let s, s' be Complex_Sequence; :: thesis: ( s is convergent & s' is convergent implies lim ((s + s') *' ) = ((lim s) *' ) + ((lim s') *' ) )
assume A1: ( s is convergent & s' is convergent ) ; :: thesis: lim ((s + s') *' ) = ((lim s) *' ) + ((lim s') *' )
then s + s' is convergent by Th13;
hence lim ((s + s') *' ) = (lim (s + s')) *' by Th12
.= ((lim s) + (lim s')) *' by A1, Th14
.= ((lim s) *' ) + ((lim s') *' ) by COMPLEX1:118 ;
:: thesis: verum