reserve T for TopSpace,
  B for Subset of T;

theorem Th8:
  SO T /\ D(c,s)(T) = the topology of T
proof
  thus SO T /\ D(c,s)(T) c= the topology of T
  proof
    let x be object;
    assume
A1: x in SO T /\ D(c,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(c,s)(T) by A1,XBOOLE_0:def 4;
    then consider Z being Subset of T such that
A4: x = Z and
A5: Int Z = sInt Z;
    A = sInt A by A3,Th3;
    hence thesis by A4,PRE_TOPC:def 2,A2,A5;
  end;
  let x be object;
  assume
A6: x in the topology of T;
  then reconsider K = x as Subset of T;
  K is open by A6,PRE_TOPC:def 2;
  then
A7: K = Int K by TOPS_1:23;
  then
A8: K is semi-open by PRE_TOPC:18;
  then Int K = sInt K by A7,Th3;
  then
A9: K in {B: Int B = sInt B};
  K in SO T by A8;
  hence thesis by A9,XBOOLE_0:def 4;
end;
