
theorem
  for L be add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr holds the doubleLoopStr of
  Polynom-Algebra L = Polynom-Ring L
proof
  let L be add-associative right_zeroed right_complementable distributive non
  empty doubleLoopStr;
A1: ex A being non empty Subset of Formal-Series L st A = the carrier of
  Polynom-Ring L & Polynom-Algebra L = GenAlg A by Def6;
  then
A2: the carrier of Polynom-Algebra L = the carrier of Polynom-Ring L by Th21
,Th22;
A3: the carrier of Polynom-Algebra L c= the carrier of Formal-Series L by A1
,Def3;
A4: for x being Element of Polynom-Algebra L for y being Element of
  Polynom-Ring L holds (the addF of Polynom-Algebra L).(x,y)=(the addF of
  Polynom-Ring L).(x,y)
  proof
    let x be Element of Polynom-Algebra L, y be Element of Polynom-Ring L;
    reconsider y1=y as Element of Polynom-Algebra L by A1,Th21,Th22;
    reconsider y9=y1 as Element of Formal-Series L by A1,TARSKI:def 3;
    reconsider x9=x as Element of Formal-Series L by A3;
    reconsider p=x as AlgSequence of L by A2,POLYNOM3:def 10;
    reconsider x1=x as Element of Polynom-Ring L by A1,Th21,Th22;
    reconsider q=y as AlgSequence of L by POLYNOM3:def 10;
    thus (the addF of Polynom-Algebra L).(x,y) = x+y1 .= x9 + y9 by A1,Th15
      .= p+q by Def2
      .= x1+y by POLYNOM3:def 10
      .= (the addF of Polynom-Ring L).(x,y);
  end;
  now
    let x be Element of Polynom-Algebra L, y be Element of Polynom-Ring L;
    reconsider y1=y as Element of Polynom-Algebra L by A1,Th21,Th22;
    reconsider y9=y1 as Element of Formal-Series L by A1,TARSKI:def 3;
    reconsider x9=x as Element of Formal-Series L by A3;
    reconsider p=x as AlgSequence of L by A2,POLYNOM3:def 10;
    reconsider x1=x as Element of Polynom-Ring L by A1,Th21,Th22;
    reconsider q=y as AlgSequence of L by POLYNOM3:def 10;
    thus (the multF of Polynom-Algebra L).(x,y) = x*y1 .= x9 * y9 by A1,Th16
      .= p*'q by Def2
      .= x1*y by POLYNOM3:def 10
      .= (the multF of Polynom-Ring L).(x,y);
  end;
  then
A5: the multF of Polynom-Algebra L = the multF of Polynom-Ring L by A2,
BINOP_1:2;
A6: 1.Polynom-Algebra L = 1.Formal-Series L by A1,Def3
    .= 1_.(L) by Def2
    .= 1.Polynom-Ring L by POLYNOM3:def 10;
  0.Polynom-Algebra L = 0.(Formal-Series L) by A1,Def3
    .= 0_.(L) by Def2
    .= 0.Polynom-Ring L by POLYNOM3:def 10;
  hence thesis by A2,A4,A5,A6,BINOP_1:2;
end;
