reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem Th12:
  for X be non empty set,x be Element of X, i be Element of NAT holds
   (EmptyBag X) +*(x,i) = ({x},i)-bag
proof
  let X be non empty set,x be Element of X, i be Element of NAT;
  dom (EmptyBag X) = X by PARTFUN1:def 2;
  hence (EmptyBag X) +*(x,i) = (EmptyBag X)+* (x.-->i) by FUNCT_7:def 3
  .= (EmptyBag X)+* ({x} -->i) by FUNCOP_1:def 9
  .= ({x},i)-bag by UPROOTS:def 2;
end;
