
theorem
  for V being RealUnitarySpace, A being Subset of V st Lin(A) = V holds
  ex I being Basis of V st I c= A
proof
  let V be RealUnitarySpace;
  let A be Subset of V;
  assume Lin(A) = V;
  then consider B being Subset of V such that
A1: B c= A and
A2: B is linearly-independent & Lin(B) = V by Th12;
  reconsider B as Basis of V by A2,Def2;
  take B;
  thus thesis by A1;
end;
