let X be ComplexNormSpace; :: thesis: for seq1, seq2 being sequence of X st seq1 is summable & seq2 is summable holds
( seq1 + seq2 is summable & Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2) )
let seq1, seq2 be sequence of X; :: thesis: ( seq1 is summable & seq2 is summable implies ( seq1 + seq2 is summable & Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2) ) )
assume ( seq1 is summable & seq2 is summable ) ; :: thesis: ( seq1 + seq2 is summable & Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2) )
then A1: ( Partial_Sums seq1 is convergent & Partial_Sums seq2 is convergent ) ;
then A2: (Partial_Sums seq1) + (Partial_Sums seq2) is convergent by CLVECT_1:113;
A3: (Partial_Sums seq1) + (Partial_Sums seq2) = Partial_Sums (seq1 + seq2) by Th15;
Sum (seq1 + seq2) = lim ((Partial_Sums seq1) + (Partial_Sums seq2)) by Th15
.= (lim (Partial_Sums seq1)) + (lim (Partial_Sums seq2)) by A1, CLVECT_1:119 ;
hence ( seq1 + seq2 is summable & Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2) ) by A2, A3; :: thesis: verum