
theorem Th31:
  for R being commutative associative non empty multLoopStr, A
  being non empty Subset of R, f being LinearCombination of A holds
  f is LeftLinearCombination of A & f is RightLinearCombination of A
proof
  let R being commutative associative non empty multLoopStr, A being non
  empty Subset of R, f being LinearCombination of A;
  hereby
    let i be set;
    assume i in dom f;
    then consider r,s being Element of R, a being Element of A such that
A1: f/.i = r*a*s by Def8;
    f/.i = (r*s)*a by A1,GROUP_1:def 3;
    hence ex r being Element of R, a being Element of A st f/.i = r*a;
  end;
  let i be set;
  assume i in dom f;
  then consider r,s being Element of R, a being Element of A such that
A2: f/.i = r*a*s by Def8;
  f/.i = a*(r*s) by A2,GROUP_1:def 3;
  hence ex r being Element of R, a being Element of A st f/.i = a*r;
end;
