reconsider F = f, G = g as continuous Function of T,R^1 by TOPMETR:24, TOPREAL6:83;
consider H being Function of T,R^1 such that
A1: for p being Point of T
for r1, r2 being real number st F . p = r1 & G . p = r2 holds
H . p = r1 + r2 and
A2: H is continuous by JGRAPH_2:29;
reconsider h = H as RealMap of T by TOPMETR:24;
A3: dom h = the carrier of T /\ the carrier of T by FUNCT_2:def 1
.= the carrier of T /\ (dom g) by FUNCT_2:def 1
.= (dom f) /\ (dom g) by FUNCT_2:def 1 ;
for c being set st c in dom h holds
h . c = (f . c) + (g . c) by A1;
then h = f + g by A3, VALUED_1:def 1;
hence f + g is continuous RealMap of T by A2, TOPREAL6:83; :: thesis: verum