
theorem Th52:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non trivial
doubleLoopStr, p1,p2 being Polynomial of n,L st HT(p1,T),HT(p2,T) are_disjoint
  for b1,b2 being bag of n st b1 in Support Red(p1,T) & b2 in Support Red(p2,T)
  holds not(HT(p1,T) + b2 = HT(p2,T) + b1)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, p1
  ,p2 be Polynomial of n,L;
  assume
A1: HT(p1,T),HT(p2,T) are_disjoint;
A2: Support(Red(p1,T)) c= Support(p1) & Support(Red(p2,T)) c= Support(p2) by
TERMORD:35;
  let b1,b2 be bag of n;
  assume that
A3: b1 in Support Red(p1,T) and
A4: b2 in Support Red(p2,T);
  now
    assume b1 = HT(p1,T);
    then Red(p1,T).b1 = 0.L by TERMORD:39;
    hence contradiction by A3,POLYNOM1:def 4;
  end;
  hence thesis by A1,A3,A4,A2,Lm5;
end;
