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 Th15:
  for A being Subset of V st A is linearly-independent & A c= the
  carrier of W holds A is linearly-independent Subset of W
proof
  let A be Subset of V such that
A1: A is linearly-independent and
A2: A c= the carrier of W;
  reconsider A9= A as Subset of W by A2;
  now
    assume A9 is linearly-dependent;
    then consider K being Linear_Combination of A9 such that
A3: Sum(K) = 0.W and
A4: Carrier(K) <> {} by RLVECT_3:def 1;
    consider L being Linear_Combination of V such that
A5: Carrier(L) = Carrier(K) and
A6: Sum(L) = Sum(K) by Th11;
    reconsider L as Linear_Combination of A by A5,RLVECT_2:def 6;
    Sum(L) = 0.V by A3,A6,RLSUB_1:11;
    hence contradiction by A1,A4,A5,RLVECT_3:def 1;
  end;
  hence thesis;
end;
