
theorem Th3:
  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, f,
  g being Polynomial of n,L, P,Q being Subset of Polynom-Ring(n,L) st P c= Q
  holds f reduces_to g,P,T implies f reduces_to g,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, f,g be
  Polynomial of n,L, P,Q be Subset of Polynom-Ring(n,L);
  assume
A1: P c= Q;
  assume f reduces_to g,P,T;
  then ex p being Polynomial of n,L st p in P & f reduces_to g,p,T by
POLYRED:def 7;
  hence thesis by A1,POLYRED:def 7;
end;
