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

theorem Th20:
   for n be Ordinal, L holds Formal-Series(n,L) is mix-associative
   proof
     let n be Ordinal, L;
     for a being Element of L
     for x, y being Element of Formal-Series(n,L) holds a*(x*y) = (a*x)*y
     proof
       let a be Element of L;
       for x, y being Element of Formal-Series(n,L) holds a*(x*y) = (a*x)*y
       proof
         let x, y be Element of Formal-Series(n,L);
    reconsider x1=x, y1=y as Element of Formal-Series(n,L);
    reconsider p=x1, q=y1 as Series of n,L by Def3;
A1:      a*x = a*p by Def3;
         x*y = p*'q by Def3;
         hence a*(x*y) = a*(p*'q) by Def3 .=(a*p)*'q by POLYRED:12
         .= (a*x)*y by A1,Def3;
       end;
       hence thesis;
     end;
     hence thesis by POLYALG1:def 1;
   end;
