reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;

theorem
  A is everywhere_dense iff for F being Subset of X st F <> the carrier
of X & F is closed ex H being Subset of X st F c= H & H <> the carrier of X & H
  is closed & A \/ H = the carrier of X
proof
  thus A is everywhere_dense implies for F being Subset of X st F <> the
carrier of X & F is closed ex H being Subset of X st F c= H & H <> the carrier
  of X & H is closed & A \/ H = the carrier of X
  proof
    assume A is everywhere_dense;
    then
A1: A` is nowhere_dense by Th39;
    let F be Subset of X;
    assume F <> the carrier of X;
    then
A2: [#]X \ F <> {} by PRE_TOPC:4;
    assume F is closed;
    then consider G being Subset of X such that
A3: G c= F` and
A4: G <> {} and
A5: G is open and
A6: ( A`) misses G by A1,A2,Th28;
    take H = G`;
    F`` c= H by A3,SUBSET_1:12;
    hence F c= H;
    H` <> {} by A4;
    then [#]X \ H <> {};
    hence H <> the carrier of X by PRE_TOPC:4;
    thus H is closed by A5;
    ( A`) /\ H` = {} by A6;
    then (A \/ H)` = {}X by XBOOLE_1:53;
    hence thesis by Th2;
  end;
  assume
A7: for F being Subset of X st F <> the carrier of X & F is closed ex H
  being Subset of X st F c= H & H <> the carrier of X & H is closed & A \/ H =
  the carrier of X;
  for G being Subset of X st G <> {} & G is open ex H being Subset of X
  st H c= G & H <> {} & H is open & ( A`) misses H
  proof
    let G be Subset of X;
    assume G <> {};
    then G`` <> {};
    then
A8: G` <> [#]X by PRE_TOPC:4;
    assume G is open;
    then consider F being Subset of X such that
A9: G` c= F and
A10: F <> the carrier of X and
A11: F is closed and
A12: A \/ F = the carrier of X by A7,A8;
    take H = F`;
    H c= G`` by A9,SUBSET_1:12;
    hence H c= G;
    F <> [#]X by A10;
    hence H <> {} by PRE_TOPC:4;
    thus H is open by A11;
    (A \/ F)` = {}X by A12,Th2;
    hence ( A`) /\ H = {} by XBOOLE_1:53;
  end;
  then A` is nowhere_dense by Th28;
  hence thesis by Th39;
end;
