\ProvidesDictionary{translator-theorem-dictionary}{English} \providetranslation{Comments}{Comments} \providetranslation{comments}{comments} \providetranslation{Comment}{Comment} \providetranslation{comment}{comment} \providetranslation{Corollaries}{Corollaries} \providetranslation{corollaries}{corollaries} \providetranslation{Corollary}{Corollary} \providetranslation{corollary}{corollary} \providetranslation{Definitions}{Definitions} \providetranslation{definitions}{definitions} \providetranslation{Definition}{Definition} \providetranslation{definition}{definition} \providetranslation{Examples}{Examples} \providetranslation{examples}{examples} \providetranslation{Example}{Example} \providetranslation{example}{example} \providetranslation{Exercises}{Exercises} \providetranslation{exercises}{exercises} \providetranslation{Exercise}{Exercise} \providetranslation{exercise}{exercise} \providetranslation{Facts}{Facts} \providetranslation{facts}{facts} \providetranslation{Fact}{Fact} \providetranslation{fact}{fact} \providetranslation{Key Lemmas}{Key Lemmas} \providetranslation{key lemmas}{key lemmas} \providetranslation{Key Lemma}{Key Lemma} \providetranslation{key lemma}{key lemma} \providetranslation{Key Observations}{Key Observations} \providetranslation{key observations}{key observations} \providetranslation{Key Observation}{Key Observation} \providetranslation{key observation}{key observation} \providetranslation{Lemmas}{Lemmas} \providetranslation{lemmas}{lemmas} \providetranslation{Lemma}{Lemma} \providetranslation{lemma}{lemma} \providetranslation{Main Theorems}{Main Theorems} \providetranslation{main theorems}{main theorems} \providetranslation{Main Theorem}{Main Theorem} \providetranslation{main theorem}{main theorem} \providetranslation{Observations}{Observations} \providetranslation{observations}{observations} \providetranslation{Observation}{Observation} \providetranslation{observation}{observation} \providetranslation{Problems}{Problems} \providetranslation{problems}{problems} \providetranslation{Problem}{Problem} \providetranslation{problem}{problem} \providetranslation{Proofs}{Proofs} \providetranslation{proofs}{proofs} \providetranslation{Proof}{Proof} \providetranslation{proof}{proof} \providetranslation{Proof Sketch}{Proof Sketch} \providetranslation{Proof sketch}{Proof sketch} \providetranslation{proof sketch}{proof sketch} \providetranslation{Proof Sketches}{Proof Sketches} \providetranslation{Proof sketches}{Proof sketches} \providetranslation{proof sketches}{proof sketches} \providetranslation{Sketch of Proof}{Sketch of Proof} \providetranslation{Sketch of Proofs}{Sketch of Proofs} \providetranslation{Sketch of proof}{Sketch of proof} \providetranslation{Sketch of proofs}{Sketch of proofs} \providetranslation{sketch of proof}{sketch of proof} \providetranslation{sketch of proofs}{sketch of proofs} \providetranslation{Propositions}{Propositions} \providetranslation{propositions}{propositions} \providetranslation{Proposition}{Proposition} \providetranslation{proposition}{proposition} \providetranslation{Remarks}{Remarks} \providetranslation{remarks}{remarks} \providetranslation{Remark}{Remark} \providetranslation{remark}{remark} \providetranslation{Solutions}{Solutions} \providetranslation{solutions}{solutions} \providetranslation{Solution}{Solution} \providetranslation{solution}{solution} \providetranslation{Theorems}{Theorems} \providetranslation{theorems}{theorems} \providetranslation{Theorem}{Theorem} \providetranslation{theorem}{theorem}