
theorem
  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, f,f1,g,p being Polynomial of n,L st PolyRedRel({p},T) reduces f
  ,f1 & for b1 being bag of n st b1 in Support g holds not(HT(p,T) divides b1)
  holds PolyRedRel({p},T) reduces f+g,f1+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, f,f1,g,p be Polynomial of n,L;
  assume that
A1: PolyRedRel({p},T) reduces f,f1 and
A2: for b1 being bag of n st b1 in Support g holds not HT(p,T) divides b1;
  consider R being RedSequence of PolyRedRel({p},T) such that
A3: R.1 = f and
A4: R.len R = f1 by A1,REWRITE1:def 3;
  defpred P[Nat] means for q being Polynomial of n,L st q = R.$1
  holds PolyRedRel({p},T) reduces f+g,q+g;
A5: now
    let k be Element of NAT;
    assume
A6: 1 <= k & k < len R;
    then 1 < len R by XXREAL_0:2;
    then reconsider h = R.k as Polynomial of n,L by A6,Lm4;
    assume P[k];
    then
A7: PolyRedRel({p},T) reduces f+g,h+g;
    now
      let q be Polynomial of n,L;
      assume
A8:   q = R.(k+1);
      1 <= k+1 & k+1 <= len R by A6,NAT_1:13;
      then
A9:   k+1 in dom R by FINSEQ_3:25;
      k in dom R by A6,FINSEQ_3:25;
      then [R.k, R.(k+1)] in PolyRedRel({p},T) by A9,REWRITE1:def 2;
      then h reduces_to q,{p},T by A8,POLYRED:def 13;
      then h+g reduces_to q+g,{p},T by A2,Th46;
      then [h+g,q+g] in PolyRedRel({p},T) by POLYRED:def 13;
      then PolyRedRel({p},T) reduces h+g,q+g by REWRITE1:15;
      hence PolyRedRel({p},T) reduces f+g,q+g by A7,REWRITE1:16;
    end;
    hence P[k+1];
  end;
A10: 1 <= len R by NAT_1:14;
A11: P[1] by A3,REWRITE1:12;
  for i being Element of NAT st 1<=i & i<=len R holds P[i] from
  INT_1:sch 7(A11,A5);
  hence thesis by A4,A10;
end;
