theorem
  for F being Subset-Family of A holds F is Subset-Family of T
proof
  let F be Subset-Family of A;
  [#] A c= [#]T by PRE_TOPC:def 4;
  then bool the carrier of A c= bool the carrier of T by ZFMISC_1:67;
  hence thesis by XBOOLE_1:1;
end;
