
theorem Th39:
  for n being Ordinal, T being admissible connected TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non trivial
doubleLoopStr, f being non-zero Polynomial of n,L, P being non empty Subset of
  Polynom-Ring(n,L) holds f has_a_Standard_Representation_of P,T implies f
  is_top_reducible_wrt P,T
proof
  let n be Ordinal, T be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, f
  be non-zero Polynomial of n,L, P be non empty Subset of Polynom-Ring(n,L);
  assume f has_a_Standard_Representation_of P,T;
  then consider A being LeftLinearCombination of P such that
A1: A is_Standard_Representation_of f,P,T;
  consider i being Element of NAT, m being non-zero Monomial of n,L, p being
  non-zero Polynomial of n,L such that
A2: p in P and
  i in dom A and
  A/.i = m*'p and
A3: HT(f,T) = HT(m*'p,T) by A1,Th37;
  set s = HT(m,T);
A4: HT(f,T) = s + HT(p,T) by A3,TERMORD:31;
  set g = f - (f.HT(f,T)/HC(p,T)) * (s *' p);
A5: f <> 0_(n,L) by POLYNOM7:def 1;
  then Support f <> {} by POLYNOM7:1;
  then p <> 0_(n,L) & HT(f,T) in Support f by POLYNOM7:def 1,TERMORD:def 6;
  then f reduces_to g,p,HT(f,T),T by A5,A4,POLYRED:def 5;
  then f top_reduces_to g,p,T by POLYRED:def 10;
  then f is_top_reducible_wrt p,T by POLYRED:def 11;
  hence thesis by A2,POLYRED:def 12;
end;
