
theorem
  for V being RealUnitarySpace, A being Subset of V st Lin(A) = (0).V
  holds A = {} or A = {0.V}
proof
  let V be RealUnitarySpace;
  let A be Subset of V;
  assume that
A1: Lin(A) = (0).V and
A2: A <> {};
  thus A c= {0.V}
  proof
    let x be object;
    assume x in A;
    then x in Lin(A) by Th2;
    then x = 0.V by A1,Lm1;
    hence thesis by TARSKI:def 1;
  end;
  set y = the Element of A;
  let x be object;
  assume x in {0.V};
  then
A3: x = 0.V by TARSKI:def 1;
  y in A & y in Lin(A) by A2,Th2;
  hence thesis by A1,A3,Lm1;
end;
