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Revision as of 02:36, 13 February 2009
, 02:36, 13 February 2009→Fact 2.3
)) :D10a
)) :D10a
</pre>
</pre>
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px; text-align:center"
| style="width:80%" |
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.2}</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| width="2%" style="border-top:1px solid black" |
| width="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="66%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:50px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
| style="border-left:1px solid black" |
|}
|-
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:10px"
| width="2%" style="border-top:1px solid black" |
| width="12%" style="border-top:1px solid black" |
| width="66%" style="border-top:1px solid black" |
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:100px"
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| valign="top" | <math>\operatorname{F2.2a.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\operatorname{Der}^L
& = & \{ & (x, y) \in S \times I ~: \\
& & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& ) & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.2a~:~R11a}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.2b.}</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \upharpoonleft & \{ & (x, y) \in S \times I ~: \\
& & & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& ) & \\
\end{array} \\
& & & \} & \\
& & \upharpoonright & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2b~:~R11b}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.2c.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{cccl}
\downharpoonleft & \underset{o \in O}{\operatorname{Conj}} \\
& & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& & ) & \\
\downharpoonright & & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2c~:~R11c}</math></p>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.2d.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{cccl}
\underset{o \in O}{\operatorname{Conj}} \\
& \downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& & ) & \\
& \downharpoonright & & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2d~:~Log}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.2e.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& , & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& \underline{))} & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2e~:~Log}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.2f.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{cll}
\underset{o \in O}{\operatorname{Conj}} \\
& \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright \\
& , & \upharpoonleft \operatorname{Den}^L y \upharpoonright \\
& \underline{))}^\$ & (o) \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2f~:~$~}</math>
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