
theorem
  for R be associative non empty multLoopStr, A be non empty Subset of
  R, a be Element of R, F be RightLinearCombination of A holds
  a*F is LinearCombination of A
proof
  let R be associative non empty multLoopStr, A be non empty Subset of R, a
  be Element of R, F be RightLinearCombination of A;
  let i be set;
  reconsider c = a as Element of R;
  assume i in dom (a*F);
  then
A1: i in dom F by POLYNOM1:def 1;
  then consider u being Element of R, b being Element of A such that
A2: F/.i = b*u by Def10;
  take c, u, b;
  thus (a*F)/.i=a*(F/.i) by A1,POLYNOM1:def 1
    .= c*b*u by A2,GROUP_1:def 3;
end;
