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 Th11
.= |.((lim s) + (lim s')).| by A1, Th14 ;
:: thesis: verum