
theorem Th42:
  for T being non empty TopSpace for A being Element of InclPoset
  the topology of T for B being Subset of T st A = B holds A is dense iff B is
  everywhere_dense
proof
  let T be non empty TopSpace, A be Element of InclPoset the topology of T, B
  be Subset of T such that
A1: A = B;
A2: Bottom InclPoset the topology of T = {} by PRE_TOPC:1,YELLOW_1:13;
A3: the carrier of InclPoset the topology of T = the topology of T by
YELLOW_1:1;
  then
A4: B is open by A1;
  hereby
    assume
A5: A is dense;
    for Q being Subset of T st Q <> {} & Q is open holds B meets Q
    proof
      let Q be Subset of T such that
A6:   Q <> {} and
A7:   Q is open;
      reconsider q = Q as Element of InclPoset the topology of T by A3,A7;
      B /\ Q is open by A4,A7;
      then
A8:   A /\ q in the topology of T by A1;
      A "/\" q <> Bottom InclPoset the topology of T by A2,A5,A6;
      then B /\ Q <> {} by A1,A2,A8,YELLOW_1:9;
      hence thesis;
    end;
    then B is dense by TOPS_1:45;
    hence B is everywhere_dense by A4,TOPS_3:36;
  end;
  assume B is everywhere_dense;
  then
A9: B is dense by TOPS_3:33;
  let v be Element of InclPoset the topology of T such that
A10: v <> Bottom InclPoset the topology of T;
  v in the topology of T by A3;
  then reconsider V = v as Subset of T;
A11: V is open by A3;
  B is open by A1,A3;
  then B /\ V is open by A11;
  then
A12: B /\ V in the topology of T;
  B meets V by A2,A9,A10,A11,TOPS_1:45;
  then B /\ V <> {};
  hence A "/\" v <> Bottom InclPoset the topology of T by A1,A2,A12,YELLOW_1:9;
end;
