Orthocomplement of Subset of Orthocomplement is Superset

Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $A, B \subseteq V$ be subsets of $V$ such that $B \subseteq A^\perp$, where $A^\perp$ is the orthocomplement of $A$.

Then:


 * $A \subseteq B^\perp$.

Proof
Let $B \subseteq A^\perp$.

Then by Orthocomplement Reverses Subset:


 * $A^{\perp\perp} \subseteq B^\perp$

By Double Orthocomplement is Closed Linear Span and the definition of closed linear span:


 * $A \subseteq A^{\perp\perp}$

Hence, by Subset Relation is Transitive:


 * $A \subseteq B^\perp$