let X be non empty TopSpace; :: thesis: for S being LinearTopSpace
for f, g being Function of X,S st f is continuous & g is continuous holds
f + g is continuous
let S be LinearTopSpace; :: thesis: for f, g being Function of X,S st f is continuous & g is continuous holds
f + g is continuous
let f, g be Function of X,S; :: thesis: ( f is continuous & g is continuous implies f + g is continuous )
assume A1: ( f is continuous & g is continuous ) ; :: thesis: f + g is continuous
for x being Point of X holds f + g is_continuous_at x

proof

let x be Point of X; :: thesis: f + g is_continuous_at x
( f is_continuous_at x & g is_continuous_at x ) by A1, TMAP_1:44;
hence f + g is_continuous_at x by Th1; :: thesis: verum

end;

hence f + g is continuous by TMAP_1:44; :: thesis: verum