let S, T be TopSpace; :: thesis: for f being Function of S,T
for g being Function of TopStruct(# the carrier of S,the topology of S #),T st f = g holds
( f is continuous iff g is continuous )
let f be Function of S,T; :: thesis: for g being Function of TopStruct(# the carrier of S,the topology of S #),T st f = g holds
( f is continuous iff g is continuous )
let g be Function of TopStruct(# the carrier of S,the topology of S #),T; :: thesis: ( f = g implies ( f is continuous iff g is continuous ) )
assume Z0: f = g ; :: thesis: ( f is continuous iff g is continuous )
thus ( f is continuous implies g is continuous ) :: thesis: ( g is continuous implies f is continuous )

proof

assume Z: f is continuous ; :: thesis: g is continuous
let P1 be Subset of T; :: according to PRE_TOPC:def 12 :: thesis: ( P1 is closed implies g " P1 is closed )
assume P1 is closed ; :: thesis: g " P1 is closed
then f " P1 is closed by Z, Def12;
hence g " P1 is closed by Z0, Th60; :: thesis: verum

end;

assume Z: g is continuous ; :: thesis: f is continuous
let P1 be Subset of T; :: according to PRE_TOPC:def 12 :: thesis: ( P1 is closed implies f " P1 is closed )
assume P1 is closed ; :: thesis: f " P1 is closed
then g " P1 is closed by Z, Def12;
hence f " P1 is closed by Z0, Th60; :: thesis: verum