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

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

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

end;

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

proof

let G be Subset of Y; :: according to DECOMP_1:def 40 :: thesis: ( G is open implies f " G in D(sp,ps) X )
assume G is open ; :: thesis: f " G in D(sp,ps) X
then f " G in SPO X by A1;
then f " G in (PSO X) /\ (D(sp,ps) X) by Th17;
hence f " G in D(sp,ps) X by XBOOLE_0:def 4; :: thesis: verum

end;

end;

assume A2: ( f is ps-continuous & f is (sp,ps)-continuous ) ; :: thesis: f is sp-continuous
let V be Subset of Y; :: according to DECOMP_1:def 30 :: thesis: ( V is open implies f " V in SPO X )
assume V is open ; :: thesis: f " V in SPO X
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 f " V in SPO X by Th17; :: thesis: verum