
theorem Th26:
  for R be associative non empty multLoopStr, A be non empty
  Subset of R, a be Element of R, F be LeftLinearCombination of A holds
  a*F is LeftLinearCombination of A
proof
  let R be associative non empty multLoopStr, A be non empty Subset of R, a
  be Element of R, F be LeftLinearCombination 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 being Element of R, b being Element of A such that
A2: F/.i = u*b by Def9;
  take x = a*u, b;
  thus (a*F)/.i = a*F/.i by A1,POLYNOM1:def 1
    .= x*b by A2,GROUP_1:def 3;
end;
