let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f being PartFunc of M,V
for Y being set st f | Y is constant holds
( ||.f.|| | Y is constant & (- f) | Y is constant )
let V be ComplexNormSpace; :: thesis: for f being PartFunc of M,V
for Y being set st f | Y is constant holds
( ||.f.|| | Y is constant & (- f) | Y is constant )
let f be PartFunc of M,V; :: thesis: for Y being set st f | Y is constant holds
( ||.f.|| | Y is constant & (- f) | Y is constant )
let Y be set ; :: thesis: ( f | Y is constant implies ( ||.f.|| | Y is constant & (- f) | Y is constant ) )
assume f | Y is constant ; :: thesis: ( ||.f.|| | Y is constant & (- f) | Y is constant )
then consider r being VECTOR of V such that
A1: for c being Element of M st c in Y /\ (dom f) holds
f /. c = r by PARTFUN2:35;
A2: ||.r.|| in REAL by XREAL_0:def 1;

now :: thesis: for c being Element of M st c in Y /\ (dom ||.f.||) holds
||.f.|| . c = ||.r.||

let c be Element of M; :: thesis: ( c in Y /\ (dom ||.f.||) implies ||.f.|| . c = ||.r.|| )
assume A3: c in Y /\ (dom ||.f.||) ; :: thesis: ||.f.|| . c = ||.r.||
then A4: c in Y by XBOOLE_0:def 4;
A5: c in dom ||.f.|| by A3, XBOOLE_0:def 4;
then c in dom f by NORMSP_0:def 3;
then A6: c in Y /\ (dom f) by A4, XBOOLE_0:def 4;
thus ||.f.|| . c = ||.(f /. c).|| by A5, NORMSP_0:def 3
.= ||.r.|| by A1, A6 ; :: thesis: verum

end;

hence ||.f.|| | Y is constant by PARTFUN2:57, A2; :: thesis: (- f) | Y is constant

now :: thesis: ex p being Element of the U1 of V st
for c being Element of M st c in Y /\ (dom (- f)) holds
(- f) /. c = p

take p = - r; :: thesis: for c being Element of M st c in Y /\ (dom (- f)) holds
(- f) /. c = p
let c be Element of M; :: thesis: ( c in Y /\ (dom (- f)) implies (- f) /. c = p )
assume A7: c in Y /\ (dom (- f)) ; :: thesis: (- f) /. c = p
then c in Y /\ (dom f) by VFUNCT_1:def 5;
then A8: - (f /. c) = p by A1;
c in dom (- f) by A7, XBOOLE_0:def 4;
hence (- f) /. c = p by A8, VFUNCT_1:def 5; :: thesis: verum

end;

hence (- f) | Y is constant by PARTFUN2:35; :: thesis: verum