consider a being Element of A;
consider u, v being Element of R;
reconsider p = <*((u * a) * v)*> as FinSequence of the carrier of R ;
take p ; :: thesis: ( p is LinearCombination of A & not p is empty )

now

let i be set ; :: thesis: ( i in dom p implies ex u, v being Element of R ex a being Element of A st p /. i = (u * a) * v )
assume A1: i in dom p ; :: thesis: ex u, v being Element of R ex a being Element of A st p /. i = (u * a) * v
take u = u; :: thesis: ex v being Element of R ex a being Element of A st p /. i = (u * a) * v
take v = v; :: thesis: ex a being Element of A st p /. i = (u * a) * v
take a = a; :: thesis: p /. i = (u * a) * v
i in {1} by A1, FINSEQ_1:4, FINSEQ_1:55;
then i = 1 by TARSKI:def 1;
hence p /. i = (u * a) * v by FINSEQ_4:25; :: thesis: verum

end;

hence ( p is LinearCombination of A & not p is empty ) by Def9; :: thesis: verum