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

theorem
  for S be cap-finite-partition-closed Subset-Family of X,
  A be Element of S
  holds {{}}\/union (PARTITIONS(A)/\Fin S) is semiring_of_sets of A
  proof
    let S be cap-finite-partition-closed Subset-Family of X,
    A be Element of S;
    set A1=union (PARTITIONS(A)/\Fin S);
    set B=union (PARTITIONS(A)/\Fin S)\/{{}};
A1: A1 is cap-finite-partition-closed diff-finite-partition-closed
    Subset-Family of A by Thm99;
A2: {{}} c= B & {} in {{}} by XBOOLE_1:7,TARSKI:def 1;
    B c= bool A
    proof
      let x be object;
      assume x in B;
      then x in A1 or x in {{}} by XBOOLE_0:def 3;
      then
A3:   x in A1 or x = {} by TARSKI:def 1;
      {} c= A by XBOOLE_1:2;
      hence thesis by A1,A3;
    end;
    then reconsider B as Subset-Family of A;
A4: B is cap-finite-partition-closed
    proof
      let S1,S2 be Element of B such that
F1:   S1/\S2 is non empty;
      reconsider A1 as Subset-Family of A by Thm99;
      (S1 in A1 or S1 in {{}}) &
      (S2 in A1 or S2 in {{}}) by XBOOLE_0:def 3;
      then (S1 in A1 or S1 = {}) &
      (S2 in A1 or S2 = {}) by TARSKI:def 1;
      then S1 is Element of A1 &
      S2 is Element of A1 & S1/\S2 is non empty &
      A1 is cap-finite-partition-closed by F1,Thm99;
      then consider P be finite Subset of A1 such that
SOL:  P is a_partition of S1/\S2;
      reconsider P as finite Subset of B by XBOOLE_1:10;
      P is finite Subset of B & P is a_partition of S1/\S2 by SOL;
      hence thesis;
    end;
    B is diff-finite-partition-closed
    proof
      let S1,S2 be Element of B such that
F1:   S1\S2 is non empty;
      reconsider A1 as Subset-Family of A by Thm99;
F2aa: (S1 in A1 or S1 in {{}}) &
      (S2 in A1 or S2 in {{}}) by XBOOLE_0:def 3;
V1:   S2={} implies ex P be finite Subset of B st
      P is a_partition of S1\S2
      proof
        assume
D0:     S2={};
D3:     {S1} c= B
        proof
          let x be object;
          assume x in {S1};
          then x=S1 by TARSKI:def 1;
          hence thesis;
        end;
        {S1} is a_partition of S1 by F1,EQREL_1:39;
        hence thesis by D0,D3;
      end;
      S2 in A1 implies ex P be finite Subset of B st
      P is a_partition of S1\S2
      proof
        assume S2 in A1;
        then S1 is Element of A1 &
        S2 is Element of A1 & S1\S2 is non empty &
        A1 is diff-finite-partition-closed by Thm99,F1,F2aa,TARSKI:def 1;
        then
        consider P be finite Subset of A1 such that
SOL:    P is a_partition of S1\S2;
        reconsider P as finite Subset of B by XBOOLE_1:10;
        P is finite Subset of B & P is a_partition of S1\S2 by SOL;
        hence thesis;
      end;
      hence thesis by V1,F2aa,TARSKI:def 1;
    end;
    hence thesis by A2,A4;
  end;
