let s, s9 be Complex_Sequence; :: thesis: ( s is convergent & s9 is convergent implies lim |.(s + s9).| = |.((lim s) + (lim s9)).| )
assume A1: ( s is convergent & s9 is convergent ) ; :: thesis: lim |.(s + s9).| = |.((lim s) + (lim s9)).|
hence lim |.(s + s9).| = |.(lim (s + s9)).| by Th11
.= |.((lim s) + (lim s9)).| by A1, Th14 ;
:: thesis: verum