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## Comments

By definition of product (Def. 3.81), there are morphisms\(\pi_1:x\times y\to x\), \(\pi_2:x\times y\to 2\). So, \(x\leq(x\times y)\) and \(y\leq(x\times y)\). This means that \(x\times y\) is an upper bound for \(x\) and \(y\). By the universal property of products, any \(Z\) such that there are morphisms \(f:Z\to x\) and \(g:Z\to y\) has a unique morphism into \(x\times y\). Thus, \(x\times y\) is a least upper bound for \(x\) and \(y\), i.e., it equals \(x\wedge y\).

`By definition of product (Def. 3.81), there are morphisms\\(\pi_1:x\times y\to x\\), \\(\pi_2:x\times y\to 2\\). So, \\(x\leq(x\times y)\\) and \\(y\leq(x\times y)\\). This means that \\(x\times y\\) is an upper bound for \\(x\\) and \\(y\\). By the universal property of products, any \\(Z\\) such that there are morphisms \\(f:Z\to x\\) and \\(g:Z\to y\\) has a unique morphism into \\(x\times y\\). Thus, \\(x\times y\\) is a least upper bound for \\(x\\) and \\(y\\), i.e., it equals \\(x\wedge y\\).`