theorem Th24:
  for B being Subset of A holds B is open iff ex C being Subset of
  T st C is open & C /\ [#](A) = B
proof
  let B be Subset of A;
  hereby
    assume B is open;
    then B in the topology of A;
    then consider C being Subset of T such that
A1: C in the topology of T & C /\ [#](A) = B by PRE_TOPC:def 4;
    reconsider C as Subset of T;
    take C;
    thus C is open & C /\ [#] A = B by A1;
  end;
  given C being Subset of T such that
A2: C is open and
A3: C /\ [#](A) = B;
  C in the topology of T by A2;
  then B in the topology of A by A3,PRE_TOPC:def 4;
  hence thesis;
end;
