
theorem Th36:
  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_MonomialRepresentation_of f ex i being Element of NAT, m being non-zero
Monomial of n,L, p being non-zero Polynomial of n,L st i in dom A & p in P & A.
  i = m *' p & HT(f,T) <= HT(m*'p,T),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;
  HC(f,T) <> 0.L;
  then
A1: f.(HT(f,T)) <> 0.L by TERMORD:def 7;
  assume A is_MonomialRepresentation_of f;
  then consider i being Element of NAT such that
A2: i in dom A and
A3: ex m being Monomial of n,L, p being Polynomial of n,L st A.i = m *'
  p & p in P & (m*'p).HT(f,T) <> 0.L by A1,Lm4;
  consider m being Monomial of n,L, p being Polynomial of n,L such that
A4: A.i = m *' p and
A5: (m*'p).HT(f,T) <> 0.L and
A6: p in P by A3;
A7: m*'p <> 0_(n,L) by A5,POLYNOM1:22;
  then
A8: m <> 0_(n,L) by POLYRED:5;
  p <> 0_(n,L) by A7,POLYNOM1:28;
  then reconsider p as non-zero Polynomial of n,L by POLYNOM7:def 1;
  reconsider m as non-zero Monomial of n,L by A8,POLYNOM7:def 1;
  HT(f,T) in Support(m*'p) by A5,POLYNOM1:def 4;
  then HT(f,T) <= HT(m*'p,T),T by TERMORD:def 6;
  hence thesis by A2,A4,A6;
end;
