reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem Th4:
  for T being non empty TopStruct, B being Basis of T, V being
Subset of T st V is open & V <> {} ex W being Subset of T st W in B & W c= V &
  W <> {}
proof
  let T be non empty TopStruct, B be Basis of T, V be Subset of T;
  assume that
A1: V is open and
A2: V <> {};
  consider x being object such that
A3: x in V by A2,XBOOLE_0:def 1;
  V = union { G where G is Subset of T : G in B & G c= V } by A1,YELLOW_8:9;
  then consider Y being set such that
A4: x in Y and
A5: Y in { G where G is Subset of T : G in B & G c= V } by A3,TARSKI:def 4;
  ex Z being Subset of T st Z = Y & Z in B & Z c= V by A5;
  hence thesis by A4;
end;
