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 b_{1} being RealMap of T st b_{1} = abs f holds

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

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 b

b