 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem BagSupport:
  for A being finite Subset of SetPrimes holds
    support (A-bag) = A
  proof
    let A being finite Subset of SetPrimes;
    set f = A-bag;
B1: support f c= A
    proof
      let x be object;
      assume
a10:  x in support f;
      x in A
      proof
        assume not x in A; then
        not x in dom id A; then
        f.x = (EmptyBag SetPrimes).x by FUNCT_4:11
           .= 0 by PBOOLE:5;
        hence thesis by a10,PRE_POLY:def 7;
      end;
      hence thesis;
    end;
    A c= support f
    proof
      let x be object;
      assume
E1:   x in A; then
      x in dom id A; then
      f.x = (id A).x by FUNCT_4:13
         .= x by E1,FUNCT_1:17; then
      f.x is Prime by NEWTON:def 6,E1;
      hence thesis by PRE_POLY:def 7;
    end;
    hence thesis by B1,XBOOLE_0:def 10;
  end;
