
theorem Th11:
  for F being Field, V being VectSp of F for A being Subset of
  LinearlyIndependentSubsets V holds A is independent iff A is
  linearly-independent Subset of V
proof
  let F be Field;
  let V be VectSp of F;
  set M = LinearlyIndependentSubsets V;
  let B be Subset of M;
  the_family_of M = {A where A is Subset of V: A is linearly-independent}
  by Def8;
  then B in the_family_of M iff ex A being Subset of V st B = A & A is
  linearly-independent;
  hence thesis;
end;
