reserve a,b for object, I,J for set;

theorem Th1:
  a is multiset of I iff a is bag of I
  proof
    thus a is multiset of I implies a is bag of I;
    assume a is bag of I;
    then reconsider b = a as bag of I;
    dom b = I & rng b c= NAT by PARTFUN1:def 2,VALUED_0:def 6;
    then b is Function of I,NAT by FUNCT_2:2;
    then b is Multiset of I & support b is finite by MONOID_1:27;
    hence thesis by MONOID_1:def 6;
  end;
