reserve r,s,t,u for Real;

theorem Th2:
  for T being non empty TopSpace, X being non empty Subset of T, FX
  being Subset-Family of T st FX is Cover of X for x being Point of T st x in X
  ex W being Subset of T st x in W & W in FX
proof
  let T be non empty TopSpace, X be non empty Subset of T, FX be Subset-Family
  of T;
  assume FX is Cover of X;
  then
A1: X c= union FX by SETFAM_1:def 11;
  let x be Point of T;
  assume x in X;
  then consider W being set such that
A2: x in W and
A3: W in FX by A1,TARSKI:def 4;
  reconsider W as Subset of T by A3;
  take W;
  thus thesis by A2,A3;
end;
