
theorem Th7:
  for n being Ordinal, L being add-associative right_complementable
right_zeroed well-unital distributive domRing-like non trivial doubleLoopStr,
  m1,m2 being non-zero Monomial of n,L holds term(m1*'m2) = term(m1) + term(m2)
proof
  let n be Ordinal, L be add-associative right_complementable right_zeroed
  well-unital distributive domRing-like non trivial doubleLoopStr, m1,m2 be
  non-zero Monomial of n,L;
  set T = the connected TermOrder of n;
A1: HC(m2,T) <> 0.L;
   HC(m1,T) <> 0.L;
  then reconsider
  a = coefficient(m1), b = coefficient(m2) as non zero Element of L
  by A1,TERMORD:23;
  a * b <> 0.L by VECTSP_2:def 1;
  then reconsider c = a * b as non zero Element of L by STRUCT_0:def 12;
  m1 = Monom(a,term(m1)) & m2 = Monom(b,term(m2)) by POLYNOM7:11;
  then term(m1 *' m2) = term(Monom(c,term(m1)+term(m2))) by TERMORD:3;
  hence thesis by POLYNOM7:10;
end;
