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

theorem Th16:
   for n be Ordinal,L
   for a being Element of L, p,q being Series of n,L holds a*(p+q)=a*p + a*q
   proof
     let n be Ordinal, L;
     let a be Element of L, p,q be Series of n,L;
     for i be Element of Bags n holds (a*(p+q)).i = (a*p + a*q).i
     proof
       let i be Element of Bags n;
       a*((p+q).i) = a*(p.i + q.i) by POLYNOM1:15
       .= a*(p.i) + a*(q.i) by VECTSP_1:def 7
       .= (a*p).i + a*(q.i) by POLYNOM7:def 9
       .= (a*p).i + (a*q).i by POLYNOM7:def 9
       .= (a*p + a*q).i by POLYNOM1:15;
       hence thesis by POLYNOM7:def 9;
     end;
     hence thesis;
   end;
