reconsider F = f as continuous Function of T,R^1 by JORDAN5A:27, TOPMETR:17;
set c = the carrier of T;
consider H being Function of T,R^1 such that
A1: for p being Point of T
for r1 being Real st F . p = r1 holds
H . p = r * r1 and
A2: H is continuous by JGRAPH_2:23;
reconsider h = H as RealMap of T by TOPMETR:17;
A3: dom h = the carrier of T by FUNCT_2:def 1
.= dom f by FUNCT_2:def 1 ;
for c being object st c in dom h holds
h . c = r * (f . c) by A1;
then h = r (#) f by A3, VALUED_1:def 5;
hence for b₁ being RealMap of T st b₁ = r (#) f holds
b₁ is continuous by A2, JORDAN5A:27; :: thesis: verum