Inclusion of orthogonal complements: if $U_1\subset\ U_2$, then $U_2^\bot\subset\ U_1^\bot$
Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$