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 Th90:
  for A being Subset of X holds for C, D being Subset of X st C
in the topology of X & D in {G /\ A where G is Subset of X : G in the topology
  of X} holds C \/ D in A-extension_of_the_topology_of X & C /\ D in A
  -extension_of_the_topology_of X
proof
  let A be Subset of X;
  let C, D be Subset of X;
  assume
A1: C in the topology of X;
  assume D in {G /\ A where G is Subset of X : G in the topology of X};
  then consider G being Subset of X such that
A2: D = G /\ A and
A3: G in the topology of X;
  thus C \/ D in A-extension_of_the_topology_of X by A1,A2,A3;
  thus C /\ D in A-extension_of_the_topology_of X
  proof
    {}X = ({});
    then reconsider H = {} as Subset of X;
A4: C /\ D = H \/ ((C /\ G) /\ A) & H in the topology of X by A2,PRE_TOPC:1
,XBOOLE_1:16;
    C /\ G in the topology of X by A1,A3,PRE_TOPC:def 1;
    hence thesis by A4;
  end;
end;
