reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem
  for F being Subset-Family of A st F is open ex G being Subset-Family
  of T st G is open & for AA being Subset of T st AA = [#] A holds F = G|AA
proof
  let F be Subset-Family of A;
  assume
A1: F is open;
  [#] A c= [#]T by PRE_TOPC:def 4;
  then reconsider At = [#] A as Subset of T;
  defpred X[Subset of T] means ex X1 being Subset of T st X1 = $1 & X1 is open
  & $1 /\ At in F;
  consider G being Subset-Family of T such that
A2: for X being Subset of T holds X in G iff X[X] from SUBSET_1:sch 3;
  take G;
  thus G is open
  proof
    let H be Subset of T;
    assume H in G;
    then ex X1 being Subset of T st X1 = H & X1 is open & H /\ At in F by A2;
    hence thesis;
  end;
  let AA be Subset of T;
  assume
A3: AA = [#] A;
  then F c= bool AA;
  then F c= bool [#](T|AA) by PRE_TOPC:def 5;
  then reconsider FF = F as Subset-Family of T|AA;
  for X being Subset of T|AA holds X in FF iff ex X9 being Subset of T st
  X9 in G & X9 /\ AA=X
  proof
    let X be Subset of T|AA;
    thus X in FF implies ex X9 being Subset of T st X9 in G & X9 /\ AA=X
    proof
      assume
A4:   X in FF;
      then reconsider XX=X as Subset of A;
      XX is open by A1,A4;
      then consider Y being Subset of T such that
A5:   Y is open & Y /\ [#] A = XX by Th24;
      take Y;
      thus thesis by A2,A3,A4,A5;
    end;
    given X9 being Subset of T such that
A6: X9 in G and
A7: X9 /\ AA=X;
    ex X1 being Subset of T st X1 = X9 & X1 is open & X9 /\ At in F by A2,A6;
    hence thesis by A3,A7;
  end;
  hence thesis by Def3;
end;
