
theorem
  for R being non empty multLoopStr, A being non empty Subset of R, l
  being LeftLinearCombination of A, n being Nat holds l|Seg n is
  LeftLinearCombination of A
proof
  let R be non empty multLoopStr, A be non empty Subset of R, l be
  LeftLinearCombination of A, n being Nat;
  reconsider ln = l|Seg n as FinSequence of the carrier of R by FINSEQ_1:18;
  now
    let i being set such that
A1: i in dom ln;
A2: dom ln c= dom l by RELAT_1:60;
    then consider u being Element of R, a being Element of A such that
A3: l/.i = u*a by A1,Def9;
    take u, a;
    thus ln/.i = ln.i by A1,PARTFUN1:def 6
      .= l.i by A1,FUNCT_1:47
      .= u*a by A1,A2,A3,PARTFUN1:def 6;
  end;
  hence thesis by Def9;
end;
