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 Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
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 Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
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 Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
let G be Subset of (Polynom-Ring (n,L)); :: thesis: ( ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) implies for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) )
set R = PolyRedRel (G,T);
assume A1: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ; :: thesis: for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)

now

let g1, g2 be Polynomial of n,L; :: thesis: ( g1 in G & g2 in G implies PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) )

now

per cases ( S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) or not S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ) ;

case S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ; :: thesis: S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T)

hence S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) by REWRITE1:def 14; :: thesis: verum

end;

case not S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ; :: thesis: S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T)

hence S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) by REWRITE1:47; :: thesis: verum

end;

end;

end;

then consider q being set such that
A2: q is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) by REWRITE1:def 11;
PolyRedRel (G,T) reduces S-Poly (g1,g2,T),q by A2, REWRITE1:def 6;
then reconsider q = q as Polynomial of n,L by Lm1;
assume ( g1 in G & g2 in G ) ; :: thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
then q = 0_ (n,L) by A1, A2;
hence PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A2, REWRITE1:def 6; :: thesis: verum

end;

hence for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ; :: thesis: verum