
theorem Th51:
  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,g being non-zero Polynomial of n,L holds PolyRedRel({
  g},T) reduces f*'g,0_(n,L)
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,g be non-zero Polynomial of n,L;
  defpred P[Nat] means for f being Polynomial of n,L st card(
  Support f) = $1 holds PolyRedRel({g},T) reduces f*'g,0_(n,L);
A1: ex n being Element of NAT st card(Support f) = n;
A2: now
    let k be Nat;
    assume
A3: P[k];
    now
      let f be Polynomial of n,L;
      set rf = Red(f,T);
      assume
A4:   card(Support f) = k+1;
      now
        assume f = 0_(n,L);
        then Support f = {} by POLYNOM7:1;
        hence contradiction by A4;
      end;
      then reconsider f1 = f as non-zero Polynomial of n,L by POLYNOM7:def 1;
      f1*'g reduces_to rf*'g,{g},T by Th47;
      then [f1*'g,rf*'g] in PolyRedRel({g},T) by POLYRED:def 13;
      then
A5:   PolyRedRel({g},T) reduces f*'g,rf*'g by REWRITE1:15;
      f1 <> 0_(n,L) by POLYNOM7:def 1;
      then Support f <> {} by POLYNOM7:1;
      then HT(f,T) in Support f by TERMORD:def 6;
      then for u being object st u in {HT(f,T)} holds u in Support f by
TARSKI:def 1;
      then
A6:   {HT(f,T)} c= Support f;
      Support rf = Support(f) \ {HT(f,T)} by TERMORD:36;
      then card(Support rf) = card(Support f) - card({HT(f,T)}) by A6,CARD_2:44
        .= (k + 1) - 1 by A4,CARD_1:30
        .= k + 0;
      then PolyRedRel({g},T) reduces rf*'g,0_(n,L) by A3;
      hence PolyRedRel({g},T) reduces f*'g,0_(n,L) by A5,REWRITE1:16;
    end;
    hence P[k+1];
  end;
  now
    let f be Polynomial of n,L;
    assume card(Support f) = 0;
    then Support f = {};
    then f = 0_(n,L) by POLYNOM7:1;
    then f*'g = 0_(n,L) by POLYRED:5;
    hence PolyRedRel({g},T) reduces f*'g,0_(n,L) by REWRITE1:12;
  end;
  then
A7: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A7,A2);
  hence thesis by A1;
end;
