
theorem Th50: :: lemma 5.25 (ii), p. 200
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non trivial
doubleLoopStr, P being Subset of Polynom-Ring(n,L), f,g being Polynomial of n,
  L holds PolyRedRel(P,T) reduces f-g,0_(n,L) implies f,g are_convergent_wrt
  PolyRedRel(P,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 Abelian almost_left_invertible non trivial doubleLoopStr, P be
  Subset of Polynom-Ring(n,L), f,g be Polynomial of n,L;
  assume PolyRedRel(P,T) reduces f-g,0_(n,L);
  then consider f1,g1 being Polynomial of n,L such that
A1: f1 - g1 = 0_(n,L) and
A2: PolyRedRel(P,T) reduces f,f1 & PolyRedRel(P,T) reduces g,g1 by Th49;
  g1 = (f1 - g1) + g1 by A1,Th2
    .= (f1 + -g1) + g1 by POLYNOM1:def 7
    .= f1 + (-g1 + g1) by POLYNOM1:21
    .= f1 + 0_(n,L) by Th3
    .= f1 by POLYNOM1:23;
  hence thesis by A2,REWRITE1:def 7;
end;
