
theorem ThIS1:
  for R being Ring
  for V being LeftMod of R, W1, W2, W3 being Subspace of V
  st W1 /\ W2 <> (0).V holds (W1 + W3) /\ W2 <> (0).V
  proof
    let R be Ring;
    let V be LeftMod of R, W1, W2, W3 be Subspace of V such that
    A1: W1 /\ W2 <> (0).V;
    consider v be Vector of V such that
    A2: v in W1 /\ W2 & v <> 0.V by A1,ZMODUL04:24;
    A3: v in W1 & v in W2 by A2,ZMODUL01:94;
    then v in W1 + W3 by ZMODUL01:93;
    hence thesis by A2,ZMODUL02:66,A3,ZMODUL01:94;
  end;
