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 Th24:
  for R be Ring
  for V be LeftMod of R, l be Linear_Combination of V holds
  l = l!Carrier l
  proof
    let R be Ring;
    let V be LeftMod of R, l be Linear_Combination of V;
    set f = l | (Carrier l);
    set g = (Carrier l)` --> 0.R;
    set m = f +* g;
    A2: dom l = [#]V by FUNCT_2:92;
    then dom f = Carrier l by RELAT_1:62; then
    A3: (dom f) \/ (dom g) = [#]V by XBOOLE_1:45;
    for x being object st x in dom l holds l.x = m.x
    proof
      let x be object;
      assume x in dom l;
      then reconsider x as Element of V;
      per cases;
      suppose
        A5: x in Carrier l;
        then not x in dom g by XBOOLE_0:def 5;
        then m.x = f.x by A3,FUNCT_4:def 1;
        hence thesis by A5,FUNCT_1:49;
      end;
      suppose
        A6: not x in Carrier l;
        then x in (Carrier l)` by XBOOLE_0:def 5;
        then m.x = g.x & g.x = 0.R by A3,FUNCOP_1:7,FUNCT_4:def 1;
        hence thesis by A6;
      end;
    end;
    hence thesis by A2;
  end;
