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<div class="add">

<b>let </b><font color="Maroon">SAS</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/>

<b>assume </b><a NAME="E1:1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">SAS</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a> 
 ;<br/>

<a NAME="E2:1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">SAS</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a> 
 
<b>by </b><i><a class="ref" href="aff_2.html#T23">AFF_2:23</a></i>;<br/>
<b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">b3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b3</font> )
 <b>and </b><br/><a NAME="E4:1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b2</font> )
 ;<br/>

<a NAME="E5:1"/><i><font color="Green">E19</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E7:1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E8:1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E10:1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E11:1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1"/>
( <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 ;<br/><a NAME="E2:1_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, , , <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , <a class="ref" href="aff_2.html#D13">AFF_2:def 13</a></i>;<br/><b>hence </b><a NAME="E3:1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 ;<br/><b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E3:1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/><a NAME="E2:1_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_1_1"/>
<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_1_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E2:1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><a NAME="E3:1_2_1_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:1_2_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:1_2_1_1_1"/><b>then </b>
<font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E1:1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_2_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="txt" href="pardepap.html#E1:1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_1_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_1_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:1_2_1_2_1"/><b>then </b>
<font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E5:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><a NAME="E4:1_2_1_2_1"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , , , , <a class="txt" href="pardepap.html#E1:1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E5:1_2_1_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_2_1_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E1:1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:1_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 ;<br/><a NAME="E2:1_2_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_2_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_2_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><a NAME="E3:1_2_2_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:1_2_2_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_2_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font>
 ;<br/><a NAME="E2:1_2_2_1_1"/><b>then </b>
<font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E5:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><b>hence </b><a NAME="E3:1_2_2_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E2:1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_2_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="txt" href="pardepap.html#E2:1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_2_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">b3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_2_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 <b>by </b><i>, <a class="txt" href="pardepap.html#E5:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:1_2_2_2_1"/><b>then </b>
<font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><a NAME="E4:1_2_2_2_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , , <a class="txt" href="pardepap.html#E2:1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E5:1_2_2_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_2_2_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E2:1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:1_2_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> )
 ;<br/><div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_3_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_3_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><a NAME="E3:1_2_3_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:1_2_3_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><b>hence </b><a NAME="E5:1_2_3_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_3_2"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_3_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:1_2_3_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><b>hence </b><a NAME="E4:1_2_3_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , , <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_2_3"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_4"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> )
 ;<br/><a NAME="E2:1_2_4"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_4_1"/>
<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_4_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E5:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><a NAME="E3:1_2_4_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:1_2_4_1"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E5:1_2_4_1"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><b>hence </b><a NAME="E6:1_2_4_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_4_2"/>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:1_2_4_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/><b>hence </b><a NAME="E3:1_2_4_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:1_2_4"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_5"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> )
 ;<br/><a NAME="E2:1_2_5"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2_5_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/>

<a NAME="E2:1_2_5_1"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:1_2_5_1"/><b>then </b>
<font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:1_2_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:1_2_5"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2_5_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:1_2_5_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:1_2_5_2"/><b>then </b>
<font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:1_2_5_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:1_2_5"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, , , <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , , , <a class="txt" href="pardepap.html#E4:1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pardepap.html#E5:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_2.html#D2">AFF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:1_2_5"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:1_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="txt" href="pardepap.html#E1:1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pardepap.html#E2:1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="pardepap.html#E3:1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="pardepap.html#T1" target="_self">Th1</a>, , , <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div>
