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

theorem Th12:
   for b1,b2 be bag of 1 holds (b2 in rng divisors b1 iff b2.0 <= b1.0) &
   (b2 in rng divisors b1 iff b2 divides b1)
   proof
     let b1,b2 be bag of 1;
     consider S being non empty finite Subset of (Bags 1) such that
A1:  divisors b1 = SgmX ((BagOrder 1),S) &
     (for p being bag of 1 holds (p in S iff p divides b1)) by PRE_POLY:def 16;
A2:  BagOrder 1 linearly_orders S by Th9,ORDERS_1:38;
A3:  b2 in rng(divisors b1) implies b2.0 <= b1.0
     proof
       assume b2 in rng(divisors b1); then
       b2 in S by A1,A2,PRE_POLY:def 2;
       hence thesis by A1,PRE_POLY:def 11;
     end;
     b2.0 <= b1.0 implies b2 in rng(divisors b1)
     proof
       assume b2.0 <= b1.0; then
       b2 in S by A1, Th8;
       hence thesis by A1,A2,PRE_POLY:def 2;
     end;
     hence thesis by A3,Th8;
   end;
