Let A be a non-empty set, ≃ ⊆ A × A an equivalence relation, and ⪯ ⊆ A × A a partial order, both defined on A. Consider the quotient set A/≃ and define the following relation ≪ ⊆ (A/≃) × (A/≃): (S1, S2) ∈ ≪ if and only if there exists an element a ∈ S1 such that for every b ∈ S2, a ⪯ b holds true. We are given that ≪r is a partial order over A/ ≃ where ≪r is the reflex clause of ≪: Question: Is it true that A has a minimal element according to ⪯ if and only if A/≃ has a minimal element according to ≪r? Prove your statement.

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