let n be Element of NAT ; :: 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 G being non empty Subset of (Polynom-Ring (n,L)) holds
( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds
f has_a_Standard_Representation_of G,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 G being non empty Subset of (Polynom-Ring (n,L)) holds
( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds
f has_a_Standard_Representation_of G,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 G being non empty Subset of (Polynom-Ring (n,L)) holds
( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds
f has_a_Standard_Representation_of G,T )
let P be non empty Subset of (Polynom-Ring (n,L)); :: thesis: ( P is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in P -Ideal holds
f has_a_Standard_Representation_of P,T )

A1: now

assume for f being non-zero Polynomial of n,L st f in P -Ideal holds
f has_a_Standard_Representation_of P,T ; :: thesis: P is_Groebner_basis_wrt T
then for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T by Th44;
then for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) by GROEB_1:18;
then HT ((P -Ideal),T) c= multiples (HT (P,T)) by GROEB_1:19;
then PolyRedRel (P,T) is locally-confluent by GROEB_1:20;
hence P is_Groebner_basis_wrt T by GROEB_1:def 3; :: thesis: verum

end;

A2: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def 10;

now

assume P is_Groebner_basis_wrt T ; :: thesis: for f being non-zero Polynomial of n,L st f in P -Ideal holds
f has_a_Standard_Representation_of P,T
then PolyRedRel (P,T) is locally-confluent by GROEB_1:def 3;
hence for f being non-zero Polynomial of n,L st f in P -Ideal holds
f has_a_Standard_Representation_of P,T by A2, Th43, GROEB_1:15; :: thesis: verum

end;

hence ( P is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in P -Ideal holds
f has_a_Standard_Representation_of P,T ) by A1; :: thesis: verum