
theorem Th11:
for L being Abelian add-associative right_zeroed right_complementable
            well-unital associative commutative distributive
            almost_left_invertible non degenerated doubleLoopStr
for f being FinSequence of Polynom-Ring(L)
st for i being Nat st i in dom f ex z being Element of L st f.i = rpoly(1,z)
for p being Polynomial of L st p = Product f
holds p <> 0_.(L)
proof
let L be Abelian add-associative right_zeroed right_complementable
         well-unital associative commutative distributive
         almost_left_invertible non degenerated doubleLoopStr;
let f be FinSequence of Polynom-Ring(L);
assume A1:
for i being Nat st i in dom f ex z being Element of L st f.i = rpoly(1,z);
let p be Polynomial of L;
assume A2: p = Product f;
defpred P[Nat] means
for f being FinSequence of Polynom-Ring(L)
st len f = $1 &
   for i being Nat st i in dom f ex z being Element of L st f.i = rpoly(1,z)
for p being Polynomial of L st p = Product f
holds p <> 0_.(L);

now let f be FinSequence of Polynom-Ring(L);
  assume A3: len f = 0 &
  for i being Nat st i in dom f ex z being Element of L st f.i = rpoly(1,z);
  let p be Polynomial of L;
  assume A4: p = Product f;
  f = <*>(the carrier of Polynom-Ring(L)) by A3;
  then p = 1_(Polynom-Ring(L)) by A4,GROUP_4:8
        .= 1.(Polynom-Ring(L));
  then p <> 0.(Polynom-Ring(L));
  hence p <> 0_.(L) by POLYNOM3:def 10;
  end;
then A5: P[0];
A6:now let n be Nat;
  assume A7: P[n];
  now let f be FinSequence of Polynom-Ring(L);
  assume A8: len f = n+1 &
  for i being Nat st i in dom f ex z being Element of L st f.i = rpoly(1,z);
  let p be Polynomial of L;
  assume A9: p = Product f;
  f <> {} by A8;
  then consider g being FinSequence, u being object such that
  A10: f = g ^ <*u*> by FINSEQ_1:46;
  reconsider g as FinSequence of Polynom-Ring(L) by A10,FINSEQ_1:36;
  A11: dom f = Seg(n+1) by A8,FINSEQ_1:def 3;
  1 <= n+1 by NAT_1:11;
  then A12: n+1 in dom f by A11;
  A13: n+1 = len g + len <*u*> by A8,A10,FINSEQ_1:22
          .= len g + 1 by FINSEQ_1:40;
  then f.(n+1) = u by A10,FINSEQ_1:42;
  then consider z being Element of L such that
  A14: u = rpoly(1,z) by A8,A12;
  reconsider u as Element of Polynom-Ring(L) by A14,POLYNOM3:def 10;
  reconsider q = Product g as Polynomial of L by POLYNOM3:def 10;
  A15: Product f = (Product g) * u by A10,GROUP_4:6;
  A16: u <> 0.(Polynom-Ring(L)) by A14,POLYNOM3:def 10;
  now let i be Nat;
    assume A17: i in dom g;
    then A18: g.i = f.i by A10,FINSEQ_1:def 7;
    dom g c= dom f by A10,FINSEQ_1:26;
    hence ex z being Element of L st g.i = rpoly(1,z) by A17,A18,A8;
    end;
  then q <> 0_.(L) by A7,A13;
  then q <> 0.(Polynom-Ring(L)) by POLYNOM3:def 10;
  then p <> 0.(Polynom-Ring(L)) by A9,A16,A15,VECTSP_2:def 1;
  hence p <> 0_.(L) by POLYNOM3:def 10;
  end;
  hence P[n+1];
 end;
A19: for n being Nat holds P[n] from NAT_1:sch 2(A5,A6);
 len f is Nat;
hence thesis by A1,A2,A19;
end;
