theorem
  for B be Basis of TM st TM is Lindelof ex B9 be Basis of TM st B9 c= B
  & B9 is countable
proof
  let B be Basis of TM;
  assume TM is Lindelof;
  then for F be Subset-Family of TM st F is open & F is Cover of TM ex G be
  Subset-Family of TM st G c=F & G is Cover of TM & card G c=omega by Lm8;
  then consider underB be Basis of TM such that
A1: underB c=B & card underB c=omega by Th23;
  take underB;
  thus thesis by A1;
end;
