
theorem
  for T being TopSpace, B being Basis of T holds
  B \/ {the carrier of T} is Basis of T
proof
  let T be TopSpace, B be Basis of T;
  set C = B \/ {the carrier of T};
A1: the carrier of T in the topology of T by PRE_TOPC:def 1;
A2: B c= the topology of T by TOPS_2:64;
A3: {the carrier of T} c= the topology of T by A1,ZFMISC_1:31;
  then C c= the topology of T by A2,XBOOLE_1:8;
  then reconsider C as Subset-Family of T by XBOOLE_1:1;
A4: C c= the topology of T by A2,A3,XBOOLE_1:8;
A5: UniCl B c= UniCl C by CANTOR_1:9,XBOOLE_1:7;
  the topology of T c= UniCl B by CANTOR_1:def 2;
  then the topology of T c= UniCl C by A5;
  hence thesis by A4,CANTOR_1:def 2,TOPS_2:64;
end;
