
theorem Th23:
  for T being TopSpace, K being Subset-Family of T
  holds K is prebasis of T iff FinMeetCl K is Basis of T
proof
  let T be TopSpace, BB be Subset-Family of T;
A1: T is empty implies the topology of T = bool the carrier of T
    by ZFMISC_1:1,Th8;
  thus BB is prebasis of T implies FinMeetCl BB is Basis of T
  proof
    assume
A2: BB is prebasis of T;
    then BB c= the topology of T by TOPS_2:64;
    then FinMeetCl BB c= FinMeetCl the topology of T by CANTOR_1:14;
    then
A3: FinMeetCl BB c= the topology of T by A1,CANTOR_1:5;
    consider A being Basis of T such that
A4: A c= FinMeetCl BB by A2,CANTOR_1:def 4;
A5: the topology of T c= UniCl A by CANTOR_1:def 2;
    UniCl A c= UniCl FinMeetCl BB by A4,CANTOR_1:9;
    then the topology of T c= UniCl FinMeetCl BB by A5;
    hence thesis by A3,CANTOR_1:def 2,TOPS_2:64;
  end;
  assume
A6: FinMeetCl BB is Basis of T;
A7: BB c= FinMeetCl BB by CANTOR_1:4;
  FinMeetCl BB c= the topology of T by A6,TOPS_2:64;
  then BB c= the topology of T by A7;
  hence thesis by A6,CANTOR_1:def 4,TOPS_2:64;
end;
