take s = (len p) + (len q); :: according to ALGSEQ_1:def 1 :: thesis: for b₁ being set holds
( not s <= b₁ or (p + q) . b₁ = 0. L )
let i be Nat; :: thesis: ( not s <= i or (p + q) . i = 0. L )
assume A1: i >= s ; :: thesis: (p + q) . i = 0. L
(len p) + (len q) >= len p by NAT_1:11;
then A2: p . i = 0. L by A1, ALGSEQ_1:8, XXREAL_0:2;
A3: (len p) + (len q) >= len q by NAT_1:11;
thus (p + q) . i = (p . i) + (q . i) by NORMSP_1:def 2
.= (0. L) + (0. L) by A1, A2, A3, ALGSEQ_1:8, XXREAL_0:2
.= 0. L by RLVECT_1:def 4 ; :: thesis: verum