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

theorem Th28:
   for f,g be Series of 1,R holds (f*'g)*(NBag1) = (f*(NBag1)) *' (g*(NBag1))
   proof
     let f,g be Series of 1,R;
     for o st o in NAT holds ((f*'g)*(NBag1)).o = ((f*(NBag1))*'(g*(NBag1))).o
     proof
       let o;
       assume
A1:    o in NAT; then
       reconsider m = o as Element of NAT;
A2:    NBag1.o = 1 --> m by Def1;
       reconsider b = (NBag1).o as Element of Bags 1 by A2,PRE_POLY:def 12;
A3:    0 in 1 by CARD_1:49,TARSKI:def 1; then
A4:    (b.0) = m by A2,FUNCOP_1:7;
       set F = NBag1|(Segm((b.0)+1));
       consider s being FinSequence of the carrier of R such that
A5:    (f*'g).b = Sum s & len s = len decomp b &
       for k being Element of NAT st k in dom s ex b1, b2 being bag of 1 st
       (decomp b)/.k = <*b1, b2*> & s/.k = f.b1*g.b2 by POLYNOM1:def 10;
A6:    ((f*'g)*NBag1).o = (f*'g).b by A1,FUNCT_2:15 .= Sum s by A5;
       consider r be FinSequence of the carrier of R such that
A7:    len r = m+1 & ((f*(NBag1)) *' (g*(NBag1))).m = Sum r &
       for k be Element of NAT st k in dom r holds
       r.k = (f*(NBag1)).(k-'1) * (g*(NBag1)).(m+1-'k) by POLYNOM3:def 9;
A8:    Seg len decomp b = dom decomp b by FINSEQ_1:def 3
       .= dom divisors b by PRE_POLY:def 17
       .= Seg len divisors b by FINSEQ_1:def 3; then
       len decomp b = len divisors b by FINSEQ_1:6; then
A9:   len s = (b.0) + 1 by A5,Th15
       .= len r by A2,A3,FUNCOP_1:7,A7;
A10:   dom decomp b = Seg len decomp b by FINSEQ_1:def 3
       .= Seg ((b.0) +1) by A8,Th15;
A11:   dom decomp b = Seg len s by A5,FINSEQ_1:def 3
          .= dom s by FINSEQ_1:def 3;
       for k be Nat st 1 <= k & k <= len s holds s.k = r.k
       proof
         let k be Nat;
         assume A12: 1 <= k & k <= len s; then
A13:     k in Seg len s;
A14:     k in dom s by A12,FINSEQ_3:25; then
         consider b1, b2 being bag of 1 such that
A15:     (decomp b)/.k = <*b1, b2*> & s/.k = f.b1*g.b2 by A5;
A16:     k in dom decomp b & (decomp b)/.k = <*b1, b2*>
           by A13, FINSEQ_1:def 3,A11,A15; then
         b1 = (divisors b)/.k by PRE_POLY:70; then
A17:     <*b1, b-'b1*> = (decomp b)/.k by A16,PRE_POLY:def 17
         .= <*b1, b2*> by A15;
         dom decomp b = Seg len divisors b by A8,FINSEQ_1:def 3
         .= dom divisors b by FINSEQ_1:def 3
         .= dom XFS2FS(F) by Th25; then
A18:     k in dom XFS2FS(F) by A13, FINSEQ_1:def 3,A11;
A19:     k in Seg ((b.0) +1) by A13, FINSEQ_1:def 3,A11,A10; then         
A20:     1 <= k & k <= len F by FINSEQ_1:1;
         then reconsider k1 = k-1 as Element of NAT by INT_1:3;
A21:     1 <= k <= (b.0) +1 by A19,FINSEQ_1:1;
         k - 1 < k - 0 by XREAL_1:10; then
A22:     k - 1 < (b.0) + 1 by A21,XXREAL_0:2; then
A23:     k1 in dom(NBag1|Segm((b.0)+1)) by NAT_1:44;
A24:     b1 = (divisors b)/.k by PRE_POLY:70,A16
         .= (XFS2FS(F))/.k by Th25
         .= (XFS2FS(F)).k by A18,PARTFUN1:def 6
         .= (NBag1|Segm((b.0)+1)).(k-1) by A20,AFINSQ_1:def 9
         .= (NBag1).k1 by A23,FUNCT_1:47 .= 1--> k1 by Def1;
A25:     0 in 1 by CARD_1:49,TARSKI:def 1;
aaa:     k -'1 = k -1 by A21,XREAL_0:def 2; then
A26:     (b.0) >= k - 1 by A22,NAT_1:13;
A27:     (b.0) -' (k -'1) = (b.0) -' (k-1) by A21,XREAL_0:def 2
         .= (b.0)-(k-1) by A26,XREAL_1:233
         .= (b.0)+1 - k .= (b.0)+1 -' k by A21,XREAL_1:233;
A28:     b2 = b-'b1 by A17,FINSEQ_1:77
         .= 1 --> ((b.0) -' (b1.0)) by Th7
         .= 1 --> ((b.0)+ 1 -' k) by aaa,A27,A25,A24,FUNCOP_1:7;
A29:     (f*NBag1).(k-1) = f.((NBag1).k1) by FUNCT_2:15
         .= f.b1 by Def1,A24;
A30:     (g*NBag1).((b.0)+1-'k) = g.((NBag1).((b.0)+1-'k)) by FUNCT_2:15
         .= g.b2 by A28,Def1;
         k in dom r by A9,A13,FINSEQ_1:def 3; then
         r.k = (f*(NBag1)).(k-'1) * (g*(NBag1)).((b.0)+1-'k) by A4,A7
         .= f.b1 * g.b2 by aaa,A29,A30
         .= s.k by A14,A15,PARTFUN1:def 6;
         hence thesis;
       end; then
       s = r by A9; then
       ((f*'g)*(NBag1)).o = Sum r by A6
       .= ((f*(NBag1)) *' (g*(NBag1))).o by A7;
       hence thesis;
     end;
     hence thesis;
   end;
