Line 3,824: |
Line 3,824: |
| } :D12a | | } :D12a |
| </pre> | | </pre> |
| + | |
| + | <br> |
| + | |
| + | {| 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%" |
| + | | |
| + | {| 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~1}</math> |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| + | |- style="height:50px" |
| + | | style="border-top:1px solid black; width:2%" | |
| + | | style="border-top:1px solid black; width:18%" | <math>\text{If}\!</math> |
| + | | style="border-top:1px solid black; width:60%" | <math>P, Q ~\subseteq~ X</math> |
| + | | 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" | |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| + | |- style="height:60px" |
| + | | style="width:2%; border-top:1px solid black" | |
| + | | style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math> |
| + | | style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math> |
| + | | style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" | |
| + | <math>\operatorname{F1a~:~R9a}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:40px" |
| + | | |
| + | | <math>\operatorname{F1b.}</math> |
| + | | <math>s \quad \Leftrightarrow \quad (\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright)</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <math>\operatorname{F1b~:~R9b}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{F1c.}</math> |
| + | | <math>s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <math>\operatorname{F1c~:~R9c}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{F1d.}</math> |
| + | | <math>s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <p><math>\operatorname{F1d~:~R9d}</math></p> |
| + | <p><math>\operatorname{F1d~:~R8a}</math></p> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{F1e.}</math> |
| + | | <math>\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <p><math>\operatorname{F1e~:~R8b}</math></p> |
| + | <p><math>\operatorname{F1e~:~\_\_?\_\_}</math></p> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{F1f.}</math> |
| + | | <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p> |
| + | <p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{F1g.}</math> |
| + | | <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p> |
| + | <p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:40px" |
| + | | |
| + | | <math>\operatorname{F1h.}</math> |
| + | | <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <math>\operatorname{F1h~:~~\_\_?\_\_}</math> |
| + | |} |
| + | |} |
| + | |
| + | <br> |
| | | |
| ===Fact 2.2=== | | ===Fact 2.2=== |