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

theorem
   for f1,g1 be sequence of R holds (f1+g1)*(BagN1) = f1*(BagN1) + g1*(BagN1)
   proof
     let f1,g1 be sequence of R;
A1:  dom((f1+g1)*(BagN1)) = Bags 1 by FUNCT_2:def 1;
     for o st o in dom((f1+g1)*(BagN1)) holds
     ((f1+g1)*(BagN1)).o = (f1*(BagN1) + g1*(BagN1)).o
     proof
       let o;
       assume o in dom((f1+g1)*(BagN1)); then
   reconsider b = o as Element of Bags 1;
A2:    (BagN1).o = b.0 by Def2;
       reconsider s1 = f1*(BagN1) as Series of 1,R;
       reconsider t1 = g1*(BagN1) as Series of 1,R;
A3:    (f1*BagN1).b = f1.(b.0) by A2,FUNCT_2:15;
A4:    (g1*BagN1).b = g1.(b.0) by A2,FUNCT_2:15;
       ((f1+g1)*(BagN1)).o = (f1+g1).(b.0) by A2,FUNCT_2:15
       .= f1.(b.0) + g1.(b.0) by NORMSP_1:def 2
       .= ((f1*BagN1) + (g1*BagN1)).o by A3,A4,POLYNOM1:15;
       hence thesis;
     end;
     hence thesis by A1;
   end;
