
theorem :: corollary 5.34, p. 205
  for n being Element of NAT, T being connected admissible TermOrder of
  n, L being Abelian add-associative right_complementable right_zeroed
  commutative associative well-unital distributive almost_left_invertible non
  degenerated non empty doubleLoopStr, P being Subset of Polynom-Ring(n,L) st
ex p being Polynomial of n,L st p in P & P-Ideal = {p}-Ideal holds PolyRedRel(P
  ,T) is locally-confluent
proof
  let n be Element of NAT, T be connected admissible TermOrder of n, L be
  Abelian add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, P be Subset of Polynom-Ring(n,L);
  set R = PolyRedRel(P,T);
  assume ex p being Polynomial of n,L st p in P & P-Ideal = {p}-Ideal;
  then consider p being Polynomial of n,L such that
A1: p in P and
A2: P-Ideal = {p}-Ideal;
  now
    set Rp = PolyRedRel({p},T);
    reconsider Rp as locally-confluent strongly-normalizing Relation by Th10;
    let a,b,c being object;
    assume that
A3: [a,b] in R and
A4: [a,c] in R;
    a,b are_convertible_wrt R by A3,REWRITE1:29;
    then
A5: b,a are_convertible_wrt R by REWRITE1:31;
    consider pa,pb being object such that
    pa in NonZero Polynom-Ring(n,L) and
A6: pb in the carrier of Polynom-Ring(n,L) and
A7: [a,b] = [pa,pb] by A3,ZFMISC_1:def 2;
    reconsider pb as Polynomial of n,L by A6,POLYNOM1:def 11;
A8: pb = b by A7,XTUPLE_0:1;
    consider pa9,pc being object such that
    pa9 in NonZero Polynom-Ring(n,L) and
A9: pc in the carrier of Polynom-Ring(n,L) and
A10: [a,c] = [pa9,pc] by A4,ZFMISC_1:def 2;
    reconsider pc as Polynomial of n,L by A9,POLYNOM1:def 11;
A11: pc = c by A10,XTUPLE_0:1;
    reconsider pb9 = pb, pc9 = pc as Element of Polynom-Ring(n,L) by
POLYNOM1:def 11;
    reconsider pc9,pb9 as Element of Polynom-Ring(n,L);
    a,c are_convertible_wrt R by A4,REWRITE1:29;
    then pb9,pc9 are_congruent_mod {p}-Ideal by A2,A8,A11,A5,POLYRED:57
,REWRITE1:30;
    then pb,pc are_convertible_wrt PolyRedRel({p},T) by POLYRED:58;
    then b,c are_convergent_wrt Rp by A8,A11,REWRITE1:def 23;
    then consider d being object such that
A12: Rp reduces b,d & Rp reduces c,d by REWRITE1:def 7;
    for u being object holds u in {p} implies u in P by A1,TARSKI:def 1;
    then {p} c= P;
    then R reduces b,d & R reduces c,d by A12,Th4,REWRITE1:22;
    hence b,c are_convergent_wrt R by REWRITE1:def 7;
  end;
  hence thesis by REWRITE1:def 24;
end;
