
theorem Th4:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, P,
  Q being Subset of Polynom-Ring(n,L) holds P c= Q implies PolyRedRel(P,T) c=
  PolyRedRel(Q,T)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_complementable right_zeroed commutative associative well-unital
  distributive almost_left_invertible non trivial doubleLoopStr, P,Q being
  Subset of Polynom-Ring(n,L);
  assume
A1: P c= Q;
  now
    let u be object;
    assume
A2: u in PolyRedRel(P,T);
    then consider p,q being object such that
A3: p in NonZero Polynom-Ring(n,L) and
A4: q in the carrier of Polynom-Ring(n,L) and
A5: u = [p,q] by ZFMISC_1:def 2;
    reconsider q as Polynomial of n,L by A4,POLYNOM1:def 11;
    0_(n,L) = 0.Polynom-Ring(n,L) by POLYNOM1:def 11;
    then not p in {0_(n,L)} by A3,XBOOLE_0:def 5;
    then p <> 0_(n,L) by TARSKI:def 1;
    then reconsider p as non-zero Polynomial of n,L by A3,POLYNOM1:def 11
,POLYNOM7:def 1;
    p reduces_to q,P,T by A2,A5,POLYRED:def 13;
    then p reduces_to q,Q,T by A1,Th3;
    hence u in PolyRedRel(Q,T) by A5,POLYRED:def 13;
  end;
  hence thesis;
end;
