
theorem
  for V being RealUnitarySpace, A being Subset of V st V is
  finite-dimensional & A is linearly-independent holds A is finite
proof
  let V be RealUnitarySpace;
  let A be Subset of V;
  assume that
A1: V is finite-dimensional and
A2: A is linearly-independent;
  consider B being Basis of V such that
A3: A c= B by A2,RUSUB_3:15;
  B is finite by A1,Th3;
  hence thesis by A3;
end;
