set c = the carrier of T;
reconsider F = f as continuous Function of T,R^1 by TOPMETR:24, TOPREAL6:83;
consider H being Function of T,R^1 such that
A1: for p being Point of T
for r1 being real number st F . p = r1 holds
H . p = - r1 and
A2: H is continuous by JGRAPH_4:13;
reconsider h = H as RealMap of T by TOPMETR:24;
A3: dom h = the carrier of T by FUNCT_2:def 1
.= dom f by FUNCT_2:def 1 ;
for c being set st c in dom h holds
h . c = - (f . c) by A1;
then h = - f by A3, VALUED_1:9;
hence - f is continuous RealMap of T by A2, TOPREAL6:83; :: thesis: verum