
theorem Th49:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being 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 being Polynomial
  of n,L st PolyRedRel(P,T) reduces f,g holds PolyRedRel(P,T) reduces -f,-g
proof
  let n be Ordinal, T be connected admissible 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;
  then consider R being RedSequence of PolyRedRel(P,T) such that
A1: R.1 = f and
A2: R.len R = g by REWRITE1:def 3;
  defpred P[Nat] means for q being Polynomial of n,L st q = R.$1
  holds PolyRedRel(P,T) reduces -f,-q;
A3: 1 <= len R by NAT_1:14;
A4: now
    let k be Element of NAT;
    assume
A5: 1 <= k & k < len R;
    then 1 < len R by XXREAL_0:2;
    then reconsider p = R.k as Polynomial of n,L by A5,Lm4;
    assume P[k];
    then
A6: PolyRedRel(P,T) reduces -f,-p;
    now
      let q be Polynomial of n,L;
      assume
A7:   q = R.(k+1);
      1 <= k+1 & k+1 <= len R by A5,NAT_1:13;
      then
A8:   k+1 in dom R by FINSEQ_3:25;
      k in dom R by A5,FINSEQ_3:25;
      then [R.k, R.(k+1)] in PolyRedRel(P,T) by A8,REWRITE1:def 2;
      then p reduces_to q,P,T by A7,POLYRED:def 13;
      then consider l being Polynomial of n,L such that
A9:   l in P and
A10:  p reduces_to q,l,T by POLYRED:def 7;
      -p reduces_to -q,l,T by A10,Th45;
      then -p reduces_to -q,P,T by A9,POLYRED:def 7;
      then [-p,-q] in PolyRedRel(P,T) by POLYRED:def 13;
      then PolyRedRel(P,T) reduces -p,-q by REWRITE1:15;
      hence PolyRedRel(P,T) reduces -f,-q by A6,REWRITE1:16;
    end;
    hence P[k+1];
  end;
A11: P[1] by A1,REWRITE1:12;
  for i being Element of NAT st 1<=i & i<=len R holds P[i] from
   INT_1:sch 7(A11,A4);
  hence thesis by A2,A3;
end;
