let X be RealUnitarySpace; :: thesis: for x, g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )
let x, g be Point of X; :: thesis: for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )
let seq be sequence of X; :: thesis: ( seq is convergent & lim seq = g implies ( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 ) )
assume ( seq is convergent & lim seq = g ) ; :: thesis: ( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )
then ( seq + x is convergent & lim (seq + x) = g + x ) by Th7, Th17;
hence ( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 ) by Th22; :: thesis: verum