
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, i being Element of NAT st A is_Standard_Representation_of f,P,b
  ,T & B = A|i & g = Sum(A/^i) holds 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, i be
  Element of NAT;
  assume that
A1: A is_Standard_Representation_of f,P,b,T and
A2: B = A|i and
A3: g = Sum(A/^i);
A4: Sum A = f by A1;
  dom(A|(Seg i)) c= dom A by RELAT_1:60;
  then
A5: dom B c= dom A by A2,FINSEQ_1:def 16;
A6: now
    let j being Element of NAT;
    assume
A7: j in dom B;
    then
A8: j in dom(A|(Seg i)) by A2,FINSEQ_1:def 16;
    A/.j = A.j by A5,A7,PARTFUN1:def 6
      .= (A|(Seg i)).j by A8,FUNCT_1:47
      .= B.j by A2,FINSEQ_1:def 16
      .= B/.j by A7,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 & B/.j = m *' p & HT(m*'p,T) <= b,T by A1,A5,A7;
  end;
  A = B ^ (A/^i) by A2,RFINSEQ:8;
  then Sum A = Sum B + Sum(A/^i) by RLVECT_1:41;
  then Sum A + -(Sum(A/^i)) = Sum B + (Sum(A/^i) + -Sum(A/^i)) by
RLVECT_1:def 3
    .= Sum B + 0.(Polynom-Ring(n,L)) by RLVECT_1:5
    .= Sum B by RLVECT_1:def 4;
  then Sum B = Sum A - (Sum(A/^i))
    .= f - g by A3,A4,Lm3;
  hence thesis by A6;
end;
