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

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

assume A1: f is alpha-continuous ; :: thesis: ( f is ps-continuous & f is (alpha,ps)-continuous )
thus f is ps-continuous :: thesis: f is (alpha,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 X ^alpha by A1, Def28;
then f " V in (PSO X) /\ (D(alpha,ps) X) by Th13;
hence f " V in PSO X by XBOOLE_0:def 4; :: thesis: verum

end;

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

proof

let G be Subset of Y; :: according to DECOMP_1:def 37 :: thesis: ( G is open implies f " G in D(alpha,ps) X )
assume G is open ; :: thesis: f " G in D(alpha,ps) X
then f " G in X ^alpha by A1, Def28;
then f " G in (PSO X) /\ (D(alpha,ps) X) by Th13;
hence f " G in D(alpha,ps) X by XBOOLE_0:def 4; :: thesis: verum

end;

end;

assume A2: ( f is ps-continuous & f is (alpha,ps)-continuous ) ; :: thesis: f is alpha-continuous
let V be Subset of Y; :: according to DECOMP_1:def 28 :: thesis: ( V is open implies f " V in X ^alpha )
assume A3: V is open ; :: thesis: f " V in X ^alpha
then A4: f " V in PSO X by A2, Def29;
f " V in D(alpha,ps) X by A2, A3, Def37;
then f " V in (PSO X) /\ (D(alpha,ps) X) by A4, XBOOLE_0:def 4;
hence f " V in X ^alpha by Th13; :: thesis: verum