let n be Ordinal; :: thesis: for T being connected TermOrder of n
for L being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T holds
A is_MonomialRepresentation_of f

let T be connected TermOrder of n; :: thesis: for L being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T holds
A is_MonomialRepresentation_of f

let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; :: thesis: for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T holds
A is_MonomialRepresentation_of f

let f be Polynomial of n,L; :: thesis: for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T holds
A is_MonomialRepresentation_of f

let P be non empty Subset of (Polynom-Ring (n,L)); :: thesis: for A being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T holds
A is_MonomialRepresentation_of f

let A be LeftLinearCombination of P; :: thesis: for b being bag of n st A is_Standard_Representation_of f,P,b,T holds
A is_MonomialRepresentation_of f

let b be bag of n; :: thesis: ( A is_Standard_Representation_of f,P,b,T implies A is_MonomialRepresentation_of f )
assume A1: A is_Standard_Representation_of f,P,b,T ; :: thesis:
A2: now :: thesis: for i being Element of NAT st i in dom A holds
ex m being Monomial of n,L ex p being Polynomial of n,L st
( p in P & A /. i = m *' p )
let i be Element of NAT ; :: thesis: ( i in dom A implies ex m being Monomial of n,L ex p being Polynomial of n,L st
( p in P & A /. i = m *' p ) )

assume i in dom A ; :: thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st
( p in P & A /. i = m *' p )

then ex m9 being non-zero Monomial of n,L ex 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 ex p being Polynomial of n,L st
( p in P & A /. i = m *' p ) ; :: thesis: verum
end;
Sum A = f by A1;
hence A is_MonomialRepresentation_of f by A2; :: thesis: verum