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 )
i in NAT by ORDINAL1:def 12;
then consider r being FinSequence of the carrier of L such that
A1: len r = i + 1 and
A2: (p *' q) . i = Sum r and
A3: for k being Element of NAT st k in dom r holds
r . k = (p . (k -' 1)) * (q . ((i + 1) -' k)) by Def9;
assume i >= s ; :: thesis: (p *' q) . i = 0. L
then len p <= i - (len q) by XREAL_1:19;
then A4: - (len p) >= - (i - (len q)) by XREAL_1:24;

now :: thesis: for k being Element of NAT st k in dom r holds
r . k = 0. L

let k be Element of NAT ; :: thesis: ( k in dom r implies r . b₁ = 0. L )
assume A5: k in dom r ; :: thesis: r . b₁ = 0. L
then A6: r . k = (p . (k -' 1)) * (q . ((i + 1) -' k)) by A3;
k <= i + 1 by A1, A5, FINSEQ_3:25;
then A7: (i + 1) - k >= 0 by XREAL_1:48;

per cases ( k -' 1 < len p or k -' 1 >= len p ) ;

suppose k -' 1 < len p ; :: thesis: r . b₁ = 0. L

then k - 1 < len p by XREAL_0:def 2;
then - (k - 1) > - (len p) by XREAL_1:24;
then 1 - k > (len q) - i by A4, XXREAL_0:2;
then i + (1 - k) > len q by XREAL_1:19;
then (i + 1) -' k >= len q by A7, XREAL_0:def 2;
then q . ((i + 1) -' k) = 0. L by ALGSEQ_1:8;
hence r . k = 0. L by A6; :: thesis: verum

end;

suppose k -' 1 >= len p ; :: thesis: r . b₁ = 0. L

then p . (k -' 1) = 0. L by ALGSEQ_1:8;
hence r . k = 0. L by A6; :: thesis: verum

end;

end;

end;

hence (p *' q) . i = 0. L by A2, Th1; :: thesis: verum