
theorem Th51:
  for S, T being TopSpace
  st ex K being prebasis of S, L being prebasis of T
    st [#]S in K & K = INTERSECTION(L,{[#]S})
  holds S is SubSpace of T
proof
  let S, T be TopSpace;
  given K being prebasis of S, L being prebasis of T such that
    A1: [#]S in K & K = INTERSECTION(L,{[#]S});
  consider B, S0 being set such that
    A2: B in L & S0 in {[#]S} & [#]S = B /\ S0 by A1, SETFAM_1:def 5;
  B c= the carrier of T by A2;
  then A3: B c= [#]T by STRUCT_0:def 3;
  [#]S c= B by A2, XBOOLE_1:17;
  hence thesis by A1, A3, Th50, XBOOLE_1:1;
end;
