let X be RealUnitarySpace; :: 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 (Partial_Sums seq1) + (Partial_Sums seq2) is convergent by BHSP_2:3;
then Partial_Sums (seq1 + seq2) is convergent by Th1;
hence seq1 + seq2 is summable ; :: thesis: Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2)
thus Sum (seq1 + seq2) = lim ((Partial_Sums seq1) + (Partial_Sums seq2)) by Th1
.= (Sum seq1) + (Sum seq2) by A1, BHSP_2:13 ; :: thesis: verum