
theorem Th22:
  for L being add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr for A being non empty Subset of
  Formal-Series L st A = the carrier of Polynom-Ring L holds A is opers_closed
proof
  let L be add-associative right_zeroed right_complementable distributive non
  empty doubleLoopStr;
  set B = Formal-Series L;
  let A be non empty Subset of Formal-Series L such that
A1: A = the carrier of Polynom-Ring L;
A2: for a being Element of L, v being Element of B st v in A holds a * v in A
  proof
    let a be Element of L, v being Element of B;
    assume v in A;
    then reconsider p = v as AlgSequence of L by A1,POLYNOM3:def 10;
    reconsider a9 = a as Element of L;
    a * v = a9 * p by Def2;
    hence thesis by A1,POLYNOM3:def 10;
  end;
  for v,u being Element of B st v in A & u in A holds v + u in A
  proof
    let v,u be Element of B;
    assume v in A & u in A;
    then reconsider p = v, q = u as AlgSequence of L by A1,POLYNOM3:def 10;
    v + u = p + q by Def2;
    hence thesis by A1,POLYNOM3:def 10;
  end;
  hence A is linearly-closed by A2,VECTSP_4:def 1;
  thus for u,v being Element of B st u in A & v in A holds u*v in A
  proof
    let u,v be Element of B;
    assume u in A & v in A;
    then reconsider p = u,q = v as AlgSequence of L by A1,POLYNOM3:def 10;
    u * v = p*'q by Def2;
    hence thesis by A1,POLYNOM3:def 10;
  end;
  1.B = 1_.(L) by Def2
    .= 1.Polynom-Ring L by POLYNOM3:def 10;
  hence 1.B in A by A1;
  0.B = 0_.(L) by Def2
    .= 0.(Polynom-Ring L) by POLYNOM3:def 10;
  hence thesis by A1;
end;
