
theorem Th37:
  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 being non-zero Polynomial of n,L, P being
non empty Subset of Polynom-Ring(n,L), A being LeftLinearCombination of P st A
is_Standard_Representation_of f,P,T ex i being Element of NAT, m being non-zero
Monomial of n,L, p being non-zero Polynomial of n,L st p in P & i in dom A & A
  /.i = m *' p & HT(f,T) = HT(m*'p,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 be non-zero Polynomial of n,L, P be non empty Subset of
  Polynom-Ring(n,L), A be LeftLinearCombination of P;
  assume A is_Standard_Representation_of f,P,T;
  then
A1: A is_Standard_Representation_of f,P,HT(f,T),T;
  then consider
  i being Element of NAT, m being non-zero Monomial of n,L, p being
  non-zero Polynomial of n,L such that
A2: i in dom A and
  p in P and
A3: A.i = m *' p and
A4: HT(f,T) <= HT(m*'p,T),T by Th32,Th36;
  consider m9 being non-zero Monomial of n,L, p9 being non-zero Polynomial of
  n,L such that
A5: p9 in P and
A6: A/.i = m9 *' p9 and
A7: HT(m9*'p9,T) <= HT(f,T),T by A1,A2;
  take i,m9,p9;
  m *' p = m9 *' p9 by A2,A3,A6,PARTFUN1:def 6;
  hence thesis by A2,A4,A5,A6,A7,TERMORD:7;
end;
