Изменения

<tex>PH PH_{1} = {\bigcup \atop {k \in \mathbb{N}}} \Sigma_{i}</tex><br/><tex>PH_{2} = {\bigcup \atop {k \in \mathbb{N}}} \Pi_{i}</tex><br/><tex>PH_{3} = {\bigcup \atop {k \in \mathbb{N}}} (\Sigma_{i} \cup \Pi_{i})</tex>}} {{Теорема|statement = Все три определения класса <tex>PH</tex> эквивалентны, т.е. <tex>PH_{1} = PH_{2} = PH_{3}</tex>|proof = <tex>\Sigma_{i} \subset \Pi_{i+1} \Rightarrow PH_{1} \subset PH_{2}</tex><br/><tex>\Pi_{i} \subset (\Sigma_{i+1} \cap \Pi_{i+1}) \subset (\Sigma_{i+1} \cup \Pi_{i+1}) \Rightarrow PH_{2} \subset PH_{3}</tex><br/><tex>\Pi_{i} \subset \Sigma_{i+1}, \Sigma_{i} \subset \Sigma_{i+1} \Rightarrow PH_{3} \subset PH_{1}</tex><br/>Т.о., <tex>PH_{1} \subset PH_{2} \subset PH_{3} \subset PH_{1}</tex>