reserve X for set;
reserve S for Subset-Family of X;

theorem Thm99:
  for S be cap-finite-partition-closed Subset-Family of X,
  A be Element of S
  holds union (PARTITIONS(A)/\Fin S) is cap-finite-partition-closed
  diff-finite-partition-closed Subset-Family of A &
  union (PARTITIONS(A)/\Fin S) is with_non-empty_elements
  proof
    let S be cap-finite-partition-closed Subset-Family of X,
    A be Element of S;
A1: union (PARTITIONS(A)/\Fin S)=
    {x where x is Element of S: x in union (PARTITIONS(A)/\Fin S)} by ThmVAL1;
    then
A2: union (PARTITIONS(A)/\Fin S) is
    cap-finite-partition-closed Subset-Family of A by ThmVAL0;
    union (PARTITIONS(A)/\Fin S) is diff-c=-finite-partition-closed
    Subset-Family of A by A1,ThmVAL2;
    hence thesis by A2,LemPO;
  end;
