Is this claim about orthogonal projection true?

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.