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

theorem Th8:
   for b1,b2 be bag of 1 holds b1.0 <= b2.0 iff b1 divides b2
   proof
     let b1,b2 be bag of 1;
     b1.0 <= b2.0 implies b1 divides b2
     proof
       assume
A1:    b1.0 <= b2.0;
       for k being object st k in 1 holds b1.k <= b2.k
       proof
         let k be object;
         assume k in 1; then
         k = 0 by TARSKI:def 1, CARD_1:49;
         hence thesis by A1;
       end;
       hence thesis by PRE_POLY:46;
     end;
     hence thesis by PRE_POLY:def 11;
   end;
