
theorem Th22:
  for V being RealUnitarySpace, W being Subspace of V, A being
Subset of W st A is linearly-independent holds A is linearly-independent Subset
  of V
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let A be Subset of W;
  the carrier of W c= the carrier of V by RUSUB_1:def 1;
  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) <> {};
    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 Th20,XBOOLE_1:1;
    reconsider LW as Linear_Combination of A by A4,RLVECT_2:def 6;
    Sum(LW) = 0.W by A2,A5,RUSUB_1:4;
    hence contradiction by A1,A3,A4;
  end;
  hence thesis;
end;
