
theorem Th59: :: lemma 5.26, p. 202
  for n being Ordinal, T being connected TermOrder of n, L being
  Abelian add-associative right_complementable right_zeroed commutative
  associative well-unital distributive 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 implies f-g in P-Ideal
proof
  let n be Ordinal, T be connected TermOrder of n, L be Abelian
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, P
  be Subset of Polynom-Ring(n,L), f,g be Polynomial of n,L;
  reconsider f9 = f, g9 = g as Element of Polynom-Ring(n,L) by POLYNOM1:def 11;
  reconsider f9,g9 as Element of Polynom-Ring(n,L);
  set R = Polynom-Ring(n,L);
  reconsider x = -g as Element of R by POLYNOM1:def 11;
  reconsider x as Element of R;
  x + g9 = -g + g by POLYNOM1:def 11
    .= 0_(n,L) by Th3
    .= 0.R by POLYNOM1:def 11;
  then
A1: -g9 = -g by RLVECT_1:6;
  assume PolyRedRel(P,T) reduces f,g;
  then f,g are_convertible_wrt PolyRedRel(P,T) by REWRITE1:25;
  then
A2: f9,g9 are_congruent_mod P-Ideal by Th57;
  f - g = f + (-g) by POLYNOM1:def 7
    .= f9 + (-g9) by A1,POLYNOM1:def 11
    .= f9 - g9;
  hence thesis by A2;
end;
