<?xml version="1.0"?>
<div class="add">

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

<b>assume </b><a NAME="E1:20"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">AP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">AP</font> <a href="aff_2.html#NR5" target="_self">satisfies_DES_1</a> 
 
<b>by </b><i><a class="ref" href="aff_2.html#T16">AFF_2:16</a></i>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_3.html#D3" target="_self">AFF_3:def 3</a><br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E3:20"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> )
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green">E39</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">q</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> )
 ;<br/>

<a NAME="E7:20"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E8:20"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon">K</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>;<br/>
<a NAME="E9:20"/><i><font color="Green">E42</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E10:20"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E11:20"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:20_1"/>
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:20"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:20_2"/>
<font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:20_2"/><b>then </b>
( <font color="Maroon">c'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">a'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E4:20_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:20"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:20_3"/>
<font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E3:20_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:20"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>now </b></a><div class="add"><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_4_1"/><i><font color="Green">E54</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:20_4_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E3:20_4_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E4:20_4_1"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/><b>hence </b><a NAME="E5:20_4_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_4_2"/><i><font color="Green">E55</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font>
 ;<br/><b>set </b><font color="Maroon">D</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><a NAME="E2:20_4_2"/><i><font color="Green">E56</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">D'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E4:20_4_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E5:20_4_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">D'</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E6:20_4_2"/>
not <font color="Maroon">D'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_4_2_1"/>
<font color="Maroon">D'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:20_4_2_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:20_4_2_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="aff_3.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:20_4_2_1"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E8:20_4_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="aff_3.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E9:20_4_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="aff_3.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_3.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E10:20_4_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c'</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E11:20_4_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_4_2_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:20_4_2_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_3.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_3.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E3:20_4_2_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_3.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<a NAME="E4:20_4_2_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_3.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E5:20_4_2_2"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E6:20_4_2_2"/><i><font color="Green">E66</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E7:20_4_2_2"/><i><font color="Green">E67</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E8:20_4_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_4_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E2:20_4_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:20_4_2_2_1"/><b>then </b>
<font color="Maroon">c'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E4:20_4_2_2_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="txt" href="aff_3.html#E2:20"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:20_4_2_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_3.html#D8" target="_self">Def8</a>, <a class="txt" href="aff_3.html#E2:20"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:20_4_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_4_2_2_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:20_4_2_2_2"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_4_2_2_2_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:20_4_2_2_2_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , <a class="txt" href="aff_3.html#E1:20"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_3.html#E3:20"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:20_4_2_2_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="aff_3.html#E1:20"><i><font color="Green">E33</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="E3:20_4_2_2_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="aff_3.html#E1:20"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_3.html#E4:20"><i><font color="Green">E36</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="E10:20_4_2_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_3.html#D8" target="_self">Def8</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:20_4_2"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E13:20_4_2"/><b>then </b>
<font color="Maroon">c'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E14:20_4_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">d</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="aff_3.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_3.html#D6" target="_self">Def6</a>, <a class="ref" href="aff_3.html#D7" target="_self">Def7</a>, <a class="ref" href="aff_2.html#D4">AFF_2:def 4</a></i>;<br/><a NAME="E15:20_4_2"/>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E16:20_4_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, <a class="ref" href="aff_3.html#D8" target="_self">Def8</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E17:20_4_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_3.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_3.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_2.html#D5">AFF_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_4"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:20"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>


</div>
