
theorem Th57:
  for L being Abelian add-associative right_zeroed
  right_complementable commutative distributive well-unital non empty
  doubleLoopStr holds poly_with_roots(EmptyBag the carrier of L) = <%1.L%>
proof
  let L be Abelian add-associative right_zeroed right_complementable
  commutative distributive well-unital non empty doubleLoopStr;
  set b = EmptyBag the carrier of L;
  consider f being FinSequence of (the carrier of Polynom-Ring L)*, s being
  FinSequence of L such that
A1: len f = card support b and
  s = canFS(support b) and
  for i being Element of NAT st i in dom f holds f.i = fpoly_mult_root(s/.
  i,b.(s/.i)) and
A2: poly_with_roots(b) = Product FlattenSeq f by Def10;
  f = <*>((the carrier of Polynom-Ring L)*) by A1;
  then
  1_Polynom-Ring L = 1.Polynom-Ring L & FlattenSeq f = <*>(the carrier of
  Polynom-Ring L);
  then Product FlattenSeq f = 1.Polynom-Ring L by GROUP_4:8
    .= 1_.(L) by POLYNOM3:37;
  hence thesis by A2,Th28;
end;
