consider H being Function of T,R^1 such that
A7: for p being Point of T
for r1, r2 being Real st F . p = r1 & G . p = r2 holds
H . p = r1 * r2 and
A8: H is continuous by JGRAPH_2:25;
reconsider h = H as RealMap of T by TOPMETR:17;
A9: 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 A7;
then h = f (#) g by A9, VALUED_1:def 4;
hence for b₁ being RealMap of T st b₁ = f (#) g holds
b₁ is continuous by A8, JORDAN5A:27; :: thesis: verum