reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem Th13:
  for Y being anti-discrete TopStruct, A being Subset of Y st A in
  the topology of Y holds (the carrier of Y) \ A in the topology of Y
proof
  let Y be anti-discrete TopStruct, A be Subset of Y;
A1: the topology of Y = {{}, the carrier of Y} by Def2;
  assume A in the topology of Y;
  then A = {} or A = the carrier of Y by A1,TARSKI:def 2;
  then
  (the carrier of Y) \ A = the carrier of Y or (the carrier of Y) \ A = {}
  by XBOOLE_1:37;
  hence thesis by A1,TARSKI:def 2;
end;
