theorem Th91:
  for A being Subset of X holds A in A -extension_of_the_topology_of X
proof
  let A be Subset of X;
  X is SubSpace of X by TSEP_1:2;
  then reconsider G = the carrier of X as Subset of X by TSEP_1:1;
A1: G in the topology of X by PRE_TOPC:def 1;
  {}X = ({});
  then reconsider H = {} as Subset of X;
  A = H \/ (G /\ A) & H in the topology of X by PRE_TOPC:1,XBOOLE_1:28;
  hence thesis by A1;
end;
