theorem
  for V be non empty addLoopStr, L be Linear_Combination of V, v be
  Element of V holds L.v = 0 iff not v in Carrier(L)
proof
  let V be non empty addLoopStr, L be Linear_Combination of V, v be Element of
  V;
  thus L.v = 0 implies not v in Carrier(L)
  proof
    assume
A1: L.v = 0;
    assume not thesis;
    then ex u be Element of V st u = v & L.u <> 0;
    hence thesis by A1;
  end;
  thus thesis;
end;
