
theorem Th25:
  for V being RealUnitarySpace, W1,W2 being strict Subspace of V
  st (for v being VECTOR of V holds v in W1 iff v in W2) holds W1 = W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be strict Subspace of V;
  assume
A1: for v being VECTOR of V holds v in W1 iff v in W2;
  for x being object holds x in the carrier of W1 iff x in the carrier of W2
  proof
    let x be object;
    thus x in the carrier of W1 implies x in the carrier of W2
    proof
      assume
A2:   x in the carrier of W1;
      the carrier of W1 c= the carrier of V by Def1;
      then reconsider v = x as VECTOR of V by A2;
      v in W1 by A2;
      then v in W2 by A1;
      hence thesis;
    end;
    assume
A3: x in the carrier of W2;
    the carrier of W2 c= the carrier of V by Def1;
    then reconsider v = x as VECTOR of V by A3;
    v in W2 by A3;
    then v in W1 by A1;
    hence thesis;
  end;
  then the carrier of W1 = the carrier of W2 by TARSKI:2;
  hence thesis by Th24;
end;
