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, g being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A, B being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds
A ^ B is_Standard_Representation_of f + g,P,b,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, g being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L))
for A, B being LeftLinearCombination of P
for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds
A ^ B is_Standard_Representation_of f + g,P,b,T

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

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

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

let A, B be LeftLinearCombination of P; :: thesis: for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds
A ^ B is_Standard_Representation_of f + g,P,b,T

let b be bag of n; :: thesis: ( A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T implies A ^ B is_Standard_Representation_of f + g,P,b,T )
assume that
A1: A is_Standard_Representation_of f,P,b,T and
A2: B is_Standard_Representation_of g,P,b,T ; :: thesis: A ^ B is_Standard_Representation_of f + g,P,b,T
A3: now :: thesis: for i being Element of NAT st i in dom (A ^ B) holds
ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T )
let i be Element of NAT ; :: thesis: ( i in dom (A ^ B) implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) )

assume A4: i in dom (A ^ B) ; :: thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T )

now :: thesis: ( ( i in dom A & ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) ) or ( ex k being Nat st
( k in dom B & i = (len A) + k ) & ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) ) )
per cases ( i in dom A or ex k being Nat st
( k in dom B & i = (len A) + k ) )
by ;
case A5: i in dom A ; :: thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T )

(A ^ B) /. i = (A ^ B) . i by
.= A . i by
.= A /. i by ;
hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) by A1, A5; :: thesis: verum
end;
case ex k being Nat st
( k in dom B & i = (len A) + k ) ; :: thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T )

then consider k being Nat such that
A6: k in dom B and
A7: i = (len A) + k ;
(A ^ B) /. i = (A ^ B) . i by
.= B . k by
.= B /. k by ;
hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) by A2, A6; :: thesis: verum
end;
end;
end;
hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) ; :: thesis: verum
end;
( f = Sum A & g = Sum B ) by A1, A2;
then f + g = (Sum A) + (Sum B) by POLYNOM1:def 11
.= Sum (A ^ B) by RLVECT_1:41 ;
hence A ^ B is_Standard_Representation_of f + g,P,b,T by A3; :: thesis: verum