reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;

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;
