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Corollary 1 Let D be a finite set and let f : D → D be a mapping that has no fixed point.
Let (D: mathcal{X}tomathcal{X}) be a mapping that belongs to (mathcal{D}_{mathcal{X}}). Then D has at least one fixed point.
Let (D: mathcal{X}tomathcal{X}) be a mapping that belongs to (mathcal{G}_{mathcal{X}}). Then D has at least one fixed point.
Lemma 3 Let D be a finite set and let f : D → D be a mapping that has no fixed point.
Let (D: mathcal{X}tomathcal{X}) be a mapping that belongs to (mathcal{F}_{mathcal{X}}). Then D has at least one fixed point.
Theorem 2 Let D be a finite set and let f : D → D be a mapping that has no fixed point.
Similar(46)
Let r : ℕ → {1,..., N} be a map that assumes every value infinitely often.
It was a map that defined the contours and forever expanded the frontiers of our continent and our imagination.
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"The Review is like a map of Bob's mind," Mr. Hederman has said, and it's a map that is unusually varied and capacious.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com