reserve T for TopSpace,
  B for Subset of T;

theorem Th12:
  SO T /\ D(alpha,s)(T) = T^alpha
proof
  thus SO T /\ D(alpha,s)(T) c= T^alpha
  proof
    let x be object;
    assume
A1: x in SO T /\ D(alpha,s)(T);
    then x in SO T by XBOOLE_0:def 4;
    then consider A being Subset of T such that
A2: x = A and
A3: A is semi-open;
    x in D(alpha,s)(T) by A1,XBOOLE_0:def 4;
    then consider Z being Subset of T such that
A4: x = Z and
A5: alphaInt Z = sInt Z;
    Z is alpha-set of T by A2,A4,A5,Th2,A3,Th3;
    hence thesis by A4;
  end;
  let x be object;
  assume x in T^alpha;
  then consider K being Subset of T such that
A6: x = K and
A7: K is alpha-set of T;
A8: Int Cl Int K c= Cl Int K by TOPS_1:16;
  K c= Int Cl Int K by A7,Def1;
  then K c= Cl Int K by A8;
  then
A9: K is semi-open;
  then K = sInt K by Th3;
  then alphaInt K = sInt K by A7,Th2;
  then
A10: K in {B: alphaInt B = sInt B};
  K in {B:B is semi-open} by A9;
  hence thesis by A6,A10,XBOOLE_0:def 4;
end;
