
theorem Th12:
  for T being non empty TopStruct, x being Point of T, B being
  Basis of x, V being Subset of T st V in B holds V is open & x in V
proof
  let T be non empty TopStruct, x be Point of T, B be Basis of x, V be Subset
  of T such that
A1: V in B;
  B c= the topology of T by TOPS_2:64;
  hence V is open by A1;
  x in Intersect B by Def1;
  hence thesis by A1,SETFAM_1:43;
end;
