reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th3:
  for L being Linear_Combination of V, x being VECTOR of V holds x
  in Carrier L iff ex v st x = v & L.v <> 0
proof
  let L be Linear_Combination of V, x be VECTOR of V;
  hereby
    assume x in Carrier(L);
    then x in {w where w is VECTOR of V : L.w <> 0} by RLVECT_2:def 4;
    hence ex v st x = v & L.v <> 0;
  end;
  given v such that
A1: x=v & L.v <> 0;
  x in {w where w is VECTOR of V : L.w <> 0 } by A1;
  hence thesis by RLVECT_2:def 4;
end;
