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 non-zero Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds
ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 )

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 non-zero Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds
ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 )

let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; :: thesis: for f being non-zero Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds
ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 )

let f be non-zero Polynomial of n,L; :: thesis: for P being non empty Subset of (Polynom-Ring (n,L))
for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds
ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 )

let P be non empty Subset of (Polynom-Ring (n,L)); :: thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds
ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 )

let A be LeftLinearCombination of P; :: thesis: ( A is_MonomialRepresentation_of f implies ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 ) )

HC (f,T) <> 0. L ;
then A1: f . (HT (f,T)) <> 0. L by TERMORD:def 7;
assume A is_MonomialRepresentation_of f ; :: thesis: ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 )

then consider i being Element of NAT such that
A2: i in dom A and
A3: ex m being Monomial of n,L ex p being Polynomial of n,L st
( A . i = m *' p & p in P & (m *' p) . (HT (f,T)) <> 0. L ) by ;
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 ;
then A8: m <> 0_ (n,L) by POLYRED:5;
p <> 0_ (n,L) by ;
then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def 1;
reconsider m = m as non-zero Monomial of n,L by ;
HT (f,T) in Support (m *' p) by ;
then HT (f,T) <= HT ((m *' p),T),T by TERMORD:def 6;
hence ex i being Element of NAT ex m being non-zero Monomial of n,L ex 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 ) by A2, A4, A6; :: thesis: verum