consider H being Function of T,R^1 such that

A1: for p being Point of T

for r1, r2 being Real st F . p = r1 & G . p = r2 holds

H . p = r1 + r2 and

A2: H is continuous by JGRAPH_2:19;

reconsider h = H as RealMap of T by TOPMETR:17;

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 object 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 for b_{1} being RealMap of T st b_{1} = f + g holds

b_{1} is continuous
by A2, JORDAN5A:27; :: thesis: verum

A1: for p being Point of T

for r1, r2 being Real st F . p = r1 & G . p = r2 holds

H . p = r1 + r2 and

A2: H is continuous by JGRAPH_2:19;

reconsider h = H as RealMap of T by TOPMETR:17;

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 object 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 for b

b