let n be Ordinal; :: thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring n,L) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
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 )
let T be connected TermOrder of n; :: thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring n,L) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
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 )
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; :: thesis: for P being Subset of (Polynom-Ring n,L) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
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 )
let P be Subset of (Polynom-Ring n,L); :: thesis: ( ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) implies 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 ) )
assume A1: for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ; :: thesis: 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 )

now

let b be bag of n; :: thesis: ( b in HT (P -Ideal ),T implies ex b9 being bag of n st
( b9 in HT P,T & b9 divides b ) )
assume b in HT (P -Ideal ),T ; :: thesis: ex b9 being bag of n st
( b9 in HT P,T & b9 divides b )
then consider p being Polynomial of n,L such that
A2: b = HT p,T and
A3: p in P -Ideal and
A4: p <> 0_ n,L ;
reconsider p = p as non-zero Polynomial of n,L by A4, POLYNOM7:def 2;
p is_top_reducible_wrt P,T by A1, A3;
then consider u being Polynomial of n,L such that
A5: u in P and
A6: p is_top_reducible_wrt u,T by POLYRED:def 12;
consider q being Polynomial of n,L such that
A7: p top_reduces_to q,u,T by A6, POLYRED:def 11;
A8: p reduces_to q,u, HT p,T,T by A7, POLYRED:def 10;
then u <> 0_ n,L by POLYRED:def 5;
then A9: HT u,T in { (HT r,T) where r is Polynomial of n,L : ( r in P & r <> 0_ n,L ) } by A5;
ex s being bag of n st
( s + (HT u,T) = HT p,T & q = p - (((p . (HT p,T)) / (HC u,T)) * (s *' u)) ) by A8, POLYRED:def 5;
hence ex b9 being bag of n st
( b9 in HT P,T & b9 divides b ) by A2, A9, PRE_POLY:50; :: thesis: verum

end;

hence 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 ) ; :: thesis: verum