# One to one correspondence in faces of convex sets

Let $C$ be a nonempty convex subset of $\mathbb{R}^{n}$, and let $L$ be a nonempty subspace contained in lin$C.$ Define $C_0$ tobe $C \cap L^{\perp}.$ Show that the faces $F$ of $C$ are in one-to-one correspondence with the faces $F_0$ and $C_0$ through the mappings $\phi (F_0) = F_0 + L, \psi (F) = F \cap L^{\perp}.$

Attempt : To establish a one - one correspondence between two sets I just have to show that there are functions $\sigma : S \to T$ and $\tau : T \to S$ such that $\tau \circ \sigma = id_S$ and $\sigma \circ \tau = id_T$ where id is the identity mappin

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