reserve T for TopStruct;
reserve GX for TopSpace;

theorem Th4:
  for T being 1-sorted, P being Subset of T holds P <> [#]T iff [#]T \ P <> {}
proof
  let T be 1-sorted, P be Subset of T;
  now
    assume that
A1: P <> [#]T and
A2: [#]T \ P = {};
    [#]T c= P by A2,XBOOLE_1:37;
    hence contradiction by A1,XBOOLE_0:def 10;
  end;
  hence thesis by XBOOLE_1:37;
end;
