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 Th88:
  for A being Subset of X holds the topology of X c= A
  -extension_of_the_topology_of X
proof
  let A be Subset of X;
  now
    {}X = ({});
    then reconsider G = {} as Subset of X;
    let W be object;
    assume
A1: W in the topology of X;
    then reconsider H = W as Subset of X;
    H = H \/ (G /\ A) & G in the topology of X by PRE_TOPC:1;
    hence W in A-extension_of_the_topology_of X by A1;
  end;
  hence thesis by TARSKI:def 3;
end;
