█-██████

In ███████████, █-██████ is, ███████ ████████, the █████ of a ████ █████████ by ██████ ███████ over a ███████████ █████ or ██████. In █████████ ████████, it is a ██████████ ██████ █████ as ███████████ █-██████. In ███████ and █████████ ████████, it is ████████ to as █████████ █-██████. It is ████ a ███████████ ████ in the █████ of ████████ ████████. It can be ████ as the █████ of ███████ █████ of ██████████ of █████ ████████.

█-██████ ████████ the ████████████ of ████████ of █-████████ that ███ from ███████████ ██████ or ███████, or to be ████ ████ ███████: ███ ██████ of a ████████ ████████ to ██████████ █████; █████ █████ ███████ some ███████ of the █████████ of the ████████ ██████ or ███████. As with ████████ to ██████ in █████████ ████████, the ██████ for this ██████████ ███████ is that it is ██████ to ███████ some ███████████ ██████████ from the ██████ █████ than from the ████████ ██████ or ███████. ████████ of ███████ ███████ from the █-██████ ████████ ███████ the ████████████–███████–████ ███████, ████ ███████████, the ██████–██████ █████ ███████, and the █████ ██████████.

In ████ ██████ ███████, █-██████ and in ██████████ ███████ █-██████ have ████████ in ████ ██ ██████ ██████ █████ it has been ███████████ that they ████████ █-██████, ██████–██████ █████ █████████ and ████ ███████ ███████ on ███████████ ███████ █████████. In █████████ ██████ ███████ █-██████ has been used to ████████ ███████████ ██████████, ███████████████ and ██████ █████ ████████. For ████ ███████, ███ █-██████ (███████).

The ████████████ ██████████ of an ███████ ██████ into an ███████ █████ is a █████████ ██████████ for ████████ █-██████ since ███ ███████████ █████ by ████████████ an ███████ ██████ from a ████████ ████████ and ███████ it into an ███████ █████ through this █████████ ████████████. █████ an ███████ ██████ ${\displaystyle (A,{+}^{\prime })}$ ███ ${\displaystyle \sim }$ be the ████████ on ${\displaystyle A^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=A\times A}$ ███████ by

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

if there ██████ a ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in A}$ ████ that ${\displaystyle a_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{+}^{\prime }{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{+}^{\prime }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=a_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{+}^{\prime }{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{+}^{\prime }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$ ████, the ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(A)=A^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}/\sim }$ has the █████████ of a █████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(A),+)}$ █████:

${\displaystyle [(a_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},a_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})]+[({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})]=[(a_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{+}^{\prime }{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},a_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{+}^{\prime }{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})].}$

███████████ ███████ in this █████ ██████ be ███████ of as ██████ ███████████ of ████████ in the ███████ ██████. This █████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(A),+)}$ is ████ ██████████ with a ██████ ████████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:A\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(A)}$ █████ by ${\displaystyle a\mapsto [(a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})],}$ which has a ███████ █████████ ████████.

To ███ a ██████ █████████████ of this █████, ████████ some ███████████ ███████ of the ███████ ██████ ${\displaystyle (A,+)}$. ████ ██ will ██████ the ████████ ███████ of ${\displaystyle A}$ by ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██ that ${\displaystyle [(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})]}$ will be the ████████ ███████ of ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(A),+).}$ █████, ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\sim (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ for ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in A}$ since ██ can ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and █████ the ████████ from the ███████████ ████████ to ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$ This ███████

${\displaystyle [(a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})]+[(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},a)]=[(a+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},a+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})]=[(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})]}$

█████ ██ have an ████████ ███████ ${\displaystyle [(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},a)]}$ for ████ ${\displaystyle [(a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})]\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(A)}$. This ██████ ████ ██ the ████ that ██ ██████ be ████████ of the ███████████ ███████ ${\displaystyle [(a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})]}$ as ██████ ███████████ ${\displaystyle a-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$ ███████ ██████ ███████████ is the ██████████ of ███████████ ███████ under ███████:

${\displaystyle (a,\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\sim (a+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ for ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in A.}$