
theorem Th29:
  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
  F*a is RightLinearCombination 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;
  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 = b*u by Def10;
  take x = u*a, b;
  thus (F*a)/.i=(F/.i)*a by A1,POLYNOM1:def 2
    .= b*x by A2,GROUP_1:def 3;
end;
