
theorem
for X be set, P be with_empty_element semi-diff-closed cap-closed
       Subset-Family of X,
    K be disjoint_valued Function of NAT,Ring_generated_by P
 st rng K is with_non-empty_element holds
 ex Y be non empty FinSequenceSet of P st
    Y = {F where F is disjoint_valued FinSequence of P :
           Union F in rng K & F <> {} }
  & Y is with_non-empty_elements
proof
   let X be set, P be with_empty_element semi-diff-closed cap-closed
       Subset-Family of X,
       K be disjoint_valued Function of NAT,Ring_generated_by P;
   assume A0: rng K is with_non-empty_element;
   set
   Y = {F where F is disjoint_valued FinSequence of P :
           Union F in rng K & F <> {} };

   now let a be object;
    assume a in Y; then
    ex A be disjoint_valued FinSequence of P st
      a = A & Union A in rng K & A <> {};
    hence a is FinSequence of P;
   end; then
   reconsider Y as FinSequenceSet of P by FINSEQ_2:def 3;

   consider k be non empty set such that
A2: k in rng K by A0;
   consider i be Element of NAT such that
A3: k = K.i by A2,FUNCT_2:113;

   K.i in Ring_generated_by P; then
   K.i in DisUnion P by SRINGS_3:18; then
   consider A be Subset of X such that
A4: K.i = A
  & ex F be disjoint_valued FinSequence of P st A = Union F;
   consider F be disjoint_valued FinSequence of P such that
A5: A = Union F by A4;
   now assume F = {}; then
    union rng F = {} by ZFMISC_1:2;
    hence contradiction by A5,A4,A3,CARD_3:def 4;
   end; then
   F in Y by A2,A3,A4,A5; then
   reconsider Y as non empty FinSequenceSet of P;

   take Y;
   thus Y = {A where A is disjoint_valued FinSequence of P :
               Union A in rng K & A <> {} };

   now assume {} in Y; then
    ex A be disjoint_valued FinSequence of P st
     {} = A & Union A in rng K & A <> {};
    hence contradiction;
   end;
   hence Y is with_non-empty_elements;
end;
