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

theorem Th17:
   for n be Ordinal, L
   for a,b being Element of L, p being Series of n,L holds (a+b)*p = a*p + b*p
   proof
     let n be Ordinal, L;
     let a,b be Element of L, p be Series of n,L;
     for i be Element of Bags n holds ((a+b)*p).i = (a*p + b*p).i
     proof
       let i be Element of Bags n;
       thus ((a+b)*p).i = (a+b)*p.i by POLYNOM7:def 9
       .= a*p.i + b*p.i by VECTSP_1:def 7 .= (a*p).i + b*p.i by POLYNOM7:def 9
       .= (a*p).i + (b*p).i by POLYNOM7:def 9 .= (a*p + b*p).i by POLYNOM1:15;
     end;
     hence thesis;
   end;
