let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 | Y is constant holds
f1 + f2 is_bounded_on X /\ Y
let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 | Y is constant holds
f1 + f2 is_bounded_on X /\ Y
let f1, f2 be PartFunc of M,the carrier of V; :: thesis: for X, Y being set st f1 is_bounded_on X & f2 | Y is constant holds
f1 + f2 is_bounded_on X /\ Y
let X, Y be set ; :: thesis: ( f1 is_bounded_on X & f2 | Y is constant implies f1 + f2 is_bounded_on X /\ Y )
assume A1: ( f1 is_bounded_on X & f2 | Y is constant ) ; :: thesis: f1 + f2 is_bounded_on X /\ Y
then f2 is_bounded_on Y by Th54;
hence f1 + f2 is_bounded_on X /\ Y by A1, Th46; :: thesis: verum