
theorem Th22:
  for T being non empty TopSpace, K,O being set st K c= O & O c=
  the topology of T holds (K is Basis of T implies O is Basis of T) & (K is
  prebasis of T implies O is prebasis of T)
proof
  let T be non empty TopSpace, K,O be set;
  assume that
A1: K c= O and
A2: O c= the topology of T;
  K c= the topology of T by A1,A2;
  then reconsider K9 = K, O9 = O as Subset-Family of T by A2,XBOOLE_1:1;
  reconsider K9, O9 as Subset-Family of T;
  reconsider K9, O9 as Subset-Family of T;
A3: UniCl K9 c= UniCl O9 by A1,CANTOR_1:9;
A4: UniCl the topology of T = the topology of T by CANTOR_1:6;
  then
A5: UniCl O9 c= the topology of T by A2,CANTOR_1:9;
  hereby
    assume K is Basis of T;
    then UniCl K9 = the topology of T by YELLOW_9:22;
    then UniCl O9 = the topology of T by A5,A3;
    hence O is Basis of T by YELLOW_9:22;
  end;
  FinMeetCl the topology of T = the topology of T by CANTOR_1:5;
  then FinMeetCl O9 c= the topology of T by A2,CANTOR_1:14;
  then
A6: UniCl FinMeetCl O9 c= the topology of T by A4,CANTOR_1:9;
  assume K is prebasis of T;
  then FinMeetCl K9 is Basis of T by YELLOW_9:23;
  then
A7: UniCl FinMeetCl K9 = the topology of T by YELLOW_9:22;
  FinMeetCl K9 c= FinMeetCl O9 by A1,CANTOR_1:14;
  then UniCl FinMeetCl K9 c= UniCl FinMeetCl O9 by CANTOR_1:9;
  then UniCl FinMeetCl O9 = the topology of T by A7,A6;
  then FinMeetCl O9 is Basis of T by YELLOW_9:22;
  hence thesis by YELLOW_9:23;
end;
