Let $V$ be a finite dimensional vector space endowed with an inner product. Let $V_1 \subset V_2 \subset V $ be subvector spaces of $V$.
Let $v \in V$, is it true that $\operatorname{proj\,}(v,{V_1}) = \operatorname{proj\,}(v,{V_2})$?
$\operatorname{proj\,}(v,{V_1}) ,\operatorname{proj\,}(v,{V_2})$ are the projections of $v$ onto $V_1$ and onto $V_2$ respectively, and this projection is w.r.t. the inner product.