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 Th29;
hence lim |.(s (#) s').| = |.(lim (s (#) s')).| by Th11
.= |.((lim s) * (lim s')).| by A1, Th30
.= |.(lim s).| * |.(lim s').| by COMPLEX1:151 ;
:: thesis: verum