
theorem Th5:
  for V being RealUnitarySpace, W being strict Subspace of V, A
  being Subset of V st A = the carrier of W holds Lin(A) = W
proof
  let V be RealUnitarySpace;
  let W be strict Subspace of V;
  let A be Subset of V;
  assume
A1: A = the carrier of W;
  now
    let v be VECTOR of V;
    thus v in Lin(A) implies v in W
    proof
      assume v in Lin(A);
      then
A2:   ex l being Linear_Combination of A st v = Sum(l) by Th1;
      A is linearly-closed by A1,RUSUB_1:28;
      then v in the carrier of W by A1,A2,RLVECT_2:29;
      hence thesis by STRUCT_0:def 5;
    end;
    v in W iff v in the carrier of W by STRUCT_0:def 5;
    hence v in W implies v in Lin(A) by A1,Th2;
  end;
  hence thesis by RUSUB_1:25;
end;
