
theorem
  for n being Ordinal, T being connected TermOrder of n, L being
right_zeroed add-associative right_complementable well-unital distributive non
  trivial non empty doubleLoopStr, f,g being Polynomial of n,L, P being non
  empty Subset of Polynom-Ring(n,L), A,B being LeftLinearCombination of P, b
  being bag of n st A is_Standard_Representation_of f,P,b,T & B
is_Standard_Representation_of g,P,b,T holds A^B is_Standard_Representation_of f
  +g,P,b,T
proof
  let n be Ordinal, T be connected TermOrder of n, L be right_zeroed
add-associative right_complementable well-unital distributive non trivial non
  empty doubleLoopStr, f,g be Polynomial of n,L, P be non empty Subset of
  Polynom-Ring(n,L), A,B be LeftLinearCombination of P, b be bag of n;
  assume that
A1: A is_Standard_Representation_of f,P,b,T and
A2: B is_Standard_Representation_of g,P,b,T;
A3: now
    let i be Element of NAT;
    assume
A4: i in dom(A^B);
    now
      per cases by A4,FINSEQ_1:25;
      case
A5:     i in dom A;
        (A^B)/.i = (A^B).i by A4,PARTFUN1:def 6
          .= A.i by A5,FINSEQ_1:def 7
          .= A/.i by A5,PARTFUN1:def 6;
        hence ex m being non-zero Monomial of n,L, p being non-zero Polynomial
        of n,L st p in P & (A^B)/.i = m *' p & HT(m*'p,T) <= b,T by A1,A5;
      end;
      case
        ex k being Nat st k in dom B & i = len A + k;
        then consider k being Nat such that
A6:     k in dom B and
A7:     i = len A + k;
        (A^B)/.i = (A^B).i by A4,PARTFUN1:def 6
          .= B.k by A6,A7,FINSEQ_1:def 7
          .= B/.k by A6,PARTFUN1:def 6;
        hence ex m being non-zero Monomial of n,L, p being non-zero Polynomial
        of n,L st p in P & (A^B)/.i = m *' p & HT(m*'p,T) <= b,T by A2,A6;
      end;
    end;
    hence
    ex m being non-zero Monomial of n,L, p being non-zero Polynomial of n
    ,L st p in P & (A^B)/.i = m *' p & HT(m*'p,T) <= b,T;
  end;
  f = Sum A & g = Sum B by A1,A2;
  then f + g = Sum A + Sum B by POLYNOM1:def 11
    .= Sum(A^B) by RLVECT_1:41;
  hence thesis by A3;
end;
