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 Th36:
  for R being Ring
  for V,W being LeftMod of R
  for l being Linear_Combination of V,
      T being linear-transformation of V,W
  for w being Element of W st w in Carrier (T@*l) holds T"{w} meets Carrier l
  proof
    let R be Ring;
    let V,W be LeftMod of R;
    let l be Linear_Combination of V,
        T be linear-transformation of V,W;
    let w be Element of W;
    assume w in Carrier (T@*l); then
    A1: (T@*l).w <> 0.R by ZMODUL02:8;
    assume T"{w} misses Carrier l; then
    lCFST(l,T,w) = <*>the carrier of R; then
    Sum(lCFST(l,T,w)) = 0.R by RLVECT_1:43;
    hence thesis by A1,LDef5;
  end;
