reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;
reserve V,W for Z_Module;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;

theorem
  for R being Ring
  for V being LeftMod of R, A being Subset of V,
      l being Linear_Combination of A
  for X being Subset of V st X misses Carrier l & X <> {} holds
    l .: X = {0.R}
  proof
    let R be Ring;
    let V be LeftMod of R, A be Subset of V;
    let l be Linear_Combination of A;
    let X be Subset of V such that
    A1: X misses Carrier l and
    A2: X <> {};
    dom l = [#]V by FUNCT_2:92;
    hence thesis by A1,A2,Th28,ZFMISC_1:33;
  end;
