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 = |.r1.| and
A2: H is continuous by JGRAPH_4:7;
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 = |.(f . c).| by A1;
then h = abs f by A3, VALUED_1:def 11;
hence for b₁ being RealMap of T st b₁ = abs f holds
b₁ is continuous by A2, JORDAN5A:27; :: thesis: verum