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 for a,b being set st a in A & b in B
  holds a meets 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;
  let a,b be set;
  assume that
A4: a in A and
A5: b in B;
  set h2 = (A,B) --> (a,b);
A6: for d being set st d in G holds h2.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,A4,A5,FUNCT_4:63;
  end;
  {a,b} c= bool Y
  proof
    let x be object;
    assume x in {a,b};
    then x = a or x = b by TARSKI:def 2;
    hence thesis by A4,A5;
  end;
  then reconsider SS = {a,b} as Subset-Family of Y;
A7: dom h2 = {A,B} by FUNCT_4:62;
  rng h2 = SS by A3,FUNCT_4:64;
  then Intersect SS <> {} by A1,A2,A7,A6,BVFUNC_2:def 5;
  hence a /\ b <> {} by A4,A5,MSSUBFAM:10;
end;
