<?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="E1:21"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">AP</font> <a href="aff_3.html#NR3" target="_self">satisfies_DES1_2</a> 
 ;<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#D4" target="_self">AFF_3:def 4</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="E2:21"/><i><font color="Green">E34</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="E3:21"/><i><font color="Green">E35</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="E4:21"/><i><font color="Green">E36</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">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">o</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">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">b</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">a'</font> )
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green">E37</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/>

<b>assume </b><a NAME="E6:21"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E7:21"/><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="E8:21"/><i><font color="Green">E41</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon">a</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>hence </b><a NAME="E3:21_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>
<a NAME="E9:21"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:21_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</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">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#T33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:21_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:21"/><i><font color="Green">E43</font></i>: 
<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="E11:21"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q</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="21_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_3"/><i><font color="Green">E45</font></i>: 
(  not <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or  not <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> or  not <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or  not <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> )
 ;<br/>

<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3_1"/><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="E2:21_3_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="E3:21_3_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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:21_3_2"/>
( 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="E3:21_3_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>
<b>hence </b><a NAME="E4:21_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="E12:21"/><i><font color="Green">E49</font></i>: 
(  not <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> or  not <font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_4"/>
( <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:21_4"/><b>then </b>
<font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:21_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<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/>
<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">p</font>,<font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font>;<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/>
<a NAME="E13:21"/><i><font color="Green">E50</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;  <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">p</font>,<font color="Maroon">q</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <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;  <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">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">p</font>,<font color="Maroon">q</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">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">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> &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> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E14:21"/><b>then </b><i><font color="Green">E51</font></i>: 
( <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> &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">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#T39">AFF_1:39</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_5"/>
not <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E3:21_5"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E4:21_5"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</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="21_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:21_5_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</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="E3:21_5_1"/>
 <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> <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#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:21_5_1"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="aff_3.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:21_5"/>
<font color="Maroon">x</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="21_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:21_5_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E3:21_5_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E4:21_5_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E5:21_5_2"/><b>then </b>
( <font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">q</font> or <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:21_5_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E7:21_5_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font>,<font color="Maroon">c'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E8:21_5_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E9:21_5_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>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E10:21_5_2"/><b>then </b>
<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>hence </b><a NAME="E11:21_5_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:21_5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="aff_3.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E7:21_5"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_6"/>
not <font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E3:21_6"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E4:21_6"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_6_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:21_6_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</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="E3:21_6_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b'</font>,<font color="Maroon">c</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E4:21_6_1"/><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>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E5:21_6_1"/><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>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E6:21_6_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_3.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E7:21_6_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">q</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_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E8:21_6_1"/><b>then </b>
<font color="Maroon">q</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/>
<a NAME="E9:21_6_1"/><b>then </b>
<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#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E10:21_6_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_3.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:21_6"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</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="21_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_6_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:21_6_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</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="E3:21_6_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">q</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E4:21_6_2"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E5:21_6_2"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> or <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/>
<a NAME="E6:21_6_2"/><b>then </b>
<font color="Maroon">c'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<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#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E7:21_6_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">a'</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="E8:21_6_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">c'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">a'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E9:21_6_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E10:21_6_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E11:21_6_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E12:21_6_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_3.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:21_6"/><i><font color="Green">E69</font></i>: 
( 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; 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#T15">AFF_1:15</a></i>;<br/><a NAME="E7:21_6"/>
( <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; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <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></i>;<br/><a NAME="E8:21_6"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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#D3" target="_self">Def3</a>, </i>;<br/><b>hence </b><a NAME="E9:21_6"/>
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>hence </b><a NAME="E17:21"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div>
