let n be Element of NAT ; :: thesis: for T being connected admissible TermOrder of n
for L being non degenerated non trivial 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 not 0_ n,L in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint holds
PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L
let T be connected admissible TermOrder of n; :: thesis: for L being non degenerated non trivial 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 not 0_ n,L in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint holds
PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L
let L be non degenerated non trivial 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 not 0_ n,L in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint holds
PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L
let G be Subset of (Polynom-Ring n,L); :: thesis: ( not 0_ n,L in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint holds
PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ) implies for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L )
assume A1: not 0_ n,L in G ; :: thesis: ( ex g1, g2 being Polynomial of n,L st
( g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & not PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ) or for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L )
assume A2: for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint holds
PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ; :: thesis: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L
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

proof

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 )
assume A3: ( g1 in G & g2 in G ) ; :: thesis: PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L

now

per cases ( HT g1,T, HT g2,T are_disjoint or not HT g1,T, HT g2,T are_disjoint ) ;

case HT g1,T, HT g2,T are_disjoint ; :: thesis: PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L

then A4: PolyRedRel {g1,g2},T reduces S-Poly g1,g2,T, 0_ n,L by Th56;

now

let u be set ; :: thesis: ( u in {g1,g2} implies u in G )
assume A5: u in {g1,g2} ; :: thesis: u in G

now

per cases ( u = g1 or u = g2 ) by A5, TARSKI:def 2;

case u = g1 ; :: thesis: u in G

hence u in G by A3; :: thesis: verum

end;

case u = g2 ; :: thesis: u in G

hence u in G by A3; :: thesis: verum

end;

end;

end;

hence u in G ; :: thesis: verum

end;

then {g1,g2} c= G by TARSKI:def 3;
hence PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L by A4, GROEB_1:4, REWRITE1:23; :: thesis: verum

end;

case not HT g1,T, HT g2,T are_disjoint ; :: thesis: PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L

hence PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L by A2, A3; :: thesis: verum

end;

end;

end;

hence PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L ; :: thesis: verum

end;

then G is_Groebner_basis_wrt T by A1, GROEB_2:30;
hence for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT g1,T, HT g2,T are_disjoint & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T holds
h = 0_ n,L by GROEB_2:28; :: thesis: verum