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 Th26:
  for R being Ring, V being LeftMod of R
  for l being Linear_Combination of V
  for A being Subset of V, v being Element of V st not v in A holds
    (l!A).v = 0.R
  proof
    let R be Ring, V be LeftMod of R;
    let l be Linear_Combination of V;
    let A be Subset of V, v be Element of V;
    assume not v in A; then
    A1: v in A` by XBOOLE_0:def 5;
    A2: dom (l!A) = [#]V by FUNCT_2:92;
    dom ((A`) --> 0.R) = A` &
      dom (l!A) = (dom (l|A)) \/ (dom ((A`) --> 0.R))
      by FUNCT_4:def 1;
    then (l!A).v = ((A`) --> 0.R).v by A1,A2,FUNCT_4:def 1
    .= 0.R by A1,FUNCOP_1:7;
    hence thesis;
  end;
