
theorem Th51: :: 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_convertible_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;
  set R = PolyRedRel(P,T);
  assume PolyRedRel(P,T) reduces f-g,0_(n,L);
  then f,g are_convergent_wrt R by Th50;
  then consider h being object such that
A1: R reduces f,h and
A2: R reduces g,h by REWRITE1:def 7;
  R~ reduces h,g by A2,REWRITE1:24;
  then
A3: R \/ R~ reduces h,g by REWRITE1:22,XBOOLE_1:7;
  R \/ R~ reduces f,h by A1,REWRITE1:22,XBOOLE_1:7;
  then R \/ R~ reduces f,g by A3,REWRITE1:16;
  hence thesis by REWRITE1:def 4;
end;
