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

now :: thesis: for i being set st i in dom p holds
ex u, v being Element of R ex a being Element of A st p /. i = (u * a) * v

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 = the Element of R; :: thesis: ex v being Element of R ex a being Element of A st p /. i = (u * a) * v
take v = the Element of R; :: thesis: ex a being Element of A st p /. i = (u * a) * v
take a = the Element of A; :: thesis: p /. i = (u * a) * v
i in {1} by A1, FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by TARSKI:def 1;
hence p /. i = (u * a) * v by FINSEQ_4:16; :: thesis: verum

end;

hence ( p is LinearCombination of A & not p is empty ) by Def8; :: thesis: verum