reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;

theorem
  for R being Ring, V being LeftMod of R
  for A being Subset of V, l being Linear_Combination of A,
      x being Element of V, a being Element of R holds
    l +* (x,a) is Linear_Combination of A \/ {x}
  proof
    let R be Ring, V be LeftMod of R;
    let A be Subset of V, l be Linear_Combination of A,
        x be Element of V, a be Element of R;
    set m = l +* (x,a);
    A1: dom m = dom l by FUNCT_7:30
    .= [#]V by FUNCT_2:def 1;
    rng m c= the carrier of R;
    then reconsider m as Element of Funcs ([#]V,the carrier of R)
      by A1,FUNCT_2:def 2;
    set T = Carrier l \/ {x};
    for v being Element of V st not v in T holds m.v = 0.R
    proof
      let v be Element of V such that
      A7: not v in T;
      not v in {x} by A7,XBOOLE_0:def 3;
      then v <> x by TARSKI:def 1; then
      A8: m.v = l.v by FUNCT_7:32;
      not v in Carrier l by A7,XBOOLE_0:def 3;
      hence thesis by A8;
    end;
    then reconsider m as Linear_Combination of V by VECTSP_6:def 1;
    A9: Carrier m c= T
    proof
      let y be object;
      assume y in Carrier m;
      then consider z being Element of V such that
      A10: y = z and
      A11: m.z <> 0.R;
      per cases;
      suppose
        A12: z = x;
        x in {x} & {x} c= T by TARSKI:def 1,XBOOLE_1:7;
        hence thesis by A10,A12;
      end;
      suppose
        z <> x;
        then m.z = l.z by FUNCT_7:32; then
        A13: z in Carrier l by A11;
        Carrier l c= T by XBOOLE_1:7;
        hence thesis by A10,A13;
      end;
    end;
    T c= A \/ {x} by XBOOLE_1:9,VECTSP_6:def 4;
    then Carrier m c= A \/ {x} by A9;
    hence thesis by VECTSP_6:def 4;
  end;
