reserve a,b for object, I,J for set;
reserve b for bag of I;
reserve R for asymmetric transitive non empty RelStr,
  a,b,c for bag of the carrier of R,
  x,y,z for Element of R;
reserve p for partition of b-'a, q for partition of b;
reserve J for set, m for bag of I;

theorem
  b = EmptyBag the carrier of R iff OrderedPartition b = {}
  proof
    set I = the carrier of R;
    set E = EmptyBag I;
    set p = OrderedPartition b;
    consider F,G being Function of NAT, Bags the carrier of R such that
A1: F.0 = b & G.0 = E &
    (for i being Nat holds G.(i+1) = (F.i)|{x:x is_maximal_in support (F.i)} &
    F.(i+1) = (F.i)-'(G.(i+1))) &
    ex i being Nat st F.i = E & p = G|Seg i &
    for j being Nat st j < i holds F.j <> E by OP;
    consider i being Nat such that
A2: F.i = E & p = G|Seg i & for j being Nat st j < i holds F.j <> E by A1;
    hereby assume b = E;
      then i <= 0 by A1,A2;
      then i = 0;
      hence p = {} by A2;
    end;
    assume p = {};
    then p = <*>Bags I;
    then Sum p = E by Th21;
    hence thesis by PART;
  end;
