reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;

theorem
  for A being Subset of V, l being Linear_Combination of A,
      x being Element of V, a being Element of F holds
      l +* (x,a) is Linear_Combination of A \/ {x}
proof
  let A be Subset of V, l be Linear_Combination of A,
      x be Element of V, a be Element of F;
  set m = l +* (x,a);
A1: dom m = [#]V by FUNCT_2:def 1;
  rng m c= [#]F
  proof
    let y be object;
    assume y in rng m;
    then consider x9 being object such that
A2: x9 in dom m and
A3: m.x9 = y by FUNCT_1:def 3;
A4: x9 in dom l by A1,A2,FUNCT_2:92;
    per cases;
    suppose
      x9 = x;
      then m.x9 = a by A4,FUNCT_7:31;
      hence thesis by A3;
    end;
    suppose
A5:   x9 <> x;
A6:   l.x9 in rng l & rng l c= [#]F by A4,FUNCT_1:3,FUNCT_2:92;
      m.x9 = l.x9 by A5,FUNCT_7:32;
      hence thesis by A3,A6;
    end;
  end;
  then reconsider m as Element of Funcs ([#]V,[#]F) 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.F
  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.F;
    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;
  Carrier l c= A by VECTSP_6:def 4;
  then T c= A \/ {x} by XBOOLE_1:9;
  then Carrier m c= A \/ {x} by A9;
  hence thesis by VECTSP_6:def 4;
end;
