 reserve o,o1,o2 for object;
 reserve n for Ordinal;
 reserve R,L for non degenerated comRing;
 reserve b for bag of 1;

theorem Th45:
   for R holds BSPoly(R) is one-to-one onto
   proof
     let R;
      BSPoly(R) is onto
      proof
        for o holds o in [#]Polynom-Ring R implies o in rng (BSPoly(R))
        proof
          let o;
          assume
A1:       o in [#]Polynom-Ring R; then
A2:       o is finite-Support sequence of R by POLYNOM3:def 10;
          o in Formal-Series R by A2,POLYALG1:def 2; then
          consider f1 being sequence of R such that
A3:       f1 = o & (SBFSeri(R)).o = f1*(BagN1) by Def5;
          f1 is finite-Support sequence of R by A3,A1,POLYNOM3:def 10; then
A4:       Support f1 is finite by Th3;
          card (Support f1) = card Support(f1*BagN1) by Th41; then
          Support(f1*BagN1) is finite by A4; then
A5:       f1*BagN1 is Polynomial of 1,R by POLYNOM1:def 5; then
A6:       (SBFSeri(R)).o in Polynom-Ring(1,R) by A3,POLYNOM1:def 11;
          set p = (SBFSeri(R)).o;
          dom((BSFSeri(R))*(SBFSeri(R)))
          = [#]Formal-Series R by FUNCT_2:def 1; then
A7:       o in dom((BSFSeri(R))*(SBFSeri(R))) by A2,POLYALG1:def 2; then
          o in dom SBFSeri(R) by FUNCT_2:123; then
          o in dom((BSFSeri(R))") by Th32; then
A8:       o in rng BSFSeri(R) by FUNCT_1:33;
          p in Polynom-Ring(1,R) by A5,A3,POLYNOM1:def 11; then
A9:      p in dom BSPoly(R) by FUNCT_2:def 1;
          (BSPoly(R)).p = (BSFSeri(R)).p by A6,FUNCT_1:49
          .= ((BSFSeri(R))*(SBFSeri(R))).o by A7,FUNCT_2:15
          .= ((BSFSeri(R))*(BSFSeri(R))").o by Th32 .= o by A8,FUNCT_1:35;
          hence thesis by A9,FUNCT_1:def 3;
        end; then
        [#]Polynom-Ring R c= rng(BSPoly(R));
        hence thesis by XBOOLE_0:def 10;
      end;
      hence thesis;
    end;
