
theorem Th15:
  for V being RealUnitarySpace, A being Subset of V st A is
  linearly-independent holds ex I being Basis of V st A c= I
proof
  let V be RealUnitarySpace, A be Subset of V;
  assume A is linearly-independent;
  then consider B being Subset of V such that
A1: A c= B and
A2: B is linearly-independent & Lin(B) = the UNITSTR of V by Th11;
  reconsider B as Basis of V by A2,Def2;
  take B;
  thus thesis by A1;
end;
