let X be ComplexUnitarySpace; :: thesis: for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )
let g1, g2 be Point of X; :: thesis: for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )
let seq1, seq2 be sequence of X; :: thesis: ( seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 implies ( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 ) )
assume ( seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 ) ; :: thesis: ( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )
then ( seq1 + seq2 is convergent & lim (seq1 + seq2) = g1 + g2 ) by Th3, Th13;
hence ( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 ) by Th22; :: thesis: verum