reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th22:
  for X, Y being TopSpace, H being Subset-Family of X, Y being
  Subset of X st H is Cover of Y ex F being Subset-Family of X st F c= H & F is
  Cover of Y & for C being set st C in F holds C meets Y
proof
  let X, Y be TopSpace, H be Subset-Family of X;
  let Y be Subset of X;
  assume
A1: H is Cover of Y;
  defpred P[set] means $1 in H & $1 /\ Y <> {};
  { Z where Z is Subset of X: P[Z]} is Subset-Family of X from DOMAIN_1:
  sch 7;
  then reconsider F = { Z where Z is Subset of X: Z in H & Z /\ Y <> {}} as
  Subset-Family of X;
  reconsider F as Subset-Family of X;
  take F;
  thus F c= H
  proof
    let e be object;
    assume e in F;
    then ex Z being Subset of X st e = Z & Z in H & Z /\ Y <> {};
    hence thesis;
  end;
A2: Y c= union H by A1,SETFAM_1:def 11;
  thus F is Cover of Y
  proof
    let e be object;
    assume
A3: e in Y;
    then consider u such that
A4: e in u and
A5: u in H by A2,TARSKI:def 4;
    reconsider u as Subset of X by A5;
    u /\ Y <> {} by A3,A4,XBOOLE_0:def 4;
    then u in F by A5;
    hence e in union F by A4,TARSKI:def 4;
  end;
  let C be set;
  assume C in F;
  then ex Z being Subset of X st C = Z & Z in H & Z /\ Y <> {};
  hence C /\ Y <> {};
end;
