
theorem bagset1:
for F being Field
for S1,S2 being non empty finite Subset of F
holds Bag S1 divides Bag S2 iff S1 c= S2
proof
let F be Field, S1,S2 be non empty finite Subset of F;
H1: Bag S1 = (S1,1)-bag & Bag S2 = (S2,1)-bag by RING_5:def 1;
A: now assume B: Bag S1 divides Bag S2;
   now let o be object;
     assume o in S1; then
     C: 1 = (Bag S1).o by H1,UPROOTS:7;
     (Bag S1).o <= (Bag S2).o by B,PRE_POLY:def 11;
     hence o in S2 by C,H1,UPROOTS:6;
     end;
   hence S1 c= S2;
   end;
now assume C: S1 c= S2;
  now let o be object;
    per cases;
    suppose B: o in S1; then
      (Bag S1).o = 1 by H1,UPROOTS:7 .= (Bag S2).o by C,B,H1,UPROOTS:7;
      hence (Bag S1).o <= (Bag S2).o;
      end;
    suppose not o in S1;
      hence (Bag S1).o <= (Bag S2).o by H1,UPROOTS:6;
      end;
    end;
  hence Bag S1 divides Bag S2 by PRE_POLY:def 11;
  end;
hence thesis by A;
end;
