take s = (len p) + (len q); :: according to ALGSEQ_1:def 2 :: 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_1:22, XXREAL_0:2;
A3: (len p) + (len q) >= len q by NAT_1:11;
i in NAT by ORDINAL1:def 13;
hence (p + q) . i = (p . i) + (q . i) by NORMSP_1:def 5
.= (0. L) + (0. L) by A1, A2, A3, ALGSEQ_1:22, XXREAL_0:2
.= 0. L by RLVECT_1:def 7 ;
:: thesis: verum