let X, Y be non empty TopSpace; :: thesis: for f being Function of X,Y holds
( f is p-continuous iff ( f is ps-continuous & f is (p,ps)-continuous ) )
let f be Function of X,Y; :: thesis: ( f is p-continuous iff ( f is ps-continuous & f is (p,ps)-continuous ) )

hereby :: thesis: ( f is ps-continuous & f is (p,ps)-continuous implies f is p-continuous )

assume A1: f is p-continuous ; :: thesis: ( f is ps-continuous & f is (p,ps)-continuous )
thus f is ps-continuous :: thesis: f is (p,ps)-continuous

proof

let V be Subset of Y; :: according to DECOMP_1:def 29 :: thesis: ( V is open implies f " V in PSO X )
assume V is open ; :: thesis: f " V in PSO X
then f " V in PO X by A1;
then f " V in (PSO X) /\ (D(p,ps) X) by Th15;
hence f " V in PSO X by XBOOLE_0:def 4; :: thesis: verum

end;

thus f is (p,ps)-continuous :: thesis: verum

proof

let G be Subset of Y; :: according to DECOMP_1:def 38 :: thesis: ( G is open implies f " G in D(p,ps) X )
assume G is open ; :: thesis: f " G in D(p,ps) X
then f " G in PO X by A1;
then f " G in (PSO X) /\ (D(p,ps) X) by Th15;
hence f " G in D(p,ps) X by XBOOLE_0:def 4; :: thesis: verum

end;

end;

assume A2: ( f is ps-continuous & f is (p,ps)-continuous ) ; :: thesis: f is p-continuous
let V be Subset of Y; :: according to DECOMP_1:def 27 :: thesis: ( V is open implies f " V in PO X )
assume V is open ; :: thesis: f " V in PO X
then ( f " V in PSO X & f " V in D(p,ps) X ) by A2;
then f " V in (PSO X) /\ (D(p,ps) X) by XBOOLE_0:def 4;
hence f " V in PO X by Th15; :: thesis: verum