
theorem
for F being Field
for B1,B2 being non zero bag of the carrier of F
st B2 divides B1 & card B1 = 1 holds B2 = B1
proof
let F be Field, B1,B2 be non zero bag of the carrier of F;
assume A: B2 divides B1 & card B1 = 1; then
consider a1 being Element of F such that B: B1 = Bag{a1} by bag2;
C: card B2 <= 1 by A,bag1;
support B2 <> {} by RING_5:24; then
card B2 >= 1 by RING_5:23,NAT_1:14; then
consider a2 being Element of F such that D: B2 = Bag{a2}
    by C,XXREAL_0:1,bag2;
B1 = ({a1},1)-bag & B2 = ({a2},1)-bag by B,D,RING_5:def 1; then
E: support B1 = {a1} & support B2 = {a2} by UPROOTS:8; then
a2 in support B2 by TARSKI:def 1; then
F: B2.a2 <> 0 by PRE_POLY:def 7;
now assume a1 <> a2;
  then not a2 in support B1 by E,TARSKI:def 1;
  then B1.a2 = 0 & B2.a2 > 0 by F,PRE_POLY:def 7;
  hence contradiction by A,PRE_POLY:def 11;
  end;
hence thesis by B,D;
end;
