reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem ThCarrier1:
  for K being Ring
  for V being VectSp of K
  for A being Subset of V, l1, l2 being Linear_Combination of A
  st Carrier(l1) /\ Carrier(l2) = {} holds
  Carrier(l1 + l2) = Carrier(l1) \/ Carrier(l2)
  proof
    let K be Ring;
    let V be VectSp of K;
    let A be Subset of V,
    l1, l2 be Linear_Combination of A such that
    A0: Carrier(l1) /\ Carrier(l2) = {};
    A1: Carrier(l1) misses Carrier(l2) by A0;
    Carrier(l1) \/ Carrier(l2) c= Carrier(l1 + l2)
    proof
      let x be object;
      assume B1: x in Carrier(l1) \/ Carrier(l2);
      then reconsider x as Vector of V;
      per cases by B1,XBOOLE_0:def 3;
      suppose C1: x in Carrier(l1);
        then not x in Carrier(l2) by A1,B1,XBOOLE_0:5;
        then C2: l2.x = 0.K;
        (l1 + l2).x = l1.x + l2.x by VECTSP_6:22
        .= l1.x by C2;
        then (l1 + l2).x <> 0.K by C1,VECTSP_6:2;
        hence thesis;
      end;
      suppose C1: x in Carrier(l2);
        then not x in Carrier(l1) by A1,B1,XBOOLE_0:5;
        then C2: l1.x = 0.K;
        (l1 + l2).x = l1.x + l2.x by VECTSP_6:22
        .= l2.x by C2;
        then (l1 + l2).x <> 0.K by C1,VECTSP_6:2;
        hence thesis;
      end;
    end;
    hence thesis by VECTSP_6:23;
  end;
