
theorem Th32:
  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 Polynomial of n,L, P being non empty
Subset of Polynom-Ring(n,L), A being LeftLinearCombination of P, b being bag of
  n holds A is_Standard_Representation_of f,P,b,T implies A
  is_MonomialRepresentation_of f
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 Polynomial of n,L, P be non empty Subset of
  Polynom-Ring(n,L), A be LeftLinearCombination of P, b be bag of n;
  assume
A1: A is_Standard_Representation_of f,P,b,T;
A2: now
    let i be Element of NAT;
    assume i in dom A;
    then
    ex m9 being non-zero Monomial of n,L, p9 being non-zero Polynomial of n
    ,L st p9 in P & A/.i = m9*'p9 & HT(m9*'p9,T) <= b,T by A1;
    hence
    ex m being Monomial of n,L, p being Polynomial of n,L st p in P & A/.
    i = m*'p;
  end;
  Sum A = f by A1;
  hence thesis by A2;
end;
