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

theorem Th1:
  for y being Element of Y ex X being Subset of Y st y in X & ex h
  being Function, F being Subset-Family of Y st dom h=G & rng h = F & (for d
  being set st d in G holds h.d in d) & X=Intersect F & X<>{}
proof
  let y be Element of Y;
  deffunc F(Element of PARTITIONS(Y)) = EqClass(y,$1);
  defpred P[set] means $1 in G;
  consider h being PartFunc of PARTITIONS(Y), bool Y such that
A1: for d being Element of PARTITIONS(Y) holds d in dom h iff P[d] and
A2: for d being Element of PARTITIONS(Y) st d in dom h holds h/.d = F(d)
  from PARTFUN2:sch 2;
A3: G c= dom h
  by A1;
A4: for d being set st d in G holds h.d in d
  proof
    let d be set;
    assume
A5: d in G;
    then reconsider d as a_partition of Y by PARTIT1:def 3;
    h/.d=h.d by A3,A5,PARTFUN1:def 6;
    then h.d = EqClass(y,d) by A2,A3,A5;
    hence thesis;
  end;
  dom h c= G
  by A1;
  then
A6: dom h = G by A3,XBOOLE_0:def 10;
  reconsider rr = rng h as Subset-Family of Y;
A7: for d being Element of PARTITIONS(Y) st d in dom h holds y in h.d
  proof
    let d be Element of PARTITIONS(Y);
    assume
A8: d in dom h;
    then h/.d = EqClass(y,d) by A2;
    then h.d = EqClass(y,d) by A8,PARTFUN1:def 6;
    hence thesis by EQREL_1:def 6;
  end;
A9: for c being set holds c in rng h implies y in c
  proof
    let c be set;
    assume c in rng h;
    then ex a being object st a in dom h & c = h.a by FUNCT_1:def 3;
    hence thesis by A7;
  end;
  per cases;
  suppose
    rng h ={};
    then Intersect rr = Y by SETFAM_1:def 9;
    hence thesis by A6,A4;
  end;
  suppose
    rng h <> {};
    then y in meet (rng h) & Intersect rr = meet (rng h) by A9,SETFAM_1:def 1
,def 9;
    hence thesis by A6,A4;
  end;
end;
