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

theorem Th26:
   for R be Ring, f,g be Series of 1,R holds
   (f+g)*(NBag1) = f*(NBag1) + g*(NBag1)
   proof
     let R be Ring, 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:    f.b = (f*(NBag1)).m by FUNCT_2:15;
A4:    g.b = (g*(NBag1)).m by FUNCT_2:15;
       ((f+g)*NBag1).o = (f+g).b by A1,FUNCT_2:15
       .= (f*(NBag1)).m + (g*(NBag1)).m by A3,A4,POLYNOM1:15
       .= (f*(NBag1) + g*(NBag1)).o by NORMSP_1:def 2;
       hence thesis;
     end;
     hence thesis;
   end;
