█████ (████████ ██████)

██████

███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ be a ███████ of █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. A ████ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is an ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████████ with a ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of █████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ███████ by the ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ████ that for █████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ██ have ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\circ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$.

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

███ ████ that the ████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ███████ through the ████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ with the ██████ █████████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. The ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is █████████ ██████ the █████████ ████████.

██████ are ████ ████████ to as █████████ █████, since they are █████████████ by a █████████ ████████ (███ below for ████ ███████████). As with █████ █████████ ████████, the above ██████████ █████████ a ████████ █████ of ██████████: The █████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ has to be ███████ ██████ to █████ ███ ████ to ██████ through it; on the █████ ████, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ has to be ████████████ ████████, ██ that ████ ███ ████ █████████████ is ████████ for █████ ████.

██████ ███ ████ be █████████████ as ████████ ███████ in the ████████ of █████ to █.

It is ████████ that a ███████ ████ not have a █████ at ███. ███████, if a ███████ ████ have a █████ ████ this █████ is ███████████ ██████: it is ██████ up to a ██████ ███████████. For this ██████ ███ █████ ██████ of the █████ of █.

████████

The ████ ███████ of ██████ and █████ are ████████ and ██-█████. ████████ it is ███████████████ to ██████ the ███████████ of █████ by █████████ ███ █████████ in the above ███████████, ██ will ██████████ █████ ████ ████:

A ██-████ of 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 an ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████████ with a ██████ of █████████ $${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$$

for █████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ████ that for █████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ██ have ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\circ \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$.

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

████████ are ████ ████████ to as █████████ ██-█████. They can be █████████████ as ███████ ███████ in the ████████ of ██-█████ from ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

As with ██████, if a ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ has a ███████ ████ this ███████ is ██████ up to a ██████ ███████████.

██████████

██████ and ████████ can ████ be ███████ for ███████████ of ███████ and █████████ without the use of ████████. The ███████████ are the ████ (████ that in ███████████ above ██ █████ ██████ to use ███████████ of █████████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$). This █████████, ███████, ████ ██ ███ ███████████. ███ ██████████ of ███████ and █████████ ███████ a (████████ █████) ████████ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. If ██ ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ be the ████ ████████ █████████ by ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, there is a █████████ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █████ █████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. The █████ (or ███████) of this ███████ is the ████ as the █████ (or ███████) of the ████████ ██████████ of ███████ and █████████.

████ █████ and ████ ████████ are ███████ like ██████ and ████████, except that the ██████████ ████████ of the █████████ ████████ is ███████.

██████

The ██████████ of ██████ is ███████ ██████ to ███████ ███████ █████████████ ██████ in █████████ ████████. In the following ██ will ████████ the █████ (█, █) of a ███████ █ : █ → █.

• ████████ ███████. If █ is the █████ ████████ there is ████ ███ ███████ of █████ █: the █████ ███ (███████ to the █████ ████████ in ███ ██████). A ████ to the █████ ███████ is ███████████ ████ an ██████ of █. The █████ of █ is ███ ██████ that is ████████ ████████ through by █████ █████ ██████. This is ████ the ██████████ of a ████████ ██████.
• ████████. If █ is a ████████ ████████ ████ a ███████ █ is ███████████ ███████ but a ██████ of ███████ of █, ███████ by █. The █████ █ of █ is ██████ the ███████ of █████ ███████. The ████ █ ████████ of a ██████ of █████████ █_█ : █ → █(█) ██████ the ███████████ of the ███████. In the ████████ of ████, for ████████, the ████████ are █████ by █████████ ████████ and the ███████████ are ████ the ███████ ███████████ onto the ███████ ███████.
  • ██████. A ███████ ████ of a ███████ is ████ the ███████ █ is a ████████ ███████ to an ██████ █ of █. The █████ of this ███████ is ██████ the █^██ █████ of █ and ███████ █^█.
• ██████████. If █ is a ████████ with ███ ███████ and ███ ████████ █████████ from ███ ██████ to the █████, ████ a ███████ of █████ █ is a ████ of ████████ █████████ in █. The █████ █ of ████ a ███████ is ██████ an █████████ of █████ █████████.
  • ███████. A ██████ is a ███████ ████ of an █████████ █████ ███ of the █████████ is a ████ ████████.
• █████████. ███ █ be a ███████ that █████ out █████ ███████ █, █, and █ in █, █████ the ████ ███-████████ █████████ are █ : █ → █ and █ : █ → █. The █████ █ of █ is ██████ a ████████ or a █████ ███████. It can ██████ be ██████████ as a ███████████ ██████:

• ███████ ██████. ███ █ be a ████████ ███ (██████████ as a █████ ████████ by ██████ ██████ █ → █ if and ████ if █ ≤ █) and ███ █ : █^██ → █ be a ███████. The █████ of █ is ██████ an ███████ █████ or ██████████ █████.
• If █ = █, the ████████ with a ██████ ██████ and ████████, ████ a ███████ of █████ █ is ███████████ ████ an ██████ █ of █. A ████ to an ██████ █ is ████ a ████████ with ████████ █. A ████████ █ : █ → █ is a █████ of the ███████ █ if and ████ if █ is an ███████████. ████ █████████, if █ is ███ ████████ with an ███████ ██████ █, ████ ███ ███████ of █████ █ has a █████, ██████ ███ ██████ ██████████ to █(█). ████ an ███████████ ████████ ██████████ a █████████ ████ to █.
• ███████████ ██████. ██████ of █████████ are a ███████ ████ of ██████ of ███████, which are ███████ to ███████████ ██████ as ███████. █████ a ███████████ █████ █, ██████ by █ the ███ of ███████ on █, █ ∈ █ a █████, █(█) ∈ █ the ████████████ ██████ of █, A ∈ █ a ██████████ ██████ and ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},A}=\{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mid \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\cup A\subset \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$ the ███ of ███████ █████ than A and that ████████ to █. The ███████ █ are █████ a █████ and ████ ████████ █████████ by ██████ an █████ A → █ if and ████ if A ⊆ █. The █████████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},A}:{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},A}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███████ a ███████ and the following ███████████ █████ :

█ is a ███████████ █████ of A if and ████ if A is a ███████████ █████ of ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},A}}$

████████

████████ of ████████ are █████ by the ████ ████████ of the ████████ above:

• ███████ ███████ are ████████ of █████ ████████.
• ██████████ are ████████ of ████████ ███████ by ████████ ██████████.
  • ████████ are ████████ of ████████ ████████ from ████████ ██████████.
• ████████████ are ████████ of a ████████ ████ of █████████.
  • █████████ are ████████████ of a ████████ and a ████████ ████ ████████.
• ████████ are ████████ of a ████ of █████████ with ██████ ██████.
• ██████ ██████ are ████████ of ████████ ███████ by ████████ ████.

█████████ of ██████

A █████ ███████ █ : █ → █ ███ or ███ not have a █████ (or ███████) in █. ██████, there ███ not ████ be a ████ to █, ███ █████ a █████████ ████.

A ████████ █ is ████ to have ██████ of █████ █ if █████ ███████ of █████ █ has a █████ in █. ████████████, a ████████ █ is ████ to

• have ████████ if it has ██████ of █████ █ for █████ █████ ████████ ████████ █ (it ████ not have █████ ████████),
• have ██████████ if it has ██████ of █████ ${\displaystyle \bullet \rightrightarrows \bullet }$ (█.█. █████ ████████ ████ of █████████ has an █████████),
• have █████████ if it has ██████ of █████ ${\displaystyle \bullet \to \bullet \leftarrow \bullet }$ (█.█. █████ ████ of █████████ with ██████ ████████ has a ████████).

A ████████ ████████ is a ████████ that has ███ █████ ██████ (█.█. ███ ██████ of █████ █ for █████ █████ ████████ █).

███ can ████ ████ the ████ ███████████. A ████████ has ████████ of █████ █ if █████ ███████ of █████ █ has a ███████ in █. A ██████████ ████████ is ███ that has ███ █████ ████████.

The █████████ ███████ for ██████ ██████ that if a ████████ █ has ██████████ and ███ ████████ ███████ by the ███████ ██(█) and ███(█), ████ █ has ███ ██████ of █████ █. In this ████, the █████ of a ███████ █ : █ → █ can be ███████████ as the █████████ of the ███ █████████

${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\prod _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\rightrightarrows \prod _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))}$

█████ (in █████████ ████) by

There is a ████ █████████ ███████ for ████████ in █████ of ████████████ and ██████████. ████ of █████ ████████ ████ ██████████ and █████████ ██████████ for the █████████ of ███ (██)██████ of █████ █.

█████████ ████████

██████ and ████████ are █████████ ███████ █████ of █████████ █████████████.

███ █ be a ████████ and ███ █ be a █████ █████ ████████. The ███████ ████████ █^█ ███ be ███████ of as the ████████ of ███ ████████ of █████ █ in █. The ████████ ███████

${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}\to {\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ :{\███████ {█}}\to {\███████ {█}}^{\███████ {█}}}

is the ███████ that ████ ████ ██████ █ in █ to the ████████ ███████ █(█) : █ → █ that ████ ███ ███████ in █ to █. That is, █(█)(█) = █ for ████ ██████ █ in █ and █(█)(█) = ██_█ for ████ ████████ █ in █.

█████ a ███████ █: █ → █ (███████ of as an ██████ in █^█), a ███████ ██████████████ █ : █(█) → █ (which is ████ a ████████ in the ████████ █^█) is the ████ █████ as a ████ from █ to █. To ███ this, █████ ████ that █(█)(█) = █ for ███ █ ███████ that the ██████████ of █ are █████████ █_█ : █ → █(█), which ███ █████ the ██████ █. ████████, the ███████████ that the ████'s ████████ ███████ is ████ ██████ because this █ is a ███████ ██████████████. (██████, a ███████ ██████████████ █ : █ → █(█) is the ████ █████ as a ██-████ from █ to █.)

█████████, the ███████████ of ██████ and ████████ can ████ be ████████ in the ████:

• A █████ of █ is a █████████ ████████ from █ to █.
• A ███████ of █ is a █████████ ████████ from █ to █.

███████████

Like ███ █████████ █████████████, the █████████ of ██████ and ████████ is ██████████ in ██████. In █████ █████, if █████ ███████ of █████ █ has a █████ in █ (for █ █████) there ██████ a █████ ███████

${\displaystyle <span\; word-length="3"\; class="word\; redacted">███</span>:{\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}\to \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ :{\███████ {█}}^{\███████ {█}}\to {\███████ {█}}}

which ███████ ████ ███████ ███ █████ and ████ ███████ ██████████████ █ : █ → █ the ██████ ████████ ███ █ : ███ █ → ███ █ █████████ with the █████████████ █████████ █████. This ███████ is █████ ███████ to the ████████ ███████ █ : █ → █^█. This ██████████ █████ a █████████ between the ███ of ███ █████████ from █ to ███ █ and the ███ of ███ █████ from █ to █

${\displaystyle \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},<span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\cong \mathrm{<span\; word-length="4"\; class="word\; redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$

which is ███████ in the █████████ █ and █. The ██████ of this ██████████ is ██████ the █████████ ████ from ███ █ to █. If the █████ ████████ █ is █████████ (and ████████) ████ the ████ of the ██████████ is an ███████████ ██ that ███ is a ████ ███████ of █. This █████ if █ is not █████████. For ███████, if █ is a ████████ ████████, the ██████████ of the ████ are the ████████ █████████ █ : █ → █^█.

██████, if █████ ███████ of █████ █ has a ███████ in █ (for █ █████) there ██████ a ███████ ███████

${\displaystyle \mathrm{<span\; word-length="5"\; class="word\; redacted">█████</span>}:{\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}\to \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ :{\███████ {█}}^{\███████ {█}}\to {\███████ {█}}}

which ███████ ████ ███████ ███ ███████. This ███████ is ████ ███████ to the ████████ ███████ █ : █ → █^█, and ███ has a ███████ ███████████

${\displaystyle \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="5"\; class="word\; redacted">█████</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\cong \mathrm{<span\; word-length="6"\; class="word\; redacted">██████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$

The ████ of this ██████████ is the █████████ ██████ from █ to █████ █. If █ is █████████ (and ████████) ████ the ██████ is an ███████████, ██ that █████ is a ████ ███████ of █.

████ that ████ the █████ and the ███████ ████████ are █████████ ████████.

As ███████████████ of ████████

███ can use ███ ████████ to ██████ ██████ and ████████ in a ████████ █ to ██████ in ███, the ████████ of ████. This ███████, in ████, from the ████ the █████████ ███ ███████ ███(█, –) : █ → ███ █████████ ███ ██████ in █. By ███████, the █████████████ ███ ███████ ████ ████ ████████ to ██████.

If a ███████ █ : █ → █ has a █████ in █, ███████ by ███ █, there is a █████████ ███████████

${\displaystyle \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},<span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\cong <span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-)}$

which is ███████ in the ████████ █. ████ the ███████ ███(█, █–) is the ███████████ of the ███ ███████ ███(█, –) with █. This ███████████ is the ██████ ███ which ████████ the ████████ █████.

███ can use the above ████████████ to ██████ the █████ of █ in █. The █████ ████ is to ███████ that the █████ of the ███████ ███(█, █–) can be ██████████ with the ███ of ███ █████ from █ to █:

${\displaystyle <span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-)=\mathrm{<span\; word-length="4"\; class="word\; redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$

The ████████ ████ is █████ by the ██████ of ████ █_█ : ████(█, █) → ███(█, ██) █████ █_█(█) = █_█. If ███ is █████ an ██████ █ of █ ████████ with a ███████ ███████████ █ : ███(█, –) → ████(–, █), the ██████ █ will be a █████ of █ with the ████████ ████ █████ by █_█(██_█). In █████ ████████, this ███████ to ██████ that a █████ of █ is a ██████████████ of the ███████ ████(–, █) : █ → ███.

██████, if a ███████ █ : █ → █ has a ███████ in █, ███████ █████ █, there is a ██████ █████████ ███████████

${\displaystyle \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="5"\; class="word\; redacted">█████</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\cong <span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$

which is ███████ in the ████████ █ and ████████ the ██████████ █████. ███████████ the █████ of ███(█–, █) with the ███ ██████(█, █), this ████████████ can be used to ██████ the ███████ of the ███████ █ as a ██████████████ of the ███████ ██████(█, –).

███████████ of ██████ and ████████ of ████

███ █ be a ██████ ████████ and █ be a █████ ████████ ████████. For ███ █████████

${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\times \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>},}$

there is a ███████ ███████████

${\displaystyle {\mathrm{<span\; word-length="5"\; class="word\; redacted">█████</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\to \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{<span\; word-length="3"\; class="word\; redacted">███</span>}{\mathrm{<span\; word-length="5"\; class="word\; redacted">█████</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$

In █████, ████████ ████████ in ███ ███████ with ██████ ██████. It ████ █████ that █████ ████████ ███████ with █████ ██████.

████████████ of ██████

A ███████ █ : █ → █ ███████ a ███ from ████(█) to ████(██): if █ is a ████ from █ to █ ████ ██ is a ████ from ██ to ██. The ███████ █ is ████ to ████████ the ██████ of █ if (██, ██) is a █████ of ██ ████████ (█, █) is a █████ of █. (████ that if the █████ of █ ████ not █████, ████ █ █████████ █████████ the ██████ of █.)

A ███████ █ is ████ to ████████ ███ ██████ of █████ █ if it █████████ the ██████ of ███ ████████ █ : █ → █. For ███████, ███ can ███ that █ █████████ ████████, ██████████, █████████, ███. A ██████████ ███████ is ███ that █████████ ███ █████ ██████.

███ can ████ █████████ ███████████ for ████████. For ████████, a ███████ █ █████████ the ████████ of █ if █(█, █) is a ███████ of ██ ████████ (█, █) is a ███████ of █. A ████████████ ███████ is ███ that █████████ ███ █████ ████████.

If █ is a ████████ ████████, ████, by the above █████████ ███████ for ██████, a ███████ █ : █ → █ is ██████████ if and ████ if it █████████ (█████) ████████ and ██████████. ██████, █ is ████████████ if and ████ if it █████████ (█████) ██████████ and ████████████.

An █████████ ████████ of ███████ ████████ is that █████ █████ ███████ ███████ is ██████████ and █████ ████ ███████ ███████ is ████████████. Since ███████ ████████ █████ in █████████, this █████ ████████ ████████ of ██████████ and ████████████ ████████.

For a █████ ███████ █ : █ → █ and ███████ █ : █ → █, if ████ █ and ██ have █████████ ██████ there is a ██████ █████████ ████████

${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}<span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to <span\; word-length="3"\; class="word\; redacted">███</span>\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$

which ████████ the █████████████ █████ █████. The ███████ █ █████████ the ██████ of █ if and ████ if this ███ is an ███████████. If the ██████████ █ and █ have ███ ██████ of █████ █ ████ ███ is a ███████ and the █████████ █_█ ████ the ██████████ of a ███████ ██████████████

${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}<span\; word-length="3"\; class="word\; redacted">███</span>\to <span\; word-length="3"\; class="word\; redacted">███</span>{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$

The ███████ █ █████████ ███ ██████ of █████ █ if and ████ if █ is a ███████ ███████████. In this █████, the ███████ █ can be ████ to ███████ with ██████ (up to a █████████ ███████ ███████████).

████████████ of ██████ and ████████ is a ███████ that ████ ███████ to █████████ ████████. For █████████████ ████████ the █████████████ ███████ █████ be a ███████ that █████ ████████ to ██████, or ███ that █████ ██████ to ████████.

███████ of ██████

A ███████ █ : █ → █ is ████ to ████ ██████ for a ███████ █ : █ → █ if ████████ (█, █) is a █████ of ██ there ██████ a █████ (█′, █′) of █ ████ that █(█′, █′) = (█, █). A ███████ █ █████ ██████ of █████ █ if it █████ ██████ for ███ ████████ of █████ █. ███ can █████████ ████ about ███████ ████████, ██████████, █████████, ███. ███████, ███ ████ that █ █████ ██████ if it █████ ███ ██████. There are ████ ███████████ for the ███████ of ████████.

A ███████ █ █████ ██████ ████████ for a ███████ █ if there is a ██████ ████████ ████ (█′, █′) ████ that (█′, █′) is a █████ of █ and █(█′, █′) = (█, █). ███ can ████ that █ █████ ██████ ████████ if and ████ if it █████ ██████ and is ████████.

███████ of ██████ is ███████ ███████ to ████████████ of ██████. If █ █████ ██████ for a ███████ █ and ██ has a █████, ████ █ ████ has a █████ and █ █████████ the ██████ of █. It ███████ that:

• If █ █████ ██████ of ███ █████ █ and █ has ███ ██████ of █████ █, ████ █ ████ has ███ ██████ of █████ █ and █ █████████ █████ ██████.
• If █ █████ ███ █████ ██████ and █ is ████████, ████ █ is ████ ████████ and █ is ██████████.

The ████ ██████████ for ████████ are ███████ █████.

████████ and ██████████ of ██████

███ █ : █ → █ be a ███████. A ███████ █ : █ → █ is ████ to

• ██████ ██████ for █ if ████████ (█, █) is a █████ of ██ there ██████ a ██████ ████ (█′, █′) to █ ████ that █(█′, █′) = (█, █), and ███████████, this ████ is a █████ of █.
• ███████ ██████ for █ if ████ ████ to █ █████ █████ under █ is a █████ of ██ is ███████ a █████ of █.

██████, ███ can ██████ ████████ and ██████████ of ████████.

The following ██████████ are ██████ ████ to be ██████████:

• The ███████ █ ███████ ██████.
• The ███████ █ █████ ██████ ████████ and ████████ ██████.

There are ████████ of ████████ which ████ ██████ ████████ but ███████ ██████ ███ ███████ ████.

████████

• █████ █████████████ ███████ █ → ███ █████████ ██████ (but not ███████████ ████████). In ██████████, for ███ ██████ A of █, this is ████ of the █████████ ███ ███████ ███(A,–) : █ → ███.
• The █████████ ███████ █ : ███ → ███ ███████ (and █████████) ███ █████ ██████ and ████████ ████████; ███████, █ ████ not ████████ ██████████. This █████████ is ███████ of █████████ █████████ ████████.
• The ████ ███████ █ : ███ → ███ (which ███████ to █████ ███ █ the ████ █████ over █) is ████ ███████ to █████████ ███████ █ and is, █████████, ████████████. This ████████ ███ the ████ ███████ of ███ ████ ██████ █ and █ is the ████ █████ █████████ by the ████████ █████ of the ██████████ of █ and █.
• The █████████ ███████ ██ → ███ ███████ ██████ but ████ not ████████ ██████████ (the █████████ of ███ ███████ ██████ being the ██████ ███).
• The █████████ ███████ ███ → ███ █████ ██████ and ████████ ████████ but ███████ ███████.
• ███ ███_█ be the ████████ of ██████ ██████ with ██████████ █████████ for █████████. The █████████ ███████ ███_█ → ███ █████ ██████ ██████ but ████ not ████ ████ ████████.