let S be non empty TopSpace; :: thesis: for T being non empty TopSpace-like reflexive TopRelStr
for x being set holds
( x is continuous Function of S,T iff x is Element of (ContMaps (S,T)) )
let T be non empty TopSpace-like reflexive TopRelStr ; :: thesis: for x being set holds
( x is continuous Function of S,T iff x is Element of (ContMaps (S,T)) )
let x be set ; :: thesis: ( x is continuous Function of S,T iff x is Element of (ContMaps (S,T)) )
thus ( x is continuous Function of S,T implies x is Element of (ContMaps (S,T)) ) by Def3; :: thesis: ( x is Element of (ContMaps (S,T)) implies x is continuous Function of S,T )
assume x is Element of (ContMaps (S,T)) ; :: thesis: x is continuous Function of S,T
then ex f being Function of S,T st
( x = f & f is continuous ) by Def3;
hence x is continuous Function of S,T ; :: thesis: verum