███████

In ███████████, ████████████ ████████ ██████, a ███████ is a ███████ between ██████████. ████████ were █████ ██████████ in █████████ ████████, █████ █████████ ███████ (████ as the ███████████ █████) are ██████████ to ███████████ ██████, and ████ between █████ █████████ ███████ are ██████████ to ██████████ ████ between ██████. ████████, ████████ are used ██████████ ██████ ███████████ to ██████ ███████ ██████████. ████, ████████ are █████████ in █████ ████ of ███████████ █████ ████████ ██████ is ███████.

The █████ ████████ and ███████ were ████████ by ██████████████ from the ████████████ █████████ and ██████ ██████, ████████████. The ██████ used ███████ in a ██████████ ███████; ███ ████████ ████.

███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ be ██████████. A ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ from ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is a ███████ that:

• ██████████ ████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to an ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$,
• ██████████ ████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}):\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ that the following ███ ██████████ ████:
  • ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}}$ for █████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$,
  • ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\circ \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\circ \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ for ███ █████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

That is, ████████ ████ ████████ ████████ █████████ and ███████████ of █████████.

███ █████████ ████████████ of the ███████ ██████ are:

• ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██████████ ████ ███████████ ███████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ into a ███████████ ███████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$;
• if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is an ███████████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is an ███████████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

███ can ███████ ████████, █.█. if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is a ███████ from ${\displaystyle A}$ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is a ███████ from ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ ███ can ████ the █████████ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\circ \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ from ${\displaystyle A}$ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. ███████████ of ████████ is ███████████ █████ ███████. ████████ of ███████████ of ████████ is the ████████ ███████. This █████ that ████████ can be ██████████ as █████████ in ██████████ of ██████████, for ███████ in the ████████ of █████ ██████████.

A █████ ████████ with a ██████ ██████ is the ████ █████ as a ██████: the █████████ of a ███-██████ ████████ can be ███████ of as ████████ of the ██████, and ███████████ in the ████████ is ███████ of as the ██████ █████████. ████████ between ███-██████ ██████████ ██████████ to ██████ █████████████. ██ in a █████, ████████ between █████████ ██████████ are a ████ of ██████████████ of ██████ █████████████ to ██████████ with ████ than ███ ██████.

███████

For ██████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, a ███████ of ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is a █████████ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

(████████ ███████████) ████████

For ██████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, a ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-████████ on ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is a █████████████ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. In the ███████ ████ ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ${\displaystyle \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, the ████████ of ████ and █████████, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ██████ a ████████ on ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

██████████ (over a ███████████ █████)

If █ is a ███████████ █████, ████ the ████ ████ in █ ████ a █████████ ███████ ███ ████(█) under █████████. Like █████ █████████ ███████ ███, ████(█) █████ a █████ ████████ by ██████ a ██████ █████ █ → █ if and ████ if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\subseteq \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. █████████████ ████████ on ████(█) are ██████ ██████████ on █. For ████████, by █████████ to █████ ████ ███ █ the ███████████ ███████ of ████-██████ ██████████ █████████ on █, ███ ███████ a ████████ of ████████ on █.

████████ ███████

The ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ which ████ █████ ██████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to a █████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and █████ ████████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to the ████████ ████████ on ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. ████ a ███████ is ██████ a ████████ or █████████ ███████.

███████████

A ███████ that ████ a ████████ to that ████ ████████; █.█., ██████████ ███████.

████████ ███████

The ████████ ███████ in a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ███████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ or ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$, ████ an ██████ to ██████ and a ████████ to ██████. The ████████ ███████ is an ███████████.

████████ ███████

The ████████ ███████ is ███████ as the ███████ from ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to the ███████ ████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ which █████ ████ ██████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to the ████████ ███████ at that ██████.

█████ ███████

For a █████ █████ ████████ █, if █████ ███████ █ → █ has a █████ (for ████████ if █ is ████████), ████ the █████ ███████ █^█ → █ ███████ to ████ ███████ ███ █████. The █████████ of this ███████ can be ██████ by █████████ that it is the █████-███████ to the ████████ ███████ and ████████ the █████ ███████ ███████ ███████. This ████████ a ████████ ███████ of the █████ of ██████. ███████ ███████ █████ to the ███████ ███████ (which ███████ to █████ ███████ ███ ███████, and is █████████).

█████ ████ ███████

The █████ ███ ███████ █ : ███ → ███ ████ ████ ███ to ███ █████ ███ and ████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to the ███ which █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ to ███ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\in \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$. ███ can ████ ████████ the █████████████ █████ ███ ███████ which █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to the ███ which █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\subseteq \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to ███ ███████ █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\subseteq \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$ For ███████, if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$, ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\{\{\},\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\},\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$. ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\{\}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. ████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is the ████████ which █████ ███ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ to ███ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$, in this ████ ${\displaystyle \{\}\mapsto \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\{\})=\{\}}$. For the █████ ██████,

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

████ ██████ █████

The ███ which ███████ to █████ ██████ █████ ███ ████ █████ and to █████ ██████ ███ ███ ████ or █████████ is a █████████████ ███████ from the ████████ of ███ ██████ ██████ over a █████ █████ to ██████.

███████████ █████

████████ the ████████ of ███████ ███████████ ██████, █.█. ███████████ ██████ with █████████████ ██████. The ███████ are █████ (█, █_█), █████ █ is a ███████████ █████ and █_█ is a █████ in █. A ████████ from (█, █_█) to (█, █_█) is █████ by a ██████████ ███ █ : █ → █ with █(█_█) = █_█. To █████ ███████████ █████ █ with █████████████ █████ █_█, ███ can ██████ the ███████████ █████ █████ at █_█, ███████ █_█(█, █_█). This is the █████ of ████████ ███████ of █████ █████ at █_█, with the █████ █████████ of █████████████. If █ : █ → █ is a ████████ of ███████ ██████, ████ █████ ████ in █ with ████ █████ █_█ can be ████████ with █ to █████ a ████ in █ with ████ █████ █_█. This █████████ is ██████████ with the ████████ ███████████ ████████ and the ███████████ of █████, and ██ ███ a █████ ████████████ from █(█, █_█) to █(█, █_█). ██ ████ ██████ a ███████ from the ████████ of ███████ ███████████ ██████ to the ████████ of ██████. In the ████████ of ███████████ ██████ (without █████████████ █████), ███ █████████ ████████ ███████ of ███████ ██████, but they ██████ be ████████ ██████ they █████ an ████████. ████ ███ has the ███████████ ████████ ███████ of the ███████████ █████, and this ████████████ is ██████████.

███████ of ██████████ █████████

A █████████████ ███████ from the ████████ of ███████████ ██████ (with ██████████ ████ as █████████) to the ████████ of ████ ███████████ ████████ is █████ by █████████ to █████ ███████████ █████ █ the ███████ █(█) of ███ ████-██████ ██████████ █████████ on that █████. █████ ██████████ ███ █ : █ → █ ███████ an ███████ ████████████ █(█) : █(█) → █(█) by the ████ █(█)(█) = █ ∘ █ for █████ █ in █(█).

███████ and █████████ ███████

The ███ which █████ █████ ██████████████ ████████ to ███ ███████ ██████ and █████ ██████ ███ to ███ ██████████ is a █████████ ███████ from the ████████ of ██████████████ █████████ to the ████████ of ██████ ███████. █████ this █████████████ █████████ █████ the ███████ █████, a █████████ ███████ from the ████████ of ███████ ██████████████ █████████ to the ████████ of ████ ██████ ██████. ████████, █████████ █████ is a █████████████ ███████, ███████████ the ███████████ of the ███████ █████ with the ████ █████ above.

█████ ███████/███████████████

█████ █████ █ can be ██████████ as a ████████ with a ██████ ██████ █████ █████████ are the ████████ of █. A ███████ from █ to ███ is ████ ███████ but a █████ ██████ of █ on a ██████████ ███, █.█. a █-███. ████████, a ███████ from █ to the ████████ of ██████ ██████, ████_█, is a ██████ ██████████████ of █. In ███████, a ███████ █ → █ can be ██████████ as an "██████" of █ on an ██████ in the ████████ █. If █ is a █████, ████ this ██████ is a █████ ████████████.

███ ████████

█████████ to █████ ████ (███████) ███ █████ ███ ████ (███████) ███ ███████ ███████ a ███████.

██████ ████████

If █ ███████ the ████████ of ██████ ██████ over a █████ █████, with ██████ ████ as █████████, ████ the ██████ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\otimes \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███████ a ███████ █ × █ → █ which is █████████ in ████ █████████.

█████████ ████████

The ███████ █ : ███ → ███ which ████ a █████ to ███ ██████████ ███ and a █████ ████████████ to ███ ██████████ ████████ of ████ is a ███████. ████████ like █████, which "██████" some █████████, are ██████ █████████ ████████. ███████ ███████ is the ███████ ███ → ██ which ████ a ████ to ███ ██████████ ████████ ███████ █████. █████████ in ███ (████ █████████████) ██████ █████████ in ██ (███████ █████ █████████████).

████ ████████

█████ in the opposite █████████ of █████████ ████████ are ████ ████████. The ████ ███████ █ : ███ → ███ █████ █████ ███ █ to the ████ █████ █████████ by █. █████████ ███ ██████ to █████ █████████████ between ████ ██████. ████ █████████████ █████ for ████ ██████████ █████ on ██████████ ████. ███ ████ ██████.

████████████ ██████

To █████ ████ A, █ of ███████ ██████ ███ can ██████ the ███████ █████ ███(A, █) ██████████ of ███ █████ █████████████ from A to █. This is a ███████ which is █████████████ in the █████ and █████████ in the ██████ ████████, █.█. it is a ███████ ██^██ × ██ → ██ (█████ ██ ███████ the ████████ of ███████ ██████ with █████ █████████████). If █ : A_█ → A_█ and █ : █_█ → █_█ are █████████ in ██, ████ the █████ ████████████ ███(█, █): ███(A_█, █_█) → ███(A_█, █_█) is █████ by █ ↦ █ ∘ █ ∘ █. ███ ███ ███████.

█████████████ ████████

██ can ██████████ the ████████ ███████ to ███ ████████ █. To █████ ████ █, █ of ███████ in █ ███ can ██████ the ███ ███(█, █) of █████████ from █ to █. This ███████ a ███████ to ███ which is █████████████ in the █████ ████████ and █████████ in the ██████, █.█. it is a ███████ █^██ × █ → ███. If █ : █_█ → █_█ and █ : █_█ → █_█ are █████████ in █, ████ the ███ ███(█, █) : ███(█_█, █_█) → ███(█_█, █_█) is █████ by █ ↦ █ ∘ █ ∘ █. ████████ like █████ are ██████ █████████████ ████████. An █████████ ████ in ████ ████████ is to █████████ ███████ a █████ ███████ is █████████████.