Show that g is indeed a map [on hold]

Let P1=⟨P1,≤1⟩\mathbb{P}_1 = \langle P_1, \leq_1 \rangle and P2=⟨P2,≤2⟩\mathbb{P}_2 = \langle P_2, \leq_2 \rangle be posets and let f:P1→P2f: P_1 \rightarrow P_2 be an order-preserving map. We define the map g:O(P2)→O(P1)g: \mathcal{O}(\mathbb{P}_2) \rightarrow \mathcal{O}(\mathbb{P}_1) by setting:

g(Z):=f−1[Z]g(Z) := f^{-1}[Z]

for every Z∈O(P2)Z \in \mathcal{O}(\mathbb{P}_2).

Where \mathcal{O}(\mathbb{P}_1) and \mathcal{O}(\mathbb{P}_2) are downsets.

Show that gg is indeed a map from O(P2)\mathcal{O}(\mathbb{P}_2) to O(P1)\mathcal{O}(\mathbb{P}_1)

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