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 p-continuous iff f is sp-continuous (p,sp)-continuous
proof
  hereby
    assume
A1: f is p-continuous;
    thus f is sp-continuous
    proof
      let V be Subset of Y;
      assume V is open; then
      f"V in PO X by A1; then
      f"V in SPO X /\ D(p,sp)(X) by Th14;
      hence f"V in SPO X by XBOOLE_0:def 4;
    end;
    thus f is (p,sp)-continuous
    proof
      let G be Subset of Y;
      assume G is open;
      then f"G in PO X by A1;
      then f"G in SPO X /\ D(p,sp)(X) by Th14;
      hence thesis by XBOOLE_0:def 4;
    end;
  end;
  assume
A2: f is sp-continuous (p,sp)-continuous;
  let V be Subset of Y;
  assume V is open; then
  f"V in SPO X & f"V in D(p,sp)(X) by A2; then
  f"V in SPO X /\ D(p,sp)(X) by XBOOLE_0:def 4;
  hence thesis by Th14;
end;
