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

theorem Th32:
   for R holds (BSFSeri(R))" = SBFSeri(R)
   proof
     let R;
A1:  rng (BSFSeri(R)) = the carrier of Formal-Series R by FUNCT_2:def 3
     .= dom SBFSeri(R) by FUNCT_2:def 1;
     for o st o in dom ((SBFSeri(R))*(BSFSeri(R))) holds
     ((SBFSeri(R))*(BSFSeri(R))).o = (id dom(BSFSeri(R))).o
     proof
       let o;
       assume
A2:    o in dom ((SBFSeri(R))*(BSFSeri(R))); then
A3:    o in dom BSFSeri(R) by FUNCT_2:123;
       consider x1 being Series of 1,R such that
A4:    x1 = o & (BSFSeri(R)).o = x1*(NBag1) by A2,Def4;
       reconsider y = x1*(NBag1) as Element of Formal-Series R
         by FUNCT_2:5,A2,A4;
       consider y1 being sequence of R such that
A5:    y1 = y & (SBFSeri(R)).y = y1*(BagN1) by Def5;
       ((SBFSeri(R))*(BSFSeri(R))).o = (SBFSeri(R)).(x1*(NBag1))
          by A4,A2,FUNCT_2:15
       .= (SBFSeri(R)).y .= y1*(BagN1) by A5
       .= x1*((NBag1)*(BagN1)) by A5,RELAT_1:36
       .= x1 by Th11,FUNCT_2:17 .= (id dom(BSFSeri(R))).o by A3,A4,FUNCT_1:18;
       hence thesis;
     end; then
     (SBFSeri(R))*(BSFSeri(R)) = id dom BSFSeri(R) by FUNCT_2:123;
     hence thesis by A1,FUNCT_1:41;
   end;
