reserve T for TopSpace,
  B for Subset of T;

theorem Th14:
  SPO T /\ D(p,sp)(T) = PO T
proof
  thus SPO T /\ D(p,sp)(T) c= PO T
  proof
    let x be object;
    assume x in SPO T /\ D(p,sp)(T); then
A0: x in SPO T & x in D(p,sp)(T) by XBOOLE_0:def 4; then
    consider B being Subset of T such that
A1: x = B & B is semi-pre-open;
A3: B = spInt B by A1,Th6;
    consider B1 being Subset of T such that
A2: x = B1 & pInt B1 = spInt B1 by A0;
    pInt B = B by A2,A3,A1; then
    B is pre-open by Th4;
    hence thesis by A1;
  end;
  let x be object;
  assume x in PO T;
  then consider K being Subset of T such that
A1: x = K and
A2: K is pre-open;
A3: Int Cl K c= Cl Int K \/ Int Cl K by XBOOLE_1:7;
  K c= Int Cl K by A2;
  then K c= Cl Int K \/ Int Cl K by A3;
  then
A4: K is semi-pre-open;
  then K = spInt K by Th6;
  then pInt K = spInt K by A2,Th4;
  then
A5: K in {B: pInt B = spInt B};
  K in SPO T by A4;
  hence thesis by A1,A5,XBOOLE_0:def 4;
end;
