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===Definition 13===
 
===Definition 13===
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<pre>
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Definition 13
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If R c OxSxI,
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then the following are identical subsets of SxI:
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D13a. DerR
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D13b. Der(R)
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D13c. {<x,y> C SxI : DenR|x = DenR|y}
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D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
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</pre>
      
<br>
 
<br>
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|- style="height:40px; text-align:center"
 
|- style="height:40px; text-align:center"
 
| width="80%" | &nbsp;
 
| width="80%" | &nbsp;
| width="20%" | <math>\operatorname{Definition~8}</math>
+
| width="20%" | <math>\operatorname{Definition~13}</math>
 
|}
 
|}
 
|-
 
|-
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|- style="height:40px"
 
|- style="height:40px"
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D8a.}</math>
+
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D13a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{SI}\!</math>
+
| width="80%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
|- style="height:40px"
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| &nbsp;
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| <math>\operatorname{D8b.}</math>
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| <math>\operatorname{Con}^L</math>
   
|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
| <math>\operatorname{D8c.}</math>
+
| <math>\operatorname{D13b.}</math>
| <math>\operatorname{Con}(L)</math>
+
| <math>\operatorname{Der}(L)</math>
 
|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
| <math>\operatorname{D8d.}</math>
+
| <math>\operatorname{D13c.}</math>
| <math>\operatorname{proj}_{SI}(L)</math>
+
| <math>\{ (x, y) \in S \times I ~:~ \operatorname{Den}^L|_x = \operatorname{Den}^L|_y \}</math>
 
|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
| <math>\operatorname{D8e.}</math>
+
| <math>\operatorname{D13d.}</math>
| <math>\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}</math>
+
| <math>\{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) = \operatorname{Den}(L, y) \}</math>
 
|}
 
|}
 
|}
 
|}
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