██████████ ████████

██████████ in █████ of ██████ of █████████

The ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ at some █████ █ of ███ ██████ if the █████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}),}$ as █ ██████████ █ through the ██████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ██████ and is █████ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$ In ████████████ ████████, this is ███████ as $${\displaystyle \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$$ In ██████ this █████ █████ ██████████: █████, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ has to be ███████ at █ (██████████ by the ███████████ that █ be in the ██████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$). ██████, the █████ of that ████████ ████ █████. █████, the █████ of this █████ ████ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$

(████, ██ have ███████ that the ██████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ not have ███ ████████ ██████.)

██████████ in █████ of █████████████

A ████████████ of a █████ █ is a ███ that ████████, at █████, ███ ██████ within some █████ ████████ of █. ███████████, a ████████ is ██████████ at a █████ █ if the █████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ over the ████████████ of █ ███████ to a ██████ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ as the █████ of the ████████████ around █ ███████ to ████. ████ █████████, a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ at a █████ █ of ███ ██████ if, for ███ ████████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))}$ there is a ████████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ in ███ ██████ ████ that ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\in {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))}$ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$

As █████████████ are ███████ in ███ ███████████ █████, this ██████████ of a ██████████ ████████ ███████ not ████ for ████ █████████ but ████ ████ the ██████ and the ████████ are ███████████ ██████ and is ████ the ████ ███████ ██████████. It ███████ that a ████████ is █████████████ ██████████ at █████ ████████ █████ of ███ ██████. For ███████, █████ ████-██████ ████████ on the ████████ is ██████████.

██████████ in █████ of ██████ of █████████

███ can ███████ ███████ that for ███ ████████ ${\displaystyle ({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ of ██████ in the ██████ which █████████ to █, the █████████████ ████████ ${\displaystyle {\left(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})\right)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ █████████ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$ In ████████████ ████████, $${\displaystyle \mathrm{\forall }({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}\subset \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{\infty }}{<span\; word-length="3"\; class="word\; redacted">███</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Rightarrow \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{\infty }}{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$$

███████████ and ██████ ███████████ (███████–█████) of ██████████ █████████

██████████ █████████ the ██████████ of the █████ of a ████████, ██ ██████ a ████-█████████ ██████████: █████ a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ as above and an ███████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ of the ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ████ to be ██████████ at the █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ ████ the following █████: For ███ ████████ ████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$ ███████ █████, there ██████ some ████████ ████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ that for ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in the ██████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ with the █████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ █████████

█████████████ ███████, ██████████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ at ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █████ that for █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$ there ██████ a ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ that for ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$:

████ ███████████, ██ can ███ that if ██ ████ to ███ ███ the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ██████ to ████ in some █████ ████████████ around ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\left({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right),}$ ██ ████ to ██████ a █████ ██████ ████████████ for the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██████ around ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$ If ██ can do that ██ ██████ ███ █████ the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})}$ ████████████ is, ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ at ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$

In ██████ █████, this is ███████████ by the ██████████ of ██████████ of a ████████ with ███████ to a █████ for the ████████, ████ the ██████ ████████.

███████████ had ████████ that the ████████ be ████████ within the ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, but ██████ ███████ that ███████████.

██████████ in █████ of ███████ of the █████████

In ██████ and █████████ ████████, ██ █████ ████ to ████ ███ ████ ██████ are ██████████, or in █████ █████, ███████ of the █████████. ██ can █████████ this to a ██████████ of ██████████. A ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:[\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{\infty })\to [\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{\infty }]}$ is ██████ a ███████ ████████ if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ███-██████████ and ${\displaystyle \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

A ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is █-██████████ at ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ if there ██████ a ████████████ $\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})$ ████ that $${\displaystyle |\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})|\le \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\left(\left|\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right|\right)\text{ for <span word-length="3" class="word redacted">███</span> }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\cap \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}).}$$

A ████████ is ██████████ in ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ if it is ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-██████████ for some ███████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

This ████████ █████ █████████ to ████████ the ██████ of ██████████ by ███████████ the ███ of ██████████ ███████ █████████. For a █████ ███ of ███████ █████████ ${\displaystyle \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, a ████████ is ${\displaystyle \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-██████████ if it is ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-██████████ for some ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$ For ███████, the █████████, the ██████ ██████████ █████████ of ████████ █ and the █████████ ██████████ █████████ below are ███████ by the ███ of ███████ █████████ $${\displaystyle {\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}|\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}|,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$$ $${\displaystyle {\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\text{<span word-length="6" class="word redacted">██████</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}|\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{|}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$$ $${\displaystyle {\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\text{<span word-length="7" class="word redacted">███████</span> <span word-length="4" class="word redacted">████</span>.}}=\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$$ ████████████.

██████████ using ███████████

██████████ can ████ be ███████ in █████ of ███████████: a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ at a █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ if and ████ if ███ ███████████ at that █████ is ████; in ███████, ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>.}}$ A ███████ of this ██████████ is that it ██████████ █████████████: the ███████████ █████ ███ ████ the ████████ is █████████████ at a █████.

This ██████████ is ███████ in ███████████ ███ ██████ to █████ the ███ of ███████████████ and ██████████ ██████—the ██████████ ██████ are the ████████████ of the ████ █████ the ███████████ is ████ than ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ (█████ a ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ ███)—and █████ a █████ █████ of ███ █████████ of the ████████ █████████████ █████████.

The ███████████ is ██████████ to the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██████████ by a ██████ ██-███████████ and by using a █████ (███ ███, ███ ███) to ██████ ███████████: if (at a █████ █████) for a █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ there is ██ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ that █████████ the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██████████, ████ the ███████████ is at █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},}$ and ██████████ if for █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ there is a ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$ the ███████████ is █. The ███████████ ██████████ can be █████████ ███████████ to ████ from a ███████████ █████ to a ██████ █████.

██████████ using the ██████████

██████ ███████ the ██████████ of a ████████ in the following █████████ █████: an █████████████ ██████ in the ███████████ ████████ ███████████ to an █████████████ ██████ of the █████████ ████████ (███ █████ █'███████, ████ ██). ███-████████ ████████ is a ███ of ██████ this ██████████████ ████████. The ████ ████ is █████████ by ██████ ████████ and █████████████ ███████ to ████ the █████████ ███████. In ███████████ ████████, ██████████ can be ███████ as ███████.

(███ ███████████████). In █████ █████, an █████████████ █████████ of the ███████████ ████████ ██████ ████████ an █████████████ ██████ of the █████████ ████████, ██████ a ██████ ██████████ to ████████-█████ ██████'s ██████████ of ██████████.

A ██████ █████

███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ be a ████████ that is ██████████ at a █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},}$ and ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ be a █████ ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\left({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)\ne {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$ ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\ne {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ ██████████ some ████████████ of ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$

█████: By the ██████████ of ██████████, ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\frac{|{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})|}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ , ████ there ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ that ███████ there is a █████ in the ████████████ for which ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}};}$ ████ ██ have the █████████████

████████████ █████ ███████

The ████████████ █████ ███████ is an █████████ ███████, █████ on the ████ ██████ ████████ of ████████████, and ██████:

If the ████-██████ ████████ █ is ██████████ on the ██████ ████████ ${\displaystyle [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}],}$ and █ is some ██████ between ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(a)}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}),}$ ████ there is some ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}],}$ ████ that ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$

For ███████, if a █████ █████ from █ █ to █.█ █ between the ████ of ███ and ███ █████, ████, at some ████ between ███ and ███ █████ of ███, the █████'s ██████ ████ have been █.██ █.

As a ███████████, if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ on ${\displaystyle [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}]}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(a)}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ██████ in ████, ████, at some █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}],}$ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ████ █████ ████.

███████ █████ ███████

The ███████ █████ ███████ ██████ that if a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ███████ and ██████████ on a ██████ ████████ ${\displaystyle [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}]}$ (or ███ ██████ and ███████ ███), ████ the ████████ ███████ ███ ███████, █.█. there ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}]}$ with ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\ge \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ for ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in [a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}].}$ The ████ is ████ of the ███████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. █████ ██████████ are not, in ███████, ████ if the ████████ is ███████ on an ████ ████████ ${\displaystyle (a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ (or ███ ███ that is not ████ ██████ and ███████), as, for ███████, the ██████████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},}$ ███████ on the ████ ████████ (█,█), ████ not ██████ a ███████, being █████████ above.

████████ to █████████████████ and █████████████

█████ ██████████████ ████████ $${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:(a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$$ is ██████████, as can be █████. The ████████ ████ not ████: for ███████, the ████████ █████ ████████

is ██████████ ██████████. ███████, it is not ██████████████ at ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ (but is ██ ██████████ ████). ███████████'s ████████ is ████ ██████████ ██████████ but ███████ ██████████████.

The ██████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\prime }(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ of a ██████████████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ████ not be ██████████. If ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\prime }(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is ██████████, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is ████ to be ████████████ ██████████████. The ███ of ████ █████████ is ███████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}((a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})).}$ ████ █████████, the ███ of █████████ $${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$$ (from an ████ ████████ (or ████ ██████ of ${\displaystyle \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$) ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to the █████) ████ that ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █████ ██████████████ and ████ that the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-██ ██████████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ is ███████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$; ███ █████████████████ █████. In the █████ of ████████ ████████, ██████████ ███████ (but not █████████) to ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ are █████████ ██████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ (██████████ of ████████), ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ (██████████ of ████████), and ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ (██████████ of █████████); ███ ██████████ of ██████ and ████████.

█████ ██████████ ████████ $${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:[a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}]\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$$ is ██████████ (for ███████ in the █████ of the ███████ ████████). The ████████ ████ not ████, as the (██████████ but █████████████) ████ ████████ █████.

█████████ and ███████ ██████

█████ a ████████ $${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\dots :\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$$ of █████████ ████ that the █████ $${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}):=\underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{\infty }}{<span\; word-length="3"\; class="word\; redacted">███</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$$ ██████ for ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, the █████████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is ████████ to as the █████████ █████ of the ████████ of █████████ ${\displaystyle {\left({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$ The █████████ █████ ████████ ████ not be ██████████, ████ if ███ █████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ are ██████████, as the █████████ at the █████ █████. ███████, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████████ if ███ █████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ are ██████████ and the ████████ █████████ █████████, by the ███████ ███████████ ███████. This ███████ can be used to ████ that the ███████████ █████████, ██████████, ██████ ████ ████████, and █████████████ █████████ are ██████████.