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 Th25:
  for R being Ring for V being LeftMod of R
  for l being Linear_Combination of V
  for A being Subset of V, v being Element of V st v in A holds
  (l !A).v = l.v
  proof
    let R be Ring;
    let V be LeftMod of R;
    let l be Linear_Combination of V;
    let A be Subset of V, v be Element of V such that
    A1: v in A;
    dom (l!A) = [#]V by FUNCT_2:92; then
    A2: (dom (l|A)) \/ (dom ((A`) --> 0.R)) = [#]V by FUNCT_4:def 1;
    not v in dom ((A`) --> 0.R) by A1,XBOOLE_0:def 5;
    then (l!A).v = (l|A).v by A2,FUNCT_4:def 1
    .= l.v by A1,FUNCT_1:49;
    hence thesis;
  end;
