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Revision as of 04:04, 11 February 2009
, 04:04, 11 February 2009→Fact 2.2
} :$
} :$
</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~2.1}</math>
|}
|-
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| width="2%" style="border-top:1px solid black" |
| width="10%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="68%" 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"
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| <math>\text{then}\!</math>
| <math>\text{the following are identical subsets of}~ S \times I :</math>
| style="border-left:1px solid black" |
|}
|-
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:60px"
| width="2%" style="border-top:1px solid black" |
| width="10%" style="border-top:1px solid black" | <math>\operatorname{F2.1a.}</math>
| width="68%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{F2.1a~:~D13a}</math>
|- style="height:20px"
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:40px"
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| valign="top" | <math>\operatorname{F2.1b.}</math>
| valign="top" | <math>\operatorname{Der}(L)</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1b~:~D13b}</math>
|- style="height:20px"
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|
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
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| valign="top" | <math>\operatorname{F2.1c.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \operatorname{Den}(L, x) ~=~ \operatorname{Den}(L, y) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F2.1c~:~D13c}</math></p>
<p><math>\operatorname{F2.1c~:~R9a}</math></p>
|- style="height:20px"
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
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| valign="top" | <math>\operatorname{F2.1d.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1d~:~R9b}</math>
|- style="height:20px"
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|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
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| valign="top" | <math>\operatorname{F2.1e.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \overset{O}{\underset{o}{\forall}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1e~:~R9c}</math>
|- style="height:20px"
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|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| valign="top" | <math>\operatorname{F2.1f.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1f~:~R9d}</math>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| valign="top" | <math>\operatorname{F2.1g.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) ~\underline{))} \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1g~:~R9e}</math>
|- style="height:20px"
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|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| valign="top" | <math>\operatorname{F2.1h.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright ~\underline{))}^\$ (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F2.1h~:~R9f}</math></p>
<p><math>\operatorname{F2.1h~:~D12e}</math></p>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
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| valign="top" | <math>\operatorname{F2.1i.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft L_{OS} \cdot x \upharpoonright ~,~ \upharpoonleft L_{OS} \cdot y \upharpoonright ~\underline{))}^\$ (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1i~:~D12a}</math>
|}
|}
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===Fact 2.3===
===Fact 2.3===
