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

theorem ThmVAL1:
  for S be Subset-Family of X, A be Element of S holds
  {x where x is Element of S: x in union (PARTITIONS(A)/\Fin S)} =
  union (PARTITIONS(A)/\Fin S)
  proof
    let S be Subset-Family of X, A be Element of S;
    thus {x where x is Element of S: x in union (PARTITIONS(A)/\Fin S)} c=
    union (PARTITIONS(A)/\Fin S)
    proof
      let u be object;
      assume u in {x where x is Element of S:
      x in union (PARTITIONS(A)/\Fin S)};
      then ex x be Element of S st u=x & x in union (PARTITIONS(A)/\Fin S);
      hence thesis;
    end;
    let u be object;
    assume
A5A:u in union(PARTITIONS(A)/\Fin S);
    then consider v be set such that
A6: u in v and
A7: v in PARTITIONS(A)/\Fin S by TARSKI:def 4;
    v in Fin S by A7,XBOOLE_0:def 4;
    then v c= S by FINSUB_1:def 5;
    hence thesis by A5A,A6;
  end;
