
theorem Th24:
  for R be associative non empty multLoopStr, A be non empty
  Subset of R, a be Element of R, F be LinearCombination of A holds F*a 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 LinearCombination 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, v being Element of R, b being Element of A such that
A2: F/.i = u*b*v by Def8;
  take u, x = v*a, b;
  thus (F*a)/.i = (F/.i)*a by A1,POLYNOM1:def 2
    .= u*(b*v)*a by A2,GROUP_1:def 3
    .= u*(b*v*a) by GROUP_1:def 3
    .= u*(b*(v*a)) by GROUP_1:def 3
    .= u*b*x by GROUP_1:def 3;
end;
