
theorem
  for R be non empty multLoopStr, A be non empty Subset of R,
  a be Element of R, F be LeftLinearCombination of A holds
  F*a is LinearCombination of A
proof
  let R be non empty multLoopStr, A be non empty Subset of R, a be Element of
  R, F be LeftLinearCombination of A;
  let i be set;
  reconsider c = a as Element of R;
  assume i in dom (F*a);
  then
A1: i in dom F by POLYNOM1:def 2;
  then consider u being Element of R, b being Element of A such that
A2: F/.i = u*b by Def9;
  take u, c, b;
  thus thesis by A1,A2,POLYNOM1:def 2;
end;
