let X be RealUnitarySpace; :: thesis: for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

let g be Point of X; :: thesis: for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

let a be Real; :: thesis: for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

let seq be sequence of X; :: thesis: ( seq is convergent & lim seq = g implies ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 ) )

assume ( seq is convergent & lim seq = g ) ; :: thesis: ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

then ( a * seq is convergent & lim (a * seq) = a * g ) by Th5, Th15;

hence ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 ) by Th22; :: thesis: verum

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

let g be Point of X; :: thesis: for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

let a be Real; :: thesis: for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

let seq be sequence of X; :: thesis: ( seq is convergent & lim seq = g implies ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 ) )

assume ( seq is convergent & lim seq = g ) ; :: thesis: ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

then ( a * seq is convergent & lim (a * seq) = a * g ) by Th5, Th15;

hence ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 ) by Th22; :: thesis: verum