If $\dim(A+B)=\dim(A\cap B)+1$, then $A \subset B$ or $B \subset A$ Let $A,B$ be linear subspaces of $\mathbb R^n$. Show that if
$$
\dim(A+B) = \dim(A \cap B) + 1, \tag{1}
$$
then one of the space is a subset of the other,
$$
A \subset B \text{ or } B \subset A. \tag{2}
$$
By the dimension theorem, then (1)
$$
\dim(A+B)=\dim(A)+\dim(B)-\dim(A \cap B) = \dim(A \cap B) + 1, \tag{3}
$$
so that
$$
\dim(A)+\dim(B)= 2\dim(A \cap B) + 1. \tag{4}
$$
I can't see where to go next.

Hint: To prove the contraposative, suppose neither $A \subset B$ or $B\subset A$. Then $A\cap B$ is strictly contained in both $A$ and $B$, so $\dim(A\cap B) < \dim(A), \dim(B)$.
