Line 3,832: |
Line 3,832: |
| |- style="height:50px; text-align:center" | | |- style="height:50px; text-align:center" |
| | style="width:80%" | | | | style="width:80%" | |
− | | style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~1}</math> | + | | style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.1}</math> |
| |} | | |} |
| |- | | |- |
Line 3,838: |
Line 3,838: |
| {| align="center" cellpadding="0" cellspacing="0" width="100%" | | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| |- style="height:50px" | | |- style="height:50px" |
− | | style="border-top:1px solid black; width:2%" | | + | | width="2%" style="border-top:1px solid black" | |
− | | style="border-top:1px solid black; width:18%" | <math>\text{If}\!</math> | + | | width="10%" style="border-top:1px solid black" | <math>\text{If}\!</math> |
− | | style="border-top:1px solid black; width:60%" | <math>P, Q ~\subseteq~ X</math> | + | | width="68%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math> |
− | | style="border-top:1px solid black; border-left:1px solid black" | | + | | width="20%" style="border-top:1px solid black; border-left:1px solid black" | |
| |- style="height:50px" | | |- style="height:50px" |
| | | | | |
| | <math>\text{then}\!</math> | | | <math>\text{then}\!</math> |
− | | <math>\text{the following are equivalent:}\!</math> | + | | <math>\text{the following are identical subsets of}~ S \times I :</math> |
| | style="border-left:1px solid black" | | | | style="border-left:1px solid black" | |
| |} | | |} |
Line 3,852: |
Line 3,852: |
| {| align="center" cellpadding="0" cellspacing="0" width="100%" | | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| |- style="height:60px" | | |- style="height:60px" |
− | | style="width:2%; border-top:1px solid black" | | + | | width="2%" style="border-top:1px solid black" | |
− | | style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math> | + | | width="10%" style="border-top:1px solid black" | <math>\operatorname{F2.1a.}</math> |
− | | style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math> | + | | width="68%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math> |
− | | style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" | | + | | width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{F2.1a~:~D13a}</math> |
− | <math>\operatorname{F1a~:~R9a}</math> | |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,864: |
Line 3,863: |
| |- style="height:40px" | | |- style="height:40px" |
| | | | | |
− | | <math>\operatorname{F1b.}</math> | + | | <math>\operatorname{F2.1b.}</math> |
− | | <math>s \quad \Leftrightarrow \quad (\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright)</math> | + | | <math>\operatorname{Der}(L)</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <math>\operatorname{F1b~:~R9b}</math> | + | <math>\operatorname{F2.1b~:~D13b}</math> |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,875: |
Line 3,874: |
| |- style="height:60px" | | |- style="height:60px" |
| | | | | |
− | | <math>\operatorname{F1c.}</math> | + | | <math>\operatorname{F2.1c.}</math> |
− | | <math>s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math> | + | | <math>\{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) ~=~ \operatorname{Den}(L, y) \}</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <math>\operatorname{F1c~:~R9c}</math> | + | <p><math>\operatorname{F2.1c~:~D13c}</math></p> |
| + | <p><math>\operatorname{F2.1c~:~R9a}</math></p> |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,884: |
Line 3,884: |
| | | | | |
| | style="border-left:1px solid black; text-align:center" | <math>::\!</math> | | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
− | |- style="height:60px" | + | |- style="height:40px" |
| + | | |
| + | | <math>\operatorname{F2.1d.}</math> |
| + | | <math>\{ (x, y) \in S \times I ~:~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright \}</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <math>\operatorname{F2.1d~:~R9b}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:40px" |
| | | | | |
− | | <math>\operatorname{F1d.}</math> | + | | <math>\operatorname{F2.1e.}</math> |
− | | <math>s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math> | + | | <math>\ldots</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <p><math>\operatorname{F1d~:~R9d}</math></p>
| + | <math>\operatorname{F2.1e~:~R9c}</math> |
− | <p><math>\operatorname{F1d~:~R8a}</math></p>
| |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,896: |
Line 3,906: |
| | | | | |
| | style="border-left:1px solid black; text-align:center" | <math>::\!</math> | | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
− | |- style="height:60px" | + | |- style="height:40px" |
| | | | | |
− | | <math>\operatorname{F1e.}</math> | + | | <math>\operatorname{F2.1f.}</math> |
− | | <math>\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math> | + | | <math>\ldots</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <p><math>\operatorname{F1e~:~R8b}</math></p>
| + | <math>\operatorname{F2.1f~:~R9d}</math> |
− | <p><math>\operatorname{F1e~:~\_\_?\_\_}</math></p>
| |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,908: |
Line 3,917: |
| | | | | |
| | style="border-left:1px solid black; text-align:center" | <math>::\!</math> | | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
− | |- style="height:60px" | + | |- style="height:40px" |
| | | | | |
− | | <math>\operatorname{F1f.}</math> | + | | <math>\operatorname{F2.1g.}</math> |
− | | <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math> | + | | <math>\ldots</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
| + | <math>\operatorname{F2.1g~:~R9e}</math> |
− | <p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
| |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,922: |
Line 3,930: |
| |- style="height:60px" | | |- style="height:60px" |
| | | | | |
− | | <math>\operatorname{F1g.}</math> | + | | <math>\operatorname{F2.1h.}</math> |
− | | <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math> | + | | <math>\ldots</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p> | + | <p><math>\operatorname{F2.1h~:~R9f}</math></p> |
− | <p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p> | + | <p><math>\operatorname{F2.1h~:~D12e}</math></p> |
| |- style="height:20px" | | |- style="height:20px" |
| | | | | |
Line 3,934: |
Line 3,942: |
| |- style="height:40px" | | |- style="height:40px" |
| | | | | |
− | | <math>\operatorname{F1h.}</math> | + | | <math>\operatorname{F2.1i.}</math> |
− | | <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math> | + | | <math>\ldots</math> |
| | style="border-left:1px solid black; text-align:center" | | | | style="border-left:1px solid black; text-align:center" | |
− | <math>\operatorname{F1h~:~~\_\_?\_\_}</math> | + | <math>\operatorname{F2.1i~:~D12a}</math> |
| |} | | |} |
| |} | | |} |