reserve X for TopSpace;

theorem Th9:
  for X0 being SubSpace of X, C, A being Subset of X, B being
Subset of X0 st C is open & C c= the carrier of X0 & A c= C & A = B holds B is
  open iff A is open
proof
  let X0 be SubSpace of X, C, A be Subset of X, B be Subset of X0 such that
A1: C is open and
A2: C c= the carrier of X0 and
A3: A c= C and
A4: A = B;
  thus B is open implies A is open
  proof
    assume B is open;
    then consider F being Subset of X such that
A5: F is open and
A6: F /\ [#]X0 = B by TOPS_2:24;
    A c= F by A4,A6,XBOOLE_1:17;
    then
A7: A c= F /\ C by A3,XBOOLE_1:19;
    F /\ C c= A by A2,A4,A6,XBOOLE_1:26;
    hence thesis by A1,A5,A7,XBOOLE_0:def 10;
  end;
  thus thesis by A4,TOPS_2:25;
end;
