let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f being PartFunc of M,V
for X, Y being set st Y c= X & f is_bounded_on X holds
f is_bounded_on Y
let V be ComplexNormSpace; :: thesis: for f being PartFunc of M,V
for X, Y being set st Y c= X & f is_bounded_on X holds
f is_bounded_on Y
let f be PartFunc of M,V; :: thesis: for X, Y being set st Y c= X & f is_bounded_on X holds
f is_bounded_on Y
let X, Y be set ; :: thesis: ( Y c= X & f is_bounded_on X implies f is_bounded_on Y )
assume that
A1: Y c= X and
A2: f is_bounded_on X ; :: thesis: f is_bounded_on Y
consider r being Real such that
A3: for x being Element of M st x in X /\ (dom f) holds
||.(f /. x).|| <= r by A2;
take r ; :: according to VFUNCT_2:def 3 :: thesis: for x being Element of M st x in Y /\ (dom f) holds
||.(f /. x).|| <= r
let x be Element of M; :: thesis: ( x in Y /\ (dom f) implies ||.(f /. x).|| <= r )
assume x in Y /\ (dom f) ; :: thesis: ||.(f /. x).|| <= r
then ( x in Y & x in dom f ) by XBOOLE_0:def 4;
then x in X /\ (dom f) by A1, XBOOLE_0:def 4;
hence ||.(f /. x).|| <= r by A3; :: thesis: verum