let r be Element of COMPLEX ; :: thesis: for s being Complex_Sequence st s is convergent holds
lim ((r (#) s) *') = (r *') * ((lim s) *')
let s be Complex_Sequence; :: thesis: ( s is convergent implies lim ((r (#) s) *') = (r *') * ((lim s) *') )
assume A1: s is convergent ; :: thesis: lim ((r (#) s) *') = (r *') * ((lim s) *')
hence lim ((r (#) s) *') = (lim (r (#) s)) *' by Th12
.= (r * (lim s)) *' by A1, Th18
.= (r *') * ((lim s) *') by COMPLEX1:35 ;
:: thesis: verum