
theorem
  for n being Element of NAT, T being connected admissible TermOrder of
  n, L being Abelian add-associative right_complementable right_zeroed
  commutative associative well-unital distributive almost_left_invertible non
degenerated non empty doubleLoopStr, p,q being Element of Polynom-Ring(n,L),
  G being non empty Subset of Polynom-Ring(n,L) st G is_Groebner_basis_wrt T
holds p,q are_congruent_mod G-Ideal iff nf(p,PolyRedRel(G,T)) = nf(q,PolyRedRel
  (G,T))
proof
  let n be Element of NAT, T be connected admissible TermOrder of n, L be
  Abelian add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, p,q be Element of Polynom-Ring(n,L), G be non empty
  Subset of Polynom-Ring(n,L);
  set R = PolyRedRel(G,T);
  assume G is_Groebner_basis_wrt T;
  then
A1: PolyRedRel(G,T) is locally-confluent;
  now
    nf(q,R) is_a_normal_form_of q,R by A1,REWRITE1:54;
    then R reduces q,nf(q,R) by REWRITE1:def 6;
    then
A2: nf(q,R),q are_convertible_wrt R by REWRITE1:25;
    nf(p,R) is_a_normal_form_of p,R by A1,REWRITE1:54;
    then R reduces p,nf(p,R) by REWRITE1:def 6;
    then
A3: p,nf(p,R) are_convertible_wrt R by REWRITE1:25;
    assume nf(p,PolyRedRel(G,T)) = nf(q,PolyRedRel(G,T));
    hence p,q are_congruent_mod G-Ideal by A3,A2,POLYRED:57,REWRITE1:30;
  end;
  hence thesis by A1,POLYRED:58,REWRITE1:55;
end;
