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 Th92:
  for A being Subset of X holds A in the topology of X iff the
  topology of X = A-extension_of_the_topology_of X
proof
  let A be Subset of X;
  thus A in the topology of X implies the topology of X = A
  -extension_of_the_topology_of X
  proof
    assume
A1: A in the topology of X;
    now
      let W be object;
      assume W in A-extension_of_the_topology_of X;
      then consider H, G being Subset of X such that
A2:   W = H \/ (G /\ A) and
A3:   H in the topology of X and
A4:   G in the topology of X;
      reconsider H1 = H as Subset of X;
A5:   G /\ A is open by A1,A4,PRE_TOPC:def 1;
      H1 is open by A3;
      then H1 \/ (G /\ A) is open by A5;
      hence W in the topology of X by A2;
    end;
    then
A6: A-extension_of_the_topology_of X c= the topology of X by TARSKI:def 3;
    the topology of X c= A-extension_of_the_topology_of X by Th88;
    hence thesis by A6;
  end;
  thus thesis by Th91;
end;
