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 Th8:
  for Bas be Basis of T holds Bas|A is Basis of T|A
proof
  let Bas be Basis of T;
  set BasA=Bas|A;
  set TA=T|A;
A1: for U be Subset of TA st U is open for p be Point of TA st p in U ex a
  be Subset of TA st a in BasA & p in a & a c=U
  proof
    let U be Subset of TA;
    assume U is open;
    then consider W be Subset of T such that
A2: W is open and
A3: U=W/\the carrier of TA by TSP_1:def 1;
    let p be Point of TA such that
A4: p in U;
    p in W by A3,A4,XBOOLE_0:def 4;
    then consider Wb be Subset of T such that
A5: Wb in Bas and
A6: p in Wb and
A7: Wb c=W by A2,YELLOW_9:31;
A8: Wb/\A in BasA by A5,TOPS_2:31;
    then reconsider WbA=Wb/\A as Subset of TA;
A9: [#]TA=A by PRE_TOPC:def 5;
    then p in WbA by A4,A6,XBOOLE_0:def 4;
    hence thesis by A3,A7,A8,A9,XBOOLE_1:26;
  end;
  BasA c=the topology of TA
  proof
    let x be object such that
A10: x in BasA;
    reconsider U=x as Subset of TA by A10;
    BasA is open by TOPS_2:37;
    then U is open by A10;
    hence thesis;
  end;
  hence thesis by A1,YELLOW_9:32;
end;
