A new proposition about infimum product

I’ve proved a new simple proposition about infimum product:

Let \pi^X_i be metamonovalued morphisms. If S \in \mathscr{P} (   \mathsf{FCD} ( A_0 ; B_0) \times \mathsf{FCD} ( A_1 ; B_1))
for some sets A_0, B_0, A_1, B_1 then
\bigsqcap \left\{ a \times b \,|\, ( a ; b) \in S      \right\} = \bigsqcap \mathrm{dom}\, S \times \bigsqcap \mathrm{im}\, S.

And its corollary:

( a_0 \times b_0) \sqcap ( a_1 \times b_1) = ( a_0 \sqcap a_1) \times ( b_0 \sqcap b_1).

See this online article (here there is also the dual of the above statements).

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