We could also notice that UW+VW is a subspace of (U+V)W, and therefore the following statement is valid: dim(U+V+W)<= dim(U)+dim(V)+dim(W) - dim(UV)-dim(UW)-di

We could also notice that UW+VW is a subspace of (U+V)W, and therefore the following statement is valid: dim(U+V+W)<= dim(U)+dim(V)+dim(W) - dim(UV)-dim(UW)-dim(VW) + dim(UVW). So the result is weakened, but not entirely lost.