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 ::  General Topology Th 4.1.10
  for B be Subset of T,F be Subset of T|A st F = B holds T|A|F = T|B
proof
  let B be Subset of T,F be Subset of T|A such that
A1: F = B;
  (T|A|F) is SubSpace of T & [#](T|A|F)=F by PRE_TOPC:def 5,TSEP_1:7;
  hence thesis by A1,PRE_TOPC:def 5;
end;
