reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for G being Subset of PARTITIONS(Y),PA being a_partition of Y st PA in
  G holds PA '>' ('/\' G)
proof
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
  assume
A1: PA in G;
  for x being set st x in ('/\' G) ex a being set st a in PA & x c= a
  proof
    let x be set;
    assume x in ('/\' G);
    then consider h being Function, F being Subset-Family of Y such that
A2: dom h=G and
A3: rng h =F and
A4: for d being set st d in G holds h.d in d and
A5: x=Intersect F and
    x<>{} by Def1;
    set a = h.PA;
A6: a in PA by A1,A4;
A7: a in rng h by A1,A2,FUNCT_1:def 3;
    then Intersect F = meet F by A3,SETFAM_1:def 9;
    hence thesis by A3,A5,A6,A7,SETFAM_1:3;
  end;
  hence thesis by SETFAM_1:def 2;
end;
