
theorem Th9:
  for V being finite-dimensional RealUnitarySpace, A being Subset
  of V st A is linearly-independent holds card A = dim Lin(A)
proof
  let V be finite-dimensional RealUnitarySpace;
  let A be Subset of V such that
A1: A is linearly-independent;
  set W = Lin(A);
  for x being object st x in A holds x in the carrier of W
        by STRUCT_0:def 5,RUSUB_3:2;
  then reconsider B = A as linearly-independent Subset of W by A1,RUSUB_3:23
,TARSKI:def 3;
  W = Lin(B) by RUSUB_3:28;
  then reconsider B as Basis of W by RUSUB_3:def 2;
  card B = dim W by Def2;
  hence thesis;
end;
