reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem Th22:
  for x being Point of X, F being Subset-Family of X st for A
  being Subset of X holds A in F iff A is open closed & x in A holds F <> {}
proof
A1: [#] X is open closed;
  let x be Point of X, F be Subset-Family of X;
  assume for A being Subset of X holds A in F iff A is open closed & x in A;
  hence thesis by A1;
end;
