let X be set ; :: thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let S, T be RealNormSpace; :: thesis: for f being PartFunc of S,T st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let f be PartFunc of S,T; :: thesis: ( f is_continuous_on X implies ( ||.f.|| is_continuous_on X & - f is_continuous_on X ) )
assume A1: f is_continuous_on X ; :: thesis: ( ||.f.|| is_continuous_on X & - f is_continuous_on X )
thus ||.f.|| is_continuous_on X :: thesis: - f is_continuous_on X

proof

A2: X c= dom f by A1;
hence A3: X c= dom ||.f.|| by NORMSP_0:def 3; :: according to NFCONT_1:def 8 :: thesis: for x0 being Point of S st x0 in X holds
||.f.|| | X is_continuous_in x0
let r be Point of S; :: thesis: ( r in X implies ||.f.|| | X is_continuous_in r )
assume A4: r in X ; :: thesis: ||.f.|| | X is_continuous_in r
then A5: f | X is_continuous_in r by A1;
thus ||.f.|| | X is_continuous_in r :: thesis: verum

proof

A6: r in (dom ||.f.||) /\ X by A3, A4, XBOOLE_0:def 4;
hence r in dom (||.f.|| | X) by RELAT_1:61; :: according to NFCONT_1:def 6 :: thesis: for s1 being sequence of S st rng s1 c= dom (||.f.|| | X) & s1 is convergent & lim s1 = r holds
( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
let s1 be sequence of S; :: thesis: ( rng s1 c= dom (||.f.|| | X) & s1 is convergent & lim s1 = r implies ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) ) )
assume that
A7: rng s1 c= dom (||.f.|| | X) and
A8: ( s1 is convergent & lim s1 = r ) ; :: thesis: ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
rng s1 c= (dom ||.f.||) /\ X by A7, RELAT_1:61;
then rng s1 c= (dom f) /\ X by NORMSP_0:def 3;
then A9: rng s1 c= dom (f | X) by RELAT_1:61;
then A10: (f | X) /. r = lim ((f | X) /* s1) by A5, A8;

now :: thesis: for n being Element of NAT holds ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n

let n be Element of NAT ; :: thesis: ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n
dom s1 = NAT by FUNCT_2:def 1;
then A11: s1 . n in rng s1 by FUNCT_1:3;
then s1 . n in dom (f | X) by A9;
then A12: s1 . n in (dom f) /\ X by RELAT_1:61;
then A13: s1 . n in X by XBOOLE_0:def 4;
s1 . n in dom f by A12, XBOOLE_0:def 4;
then A14: s1 . n in dom ||.f.|| by NORMSP_0:def 3;
thus ||.((f | X) /* s1).|| . n = ||.(((f | X) /* s1) . n).|| by NORMSP_0:def 4
.= ||.((f | X) /. (s1 . n)).|| by A9, FUNCT_2:109
.= ||.(f /. (s1 . n)).|| by A9, A11, PARTFUN2:15
.= ||.f.|| . (s1 . n) by A14, NORMSP_0:def 3
.= (||.f.|| | X) . (s1 . n) by A13, FUNCT_1:49
.= (||.f.|| | X) /. (s1 . n) by A7, A11, PARTFUN1:def 6
.= ((||.f.|| | X) /* s1) . n by A7, FUNCT_2:109 ; :: thesis: verum

end;

then A15: ||.((f | X) /* s1).|| = (||.f.|| | X) /* s1 by FUNCT_2:63;
A16: (f | X) /* s1 is convergent by A5, A8, A9;
hence (||.f.|| | X) /* s1 is convergent by A15, NORMSP_1:23; :: thesis: (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1)
||.((f | X) /. r).|| = ||.(f /. r).|| by A2, A4, PARTFUN2:17
.= ||.f.|| . r by A3, A4, NORMSP_0:def 3
.= ||.f.|| /. r by A3, A4, PARTFUN1:def 6
.= (||.f.|| | X) /. r by A6, PARTFUN2:16 ;
hence (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) by A16, A10, A15, LOPBAN_1:20; :: thesis: verum

end;

end;

(- 1) (#) f is_continuous_on X by A1, Th27;
hence - f is_continuous_on X by VFUNCT_1:23; :: thesis: verum