reserve T, T1, T2 for TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  A1 for Subset of T1,
  A2 for Subset of T2,
  TM, TM1, TM2 for metrizable TopSpace,
  Am, Bm for Subset of TM,
  Fm, Gm for Subset-Family of TM,
  C for Cardinal,
  iC for infinite Cardinal;

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;
