reserve Y for non empty set,
  a, b for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  A, B for a_partition of Y;

theorem
  for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G
  is independent & G={A,B} & A <> B holds '/\' G = A '/\' B
proof
  let G be Subset of PARTITIONS(Y);
  let A,B be a_partition of Y;
  assume that
A1: G is independent and
A2: G={A,B} and
A3: A <> B;
A4: '/\' G c= A '/\' B
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume x in '/\' G;
    then consider h being Function, F being Subset-Family of Y such that
A5: dom h=G and
A6: rng h = F and
A7: for d being set st d in G holds h.d in d and
A8: x=Intersect F and
A9: x<>{} by BVFUNC_2:def 1;
A10: not x in {{}} by A9,TARSKI:def 1;
A11: A in G by A2,TARSKI:def 2;
    then
A12: h.A in rng h by A5,FUNCT_1:def 3;
    then
A13: Intersect F = meet (rng h) by A6,SETFAM_1:def 9;
A14: h.A /\ h.B c= xx
    proof
A15:  rng h = {h.A,h.B}
      proof
        thus rng h c= {h.A,h.B}
        proof
          let m be object;
          assume m in rng h;
          then consider w being object such that
A16:      w in dom h and
A17:      m = h.w by FUNCT_1:def 3;
          w = A or w = B by A2,A5,A16,TARSKI:def 2;
          hence thesis by A17,TARSKI:def 2;
        end;
        let m be object;
        assume m in {h.A,h.B};
        then
A18:    m = h.A or m = h.B by TARSKI:def 2;
A19:    B in dom h by A2,A5,TARSKI:def 2;
        A in dom h by A2,A5,TARSKI:def 2;
        hence thesis by A18,A19,FUNCT_1:def 3;
      end;
      let m be object;
      assume
A20:  m in h.A /\ h.B;
      then
A21:  m in h.B by XBOOLE_0:def 4;
      m in h.A by A20,XBOOLE_0:def 4;
      then for y being set holds y in rng h implies m in y by A21,A15,
TARSKI:def 2;
      hence thesis by A8,A12,A13,SETFAM_1:def 1;
    end;
A22: B in G by A2,TARSKI:def 2;
    then
A23: h.B in B by A7;
A24: h.B in rng h by A5,A22,FUNCT_1:def 3;
    xx c= h.A /\ h.B
    proof
      let m be object;
      assume
A25:  m in xx;
      then
A26:  m in h.B by A8,A24,A13,SETFAM_1:def 1;
      m in h.A by A8,A12,A13,A25,SETFAM_1:def 1;
      hence thesis by A26,XBOOLE_0:def 4;
    end;
    then
A27: h.A /\ h.B = x by A14,XBOOLE_0:def 10;
    h.A in A by A7,A11;
    then x in INTERSECTION(A,B) by A23,A27,SETFAM_1:def 5;
    then x in INTERSECTION(A,B) \ {{}} by A10,XBOOLE_0:def 5;
    hence thesis by PARTIT1:def 4;
  end;
  A '/\' B c= '/\' G
  proof
    let x be object;
    assume x in A '/\' B;
    then x in INTERSECTION(A,B) \ {{}} by PARTIT1:def 4;
    then consider X,Z being set such that
A28: X in A and
A29: Z in B and
A30: x = X /\ Z by SETFAM_1:def 5;
    {X,Z} c= bool Y
    proof
      let m be object;
      assume m in {X,Z};
      then m = X or m = Z by TARSKI:def 2;
      hence thesis by A28,A29;
    end;
    then reconsider SS = {X,Z} as Subset-Family of Y;
A31: X /\ Z = Intersect SS by A28,A29,MSSUBFAM:10;
    set h = (A,B) --> (X,Z);
A32: for d being set st d in G holds h.d in d
    proof
      let d be set;
      assume d in G;
      then d = A or d = B by A2,TARSKI:def 2;
      hence thesis by A3,A28,A29,FUNCT_4:63;
    end;
A33: dom h = {A,B} by FUNCT_4:62;
A34: rng h = SS by A3,FUNCT_4:64;
    then Intersect SS <> {} by A1,A2,A33,A32,BVFUNC_2:def 5;
    hence thesis by A2,A30,A33,A34,A32,A31,BVFUNC_2:def 1;
  end;
  hence thesis by A4;
end;
