reserve T for TopSpace,
  B for Subset of T;
reserve X,Y for non empty TopSpace;
reserve f for Function of X,Y;

theorem
  f is sp-continuous iff f is ps-continuous (sp,ps)-continuous
proof
  hereby
    assume
A1: f is sp-continuous;
    thus f is ps-continuous
    proof
      let V be Subset of Y;
      assume V is open; then
      f"V in SPO X by A1; then
      f"V in PSO X /\ D(sp,ps)(X) by Th17;
      hence f"V in PSO X by XBOOLE_0:def 4;
    end;
    thus f is (sp,ps)-continuous
    proof
      let G be Subset of Y;
      assume G is open;
      then f"G in SPO X by A1;
      then f"G in PSO X /\ D(sp,ps)(X) by Th17;
      hence thesis by XBOOLE_0:def 4;
    end;
  end;
  assume
A2: f is ps-continuous (sp,ps)-continuous;
  let V be Subset of Y;
  assume V is open;
  then f"V in PSO X & f"V in D(sp,ps)(X) by A2;
  then f"V in PSO X /\ D(sp,ps)(X) by XBOOLE_0:def 4;
  hence thesis by Th17;
end;
