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 Th14:
  for A being Subset of W st A is linearly-independent holds A is
  linearly-independent Subset of V
proof
  let A be Subset of W;
  the carrier of W c= the carrier of V by RLSUB_1:def 2;
  then reconsider A9= A as Subset of V by XBOOLE_1:1;
  assume
A1: A is linearly-independent;
  now
    assume A9 is linearly-dependent;
    then consider L being Linear_Combination of A9 such that
A2: Sum(L) = 0.V and
A3: Carrier(L) <> {} by RLVECT_3:def 1;
    Carrier(L) c= A by RLVECT_2:def 6;
    then consider LW being Linear_Combination of W such that
A4: Carrier(LW) = Carrier(L) and
A5: Sum(LW) = Sum(L) by Th12,XBOOLE_1:1;
    reconsider LW as Linear_Combination of A by A4,RLVECT_2:def 6;
    Sum(LW) = 0.W by A2,A5,RLSUB_1:11;
    hence contradiction by A1,A3,A4,RLVECT_3:def 1;
  end;
  hence thesis;
end;
