
theorem Th2:
  for S being 1-sorted, X being Subset of S holds X` = the carrier
  of S iff X is empty
proof
  let S be 1-sorted, X be Subset of S;
  hereby
    assume X` = the carrier of S;
    then X = ([#](the carrier of S))`;
    hence X is empty by XBOOLE_1:37;
  end;
  assume X is empty;
  hence thesis;
end;
