let n be Element of NAT ; :: thesis: for CNS1, CNS2 being ComplexNormSpace
for seq being sequence of CNS1
for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
let CNS1, CNS2 be ComplexNormSpace; :: thesis: for seq being sequence of CNS1
for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
let seq be sequence of CNS1; :: thesis: for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
let h be PartFunc of CNS1,CNS2; :: thesis: ( rng seq c= dom h implies seq . n in dom h )
assume A1: rng seq c= dom h ; :: thesis: seq . n in dom h
n in NAT ;
then n in dom seq by FUNCT_2:def 1;
then n in dom (h * seq) by A1, RELAT_1:46;
hence seq . n in dom h by FUNCT_1:21; :: thesis: verum