reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;

theorem Th1:
  '/\' {} PARTITIONS Y = %O Y
proof
  for x being set holds x in %O Y iff ex h being Function, F being
Subset-Family of Y st dom h={} PARTITIONS Y & rng h = F & (for d being set st d
  in {} PARTITIONS Y holds h.d in d) & x=Intersect F & x<>{}
  proof
    let x be set;
    hereby
      reconsider h = {} as Function;
      assume x in %O Y;
      then
A1:   x in {Y} by PARTIT1:def 8;
      take h,F = {}bool Y;
      thus dom h={} PARTITIONS Y;
      thus rng h = F;
      thus for d being set st d in {} PARTITIONS Y holds h.d in d;
      x = Y by A1,TARSKI:def 1;
      hence x=Intersect F by SETFAM_1:def 9;
      thus x<>{} by A1,TARSKI:def 1;
    end;
    given h being Function, F being Subset-Family of Y such that
A2: dom h={} PARTITIONS Y & rng h = F and
    for d being set st d in {} PARTITIONS Y holds h.d in d and
A3: x=Intersect F and
    x<>{};
    F = {} by A2,RELAT_1:42;
    then x = Y by A3,SETFAM_1:def 9;
    then x in {Y} by TARSKI:def 1;
    hence thesis by PARTIT1:def 8;
  end;
  hence thesis by BVFUNC_2:def 1;
end;
