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 G is_Groebner_basis_wrt T holds
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
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 G is_Groebner_basis_wrt T holds
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
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 G is_Groebner_basis_wrt T holds
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
let G be Subset of (Polynom-Ring n,L); :: thesis: ( G is_Groebner_basis_wrt T implies 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 )
assume A1: G is_Groebner_basis_wrt T ; :: 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
set R = PolyRedRel G,T;
A2: 0_ n,L = 0. (Polynom-Ring n,L) by POLYNOM1:def 27;

per cases ( G = {} or G <> {} ) ;

suppose G = {} ; :: 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

hence 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: verum

end;

suppose G <> {} ; :: 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

then reconsider G = G as non empty Subset of (Polynom-Ring n,L) ;
A3: PolyRedRel G,T is locally-confluent by A1, GROEB_1:def 3;

now

A4: now

assume not 0_ n,L is_a_normal_form_wrt PolyRedRel G,T ; :: thesis: contradiction
then consider b being set such that
A5: [(0_ n,L),b] in PolyRedRel G,T by REWRITE1:def 5;
consider f1, f2 being set such that
A6: f1 in NonZero (Polynom-Ring n,L) and
f2 in the carrier of (Polynom-Ring n,L) and
A7: [(0_ n,L),b] = [f1,f2] by A5, ZFMISC_1:def 2;
A8: f1 = [(0_ n,L),b] `1 by A7, MCART_1:def 1
.= 0_ n,L by MCART_1:def 1 ;
not f1 in {(0_ n,L)} by A2, A6, XBOOLE_0:def 5;
hence contradiction by A8, TARSKI:def 1; :: thesis: verum

end;

let g1, g2, h be Polynomial of n,L; :: thesis: ( g1 in G & g2 in G & h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T implies h = 0_ n,L )
assume that
A9: ( g1 in G & g2 in G ) and
A10: h is_a_normal_form_of S-Poly g1,g2,T, PolyRedRel G,T ; :: thesis: h = 0_ n,L
S-Poly g1,g2,T in G -Ideal by A9, Th22;
then PolyRedRel G,T reduces S-Poly g1,g2,T, 0_ n,L by A3, GROEB_1:15;
then A11: S-Poly g1,g2,T, 0_ n,L are_convertible_wrt PolyRedRel G,T by REWRITE1:26;
PolyRedRel G,T reduces S-Poly g1,g2,T,h by A10, REWRITE1:def 6;
then h, S-Poly g1,g2,T are_convertible_wrt PolyRedRel G,T by REWRITE1:26;
then A12: h, 0_ n,L are_convertible_wrt PolyRedRel G,T by A11, REWRITE1:31;
h is_a_normal_form_wrt PolyRedRel G,T by A10, REWRITE1:def 6;
hence h = 0_ n,L by A3, A4, A12, REWRITE1:def 19; :: thesis: verum

end;

hence 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: verum

end;

end;