
  for n being Ordinal, L being add-associative right_complementable
right_zeroed well-unital distributive domRing-like non trivial doubleLoopStr,
  a,a9 being Element of L holds (a * a9) |(n,L) = (a |(n,L)) *' (a9 |(n,L))
proof
  let n be Ordinal, L be add-associative right_complementable right_zeroed
  well-unital distributive domRing-like non trivial doubleLoopStr, a,a9 be
  Element of L;
  per cases;
  suppose
A1: a is non zero & a9 is non zero;
    term((a*a9) |(n,L)) = EmptyBag n & coefficient((a*a9) |(n,L)) = a*a9
    by POLYNOM7:23;
    then
A2: (a * a9) |(n,L) = Monom(a*a9,EmptyBag n) by POLYNOM7:11;
    term(a9 |(n,L)) = EmptyBag n & coefficient(a9 |(n,L)) = a9 by POLYNOM7:23;
    then
A3: a9 |(n,L) = Monom(a9,EmptyBag n) by POLYNOM7:11;
    term(a |(n,L)) = EmptyBag n & coefficient(a |(n,L)) = a by POLYNOM7:23;
    then
A4: a |(n,L) = Monom(a,EmptyBag n) by POLYNOM7:11;
    EmptyBag n + EmptyBag n = EmptyBag n by PRE_POLY:53;
    hence thesis by A1,A2,A4,A3,Th3;
  end;
  suppose
A5: not(a is non zero & a9 is non zero);
    now
      per cases by A5,STRUCT_0:def 12;
      case
A6:     a = 0.L;
        then a * a9 = 0.L;
        then
A7:     (a * a9) |(n,L) = 0_(n,L) by POLYNOM7:19;
        a |(n,L) = 0_(n,L) by A6,POLYNOM7:19;
        hence thesis by A7,Th2;
      end;
      case
A8:     a9 = 0.L;
        then a * a9 = 0.L;
        then
A9:     (a * a9) |(n,L) = 0_(n,L) by POLYNOM7:19;
        a9 |(n,L) = 0_(n,L) by A8,POLYNOM7:19;
        hence thesis by A9,POLYNOM1:28;
      end;
    end;
    hence thesis;
  end;
end;
