Sentence examples for be a minimal essential from inspiring English sources

Proof For each w ∈ M, let m ( w ) ⊂ S ( w ) be a minimal essential subset of S ( w ).

Proof For each fixed q ∈ M 0, let m ( q ) ⊂ F ( q ) be a minimal essential subset of F ( q ).

Proof For each q ∈ M, let m ( q ) ⊂ Λ ( q ) be a minimal essential subset of Λ ( q ).

An essential set m of F ( u ) is said to be a minimal essential set of F ( u ) if m is a minimal element of the family of essential sets in F ( u ) ordered by set inclusion.

An essential subset m ( q ) ⊂ F ( q ) is said to be a minimal essential set of F ( q ) if it is a minimal element of the family of essential sets in F ( q ) ordered by set inclusion.

An essential subset m ( w ) ⊂ S ( w ) is said to be a minimal essential set of S ( w ) if it is a minimal element of the family of essential sets ordered by set inclusion.

Since m ( w ) is a minimal essential set of S ( w ), neither c 1 ( w ) nor c 2 ( w ) is essential.

Since m ( u ) is a minimal essential set of F ( u ), neither c 1 ( u ) nor c 2 ( u ) is essential.

Since m ( q ) is a minimal essential set of Λ ( q ), neither c 1 ( q ) nor c 2 ( q ) is essential.

Since m ( q ) is a minimal essential set of F ( q ), neither c 1 ( q ) nor c 2 ( q ) is essential.

Then Θ is nonempty and every decreasing chain of elements in Θ has a lower bound (because by the compactness the intersection is in Θ); therefore, by Zorn's lemma, Θ has a minimal element and it is a minimal essential set of Λ ( q ).