
theorem Th43:
  for T being non empty TopSpace st T is locally-compact for D
being countable Subset-Family of T st D is non empty dense open for O being non
  empty Subset of T st O is open ex A being irreducible Subset of T st for V
  being Subset of T st V in D holds A /\ O meets V
proof
  let T be non empty TopSpace;
  assume T is locally-compact;
  then reconsider
  L = InclPoset the topology of T as bounded continuous LATTICE by WAYBEL_3:42;
A1: the carrier of L = the topology of T by YELLOW_1:1;
  let D be countable Subset-Family of T such that
A2: D is non empty dense open;
  consider I being object such that
A3: I in D by A2;
  reconsider I as Subset of T by A3;
  I is open by A2,A3;
  then reconsider i = I as Element of L by A1;
  set D1 = {d where d is Element of L : ex d1 being Subset of T st d1 in D &
  d1 = d & d is dense};
  D1 c= the carrier of L
  proof
    let q be object;
    assume q in D1;
    then
    ex d being Element of L st q = d & ex d1 being Subset of T st d1 in D &
    d1 = d & d is dense;
    hence thesis;
  end;
  then reconsider D1 as Subset of L;
A4: D1 c= D
  proof
    let q be object;
    assume q in D1;
    then ex d being Element of L st q = d & ex d1 being Subset of T st d1 in D
    & d1 = d & d is dense;
    hence thesis;
  end;
A5: D1 is dense
  proof
    let q be Element of L;
    assume q in D1;
    then ex d being Element of L st q = d & ex d1 being Subset of T st d1 in D
    & d1 = d & d is dense;
    hence thesis;
  end;
  let O be non empty Subset of T such that
A6: O is open;
  reconsider u = O as Element of L by A6,A1;
  I is open dense by A2,A3;
  then I is everywhere_dense by TOPS_3:36;
  then i is dense by Th42;
  then u <> Bottom L & i in D1 by A3,PRE_TOPC:1,YELLOW_1:13;
  then consider p being irreducible Element of L such that
A7: p <> Top L and
A8: not p in uparrow ({u} "/\" D1) by A4,A5,Th39;
  p in the topology of T by A1;
  then reconsider P = p as Subset of T;
  reconsider A = P` as irreducible Subset of T by A7,Th41;
  take A;
  let V be Subset of T;
  assume
A9: V in D;
  then
A10: V is open by A2;
  then reconsider v = V as Element of L by A1;
A11: for d1 being Element of L st d1 in D1 holds not u "/\" d1 <= p
  proof
A12: u in {u} by TARSKI:def 1;
    let d1 be Element of L;
    assume d1 in D1;
    then u "/\" d1 in {u} "/\" D1 by A12;
    hence thesis by A8,WAYBEL_0:def 16;
  end;
  V is dense by A2,A9;
  then V is everywhere_dense by A10,TOPS_3:36;
  then v is dense by Th42;
  then v in D1 by A9;
  then not u "/\" v <= p by A11;
  then
A13: not u "/\" v c= p by YELLOW_1:3;
  O /\ V is open by A6,A10;
  then u /\ v in the topology of T;
  then not O /\ V c= p by A13,YELLOW_1:9;
  then consider x being object such that
A14: x in O /\ V and
A15: not x in A`;
  reconsider x as Element of T by A14;
  x in A by A15,XBOOLE_0:def 5;
  then O /\ V /\ A <> {} by A14,XBOOLE_0:def 4;
  hence A /\ O /\ V <> {} by XBOOLE_1:16;
end;
