
theorem Th46:
  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 f reduces_to f1,{p}
  ,T & for b1 being bag of n st b1 in Support g holds not(HT(p,T) divides b1)
  holds f + g reduces_to f1 + g,{p},T
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: f reduces_to f1,{p},T and
A2: for b1 being bag of n st b1 in Support g holds not HT(p,T) divides b1;
  consider q being Polynomial of n,L such that
A3: q in {p} and
A4: f reduces_to f1,q,T by A1,POLYRED:def 7;
  p = q by A3,TARSKI:def 1;
  then consider br being bag of n such that
A5: f reduces_to f1,p,br,T by A4,POLYRED:def 6;
  consider s being bag of n such that
A6: s + HT(p,T) = br and
A7: f1 = f -(f.br/HC(p,T))*(s*'p) by A5,POLYRED:def 5;
A8: not br in Support g by A6,TERMORD:1,A2;
A9: br is Element of Bags n by PRE_POLY:def 12;
A10: p in {p} by TARSKI:def 1;
A11: br in Support f by A5,POLYRED:def 5;
A12: (f+g).br = f.br + g.br by POLYNOM1:15
    .= f.br + 0.L by A8,A9,POLYNOM1:def 4
    .= f.br by RLVECT_1:def 4;
A13: p <> 0_(n,L) by A5,POLYRED:def 5;
  now
    per cases;
    case
      f + g = 0_(n,L);
      then (f + g) - f = -f by Th14;
      then (f + g) + -f = -f by POLYNOM1:def 7;
      then (f + -f) + g = -f by POLYNOM1:21;
      then 0_(n,L) + g = -f by POLYRED:3;
      then g = -f by POLYRED:2;
      hence contradiction by A11,A8,GROEB_1:5;
    end;
    case
A14:  f + g <> 0_(n,L);
      set g1 = (f+g) - ((f+g).br/HC(p,T))*(s*'p);
      (f+g).br <> 0.L by A11,A12,POLYNOM1:def 4;
      then br in Support(f+g) by A11,POLYNOM1:def 4;
      then f+g reduces_to g1,p,br,T by A13,A6,A14,POLYRED:def 5;
      then
A15:  f+g reduces_to g1,p,T by POLYRED:def 6;
      g1 = (f+g) + (-((f.br/HC(p,T))*(s*'p))) by A12,POLYNOM1:def 7
        .= (f + -((f.br/HC(p,T))*(s*'p))) + g by POLYNOM1:21
        .= f1 + g by A7,POLYNOM1:def 7;
      hence thesis by A10,A15,POLYRED:def 7;
    end;
  end;
  hence thesis;
end;
