
theorem
  for R being non empty multLoopStr, A being non empty Subset of R, L be
  LinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier
  of R, the carrier of R:] st E represents L
proof
  let R be non empty multLoopStr,A be non empty Subset of R, L be
  LinearCombination of A;
  set D = [:the carrier of R, the carrier of R, the carrier of R:];
  defpred P[set,set] means ex x, y, z being Element of R st $2 = [x, y, z] & y
  in A & L/.$1 = x*y*z;
A1: now
    let k be Nat;
    assume k in Seg len L;
    then k in dom L by FINSEQ_1:def 3;
    then consider u,v being Element of R, a being Element of A such that
A2: L/.k = u*a*v by Def8;
    reconsider b = a as Element of R;
    reconsider d = [u, b, v] as Element of D;
    take d;
    thus P[k, d] by A2;
  end;
  consider E being FinSequence of D such that
A3: dom E = Seg len L and
A4: for k be Nat st k in Seg len L holds P[k,E/.k] from
  RECDEF_1:sch 17(A1);
  take E;
  thus len E = len L by A3,FINSEQ_1:def 3;
  let i being set such that
A5: i in dom L;
  reconsider k = i as Element of NAT by A5;
  dom L = Seg len L by FINSEQ_1:def 3;
  then consider x, y, z being Element of R such that
A6: E/.k = [x, y, z] and
A7: y in A and
A8: L/.k = x*y*z by A4,A5;
  thus L.i = ((E/.i)`1_3)*((E/.i)`2_3)*((E/.i)`3_3)
             by A5,A6,A8,PARTFUN1:def 6;
  thus thesis by A6,A7;
end;
