
theorem
  for K be add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr for V be VectSp
of K for S, T be Subspace of V, v be Vector of V st S /\ T = (0).V & v in S & v
  in T holds v = 0.V
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V be VectSp of K, S, T be Subspace of V, v be Vector of V;
  assume that
A1: S /\ T = (0).V and
A2: v in S & v in T;
  v in the carrier of S & v in the carrier of T by A2;
  then v in (the carrier of S) /\ (the carrier of T) by XBOOLE_0:def 4;
  then v in the carrier of (S /\ T) by VECTSP_5:def 2;
  then v in {0.V} by A1,VECTSP_4:def 3;
  hence thesis by TARSKI:def 1;
end;
