let n be Ordinal; :: thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring n,L) st PolyRedRel P,T reduces f, 0_ n,L holds
f has_a_Standard_Representation_of P,T
let T be connected admissible TermOrder of n; :: thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring n,L) st PolyRedRel P,T reduces f, 0_ n,L holds
f has_a_Standard_Representation_of P,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring n,L) st PolyRedRel P,T reduces f, 0_ n,L holds
f has_a_Standard_Representation_of P,T
let f be Polynomial of n,L; :: thesis: for P being non empty Subset of (Polynom-Ring n,L) st PolyRedRel P,T reduces f, 0_ n,L holds
f has_a_Standard_Representation_of P,T
let P be non empty Subset of (Polynom-Ring n,L); :: thesis: ( PolyRedRel P,T reduces f, 0_ n,L implies f has_a_Standard_Representation_of P,T )
reconsider f9 = f as Element of (Polynom-Ring n,L) by POLYNOM1:def 27;
A1: 0_ n,L = 0. (Polynom-Ring n,L) by POLYNOM1:def 27;
assume PolyRedRel P,T reduces f, 0_ n,L ; :: thesis: f has_a_Standard_Representation_of P,T
then consider A being LeftLinearCombination of P such that
A2: f9 = (0. (Polynom-Ring n,L)) + (Sum A) and
A3: for i being Element of NAT st i in dom A holds
ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & A . i = m *' p & HT (m *' p),T <= HT f,T,T ) by A1, Lm5;

A4: now

let i be Element of NAT ; :: thesis: ( i in dom A implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & A /. i = m *' p & HT (m *' p),T <= HT f,T,T ) )
assume 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 /. i = m *' p & HT (m *' p),T <= HT f,T,T )
then ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & A . i = m *' p & HT (m *' p),T <= HT f,T,T ) by A3;
hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st
( p in P & A /. i = m *' p & HT (m *' p),T <= HT f,T,T ) by A5, PARTFUN1:def 8; :: thesis: verum

end;

f = Sum A by A2, RLVECT_1:def 7;
then A is_Standard_Representation_of f,P, HT f,T,T by A4, Def7;
then A is_Standard_Representation_of f,P,T by Def8;
hence f has_a_Standard_Representation_of P,T by Def10; :: thesis: verum