
theorem Th23:
  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 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 LinearCombination of A;
  let i be set;
  assume i in dom (a*F);
  then
A1: i in dom F by POLYNOM1:def 1;
  then consider u, v being Element of R, b being Element of A such that
A2: F/.i = u*b*v by Def8;
  take x = a*u, v, b;
  thus (a*F)/.i = a*F/.i by A1,POLYNOM1:def 1
    .= a*(u*b)*v by A2,GROUP_1:def 3
    .= x*b*v by GROUP_1:def 3;
end;
