
theorem Th38:
  for n being Ordinal, T being admissible connected TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
  degenerated non empty doubleLoopStr, f being Polynomial of n,L, P being non
  empty Subset of Polynom-Ring(n,L) holds PolyRedRel(P,T) reduces f,0_(n,L)
  implies f has_a_Standard_Representation_of P,T
proof
  let n be Ordinal, T be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non degenerated non
  empty doubleLoopStr, f be Polynomial of n,L, P be non empty Subset of
  Polynom-Ring(n,L);
  reconsider f9 = f as Element of Polynom-Ring(n,L) by POLYNOM1:def 11;
A1: 0_(n,L) = 0.(Polynom-Ring(n,L)) by POLYNOM1:def 11;
  assume PolyRedRel(P,T) reduces f,0_(n,L);
  then consider A being LeftLinearCombination of P such that
A2: f9 = 0.(Polynom-Ring(n,L)) + Sum A and
A3: for i being Element of NAT st i in dom A ex m being non-zero
Monomial of n,L, p being non-zero Polynomial of n,L st p in P & A.i = m*'p & HT
  (m*'p,T) <= HT(f,T),T by A1,Lm5;
A4: now
    let i be Element of NAT;
    assume
A5: i in dom A;
    then
    ex m being non-zero Monomial of n,L, p being non-zero Polynomial of n,L
    st p in P & A.i = m*'p & HT(m*'p,T) <= HT(f,T),T by A3;
    hence
    ex m being non-zero Monomial of n,L, p being non-zero Polynomial of n
,L st p in P & A/.i = m *' p & HT(m*'p,T) <= HT(f,T),T by A5,PARTFUN1:def 6;
  end;
  f = Sum A by A2,RLVECT_1:def 4;
  then A is_Standard_Representation_of f,P,HT(f,T),T by A4;
  then A is_Standard_Representation_of f,P,T;
  hence thesis;
end;
