
theorem Th24:
  for T being non empty TopSpace, B being Subset-Family of T
  st UniCl B is prebasis of T holds B is prebasis of T
proof
  let T be non empty TopSpace, B be Subset-Family of T;
  assume UniCl B is prebasis of T;
  then FinMeetCl UniCl B is Basis of T by Th23;
  then UniCl FinMeetCl UniCl B = the topology of T by Th22;
  then UniCl FinMeetCl B = the topology of T by Th21;
  then FinMeetCl B is Basis of T by Th22;
  hence thesis by Th23;
end;
