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

now

let i be set ; :: thesis: ( i in dom p implies ex u being Element of R ex a being Element of A st p /. i = u * a )
assume i in dom p ; :: thesis: ex u being Element of R ex a being Element of A st p /. i = u * a
then i in {1} by FINSEQ_1:4, FINSEQ_1:55;
then A2: i = 1 by TARSKI:def 1;
take u = u; :: thesis: ex a being Element of A st p /. i = u * a
take a = a; :: thesis: p /. i = u * a
thus p /. i = u * a by A2, FINSEQ_4:25; :: thesis: verum

end;

hence ( p is LeftLinearCombination of A & not p is empty ) by Def10; :: thesis: verum