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;

theorem Th28:
  for R being Ring, V being LeftMod of R,
      l be Linear_Combination of V
  for X being Subset of V st X misses Carrier l holds l .: X c= {0.R}
  proof
    let R be Ring, V be LeftMod of R, l be Linear_Combination of V;
    let X be Subset of V such that
    A1: X misses Carrier l;
    set M = l .: X;
    for y be object st y in l .: X holds y in {0.R}
    proof
      let y be object;
      assume y in M;
      then consider x being object such that
      x in dom l and
      A2: x in X and
      A3: y = l.x by FUNCT_1:def 6;
      reconsider x as Element of V by A2;
      now
        assume l.x <> 0.R;
        then x in Carrier l;
        hence contradiction by A1,A2,XBOOLE_0:def 4;
      end;
      hence thesis by A3,TARSKI:def 1;
    end;
    hence l .: X c= {0.R};
  end;
