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

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

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">AP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 ;<br/><b>thus </b><a NAME="E2:30_1"/>
<font color="Maroon">AP</font> <a href="aff_2.html#NR5" target="_self">satisfies_DES_1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<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_2.html#D5" target="_self">AFF_2:def 5</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">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">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">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/>



















<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">o</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">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">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">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">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>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</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">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</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">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</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> )
 ;<br/>

<b>assume </b><a NAME="E2:30_1_1"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <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">o</font>,<font color="Maroon">c'</font>;<br/>
<a NAME="E3:30_1_1"/><i><font color="Green">E46</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</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">o</font>,<font color="Maroon">c'</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E4:30_1_1"/><i><font color="Green">E47</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> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E5:30_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a'</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="30_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_1_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:30_1_1_1"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> or <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:30_1_1_1"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> or <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E5:30_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> or <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:30_1_1"/><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">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_2"/>
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_1_1_2"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:30_1_1_2"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:30_1_1_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</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>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:30_1_1_2"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">b</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="E6:30_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:30_1_1"/><i><font color="Green">E51</font></i>: 
<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">o</font>,<font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_1_1_3"/><b>then </b>
( <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E3:30_1_1_3"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_3"/>
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="E8:30_1_1"/>
not <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_4"/>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_1_1_4"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/>
<a NAME="E3:30_1_1_4"/><b>then </b>
<font color="Maroon">c'</font>,<font color="Maroon">a'</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T48">AFF_1:48</a></i>;<br/>
<a NAME="E4:30_1_1_4"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<a NAME="E5:30_1_1_4"/><b>then </b><i><font color="Green">E52</font></i>: 
<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#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:30_1_1_4"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</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="E7:30_1_1_4"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or <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#T62">AFF_1:62</a></i>;<br/>
<a NAME="E8:30_1_1_4"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E9:30_1_1_4"/><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">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E10:30_1_1_4"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:30_1_1_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<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="E10:30_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/>
<a NAME="E11:30_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E12:30_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">x</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="E13:30_1_1"/><i><font color="Green">E56</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_5"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:30_1_1_5"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:30_1_1_5"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_5"/>
contradiction
 <b>by </b><i>, , , , <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/>


</div><b>end;</b></div>
<a NAME="E14:30_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_6"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:30_1_1_6"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">o</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="E3:30_1_1_6"/><b>then </b>
<font color="Maroon">c</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="E4:30_1_1_6"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_1_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>
<a NAME="E15:30_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , , , , , </i>;<br/>
<a NAME="E16:30_1_1"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E17:30_1_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">o</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>
<b>hence </b><a NAME="E18:30_1_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></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">AP</font> <a href="aff_2.html#NR5" target="_self">satisfies_DES_1</a> 
 ;<br/><b>thus </b><a NAME="E2:30_2"/>
<font color="Maroon">AP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<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_2.html#D4" target="_self">AFF_2: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">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">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">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/>



















<b>assume </b><a NAME="E1:30_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">o</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">o</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">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">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">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>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</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">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</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> )
 ;<br/>

<b>assume </b><a NAME="E2:30_2_1"/><i><font color="Green">E59</font></i>: 
not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 ;<br/>

<a NAME="E3:30_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">b</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#T12">AFF_1:12</a></i>;<br/>
<a NAME="E4:30_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E5:30_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</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#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:30_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a'</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="30_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:30_2_1_1"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <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">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#T30">AFF_1:30</a></i>;<br/>
<a NAME="E3:30_2_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T12">AFF_1:12</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:30_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">a'</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="30_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_2"/>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_2_1_2"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</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#T30">AFF_1:30</a></i>;<br/>
<a NAME="E3:30_2_1_2"/><b>then </b>
( <font color="Maroon">a'</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#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2_1_2"/>
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="E8:30_2_1"/><i><font color="Green">E66</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_3"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_2_1_3"/><b>then </b>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</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">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#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:30_2_1_3"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:30_2_1_3"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</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="E5:30_2_1_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="E6:30_2_1_3"/><b>then </b>
 <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="E7:30_2_1_3"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E8:30_2_1_3"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E9:30_2_1_3"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</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">a'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E10:30_2_1_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30_2_1"/><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="30_2_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E2:30_2_1_4"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp;  <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="E3:30_2_1_4"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</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="E4:30_2_1_4"/><b>then </b>
( ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) &amp; ( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> or <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="E5:30_2_1_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:30_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">o</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="30_2_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_5"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_2_1_5"/><b>then </b>
( <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:30_2_1_5"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2_1_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>
<b>set </b><font color="Maroon">K</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">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">N</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>;<br/>
<a NAME="E11:30_2_1"/><i><font color="Green">E70</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;  <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">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> &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">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> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T38">AFF_1:38</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="E13:30_2_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">b</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="E14:30_2_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">D</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">D</font> <a href="aff_1.html#R4">//</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#T50">AFF_1:50</a></i>;<br/>
<a NAME="E15:30_2_1"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_6"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:30_2_1_6"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</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#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:30_2_1_6"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E4:30_2_1_6"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</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="E5:30_2_1_6"/><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">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E6:30_2_1_6"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:30_2_1_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<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="E17:30_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<a NAME="E18:30_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:30_2_1_7"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E2:30_2_1_7"/><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">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:30_2_1_7"/>
<font color="Maroon">a</font>,<font color="Maroon">x</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/>


</div><b>end;</b></div>
<a NAME="E19:30_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E20:30_2_1"/><i><font color="Green">E76</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="30_2_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:30_2_1_8"/><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">a'</font>
 
<b>by </b><i><a class="ref" href="aff_2.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:30_2_1_8"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:30_2_1_8"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a'</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="E5:30_2_1_8"/><b>then </b>
<font color="Maroon">c'</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="E6:30_2_1_8"/>
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="E21:30_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_9"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:30_2_1_9"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E3:30_2_1_9"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</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="E4:30_2_1_9"/><b>then </b>
<font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</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="E5:30_2_1_9"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</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/>
<b>hence </b><a NAME="E6:30_2_1_9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:30_2_1"/><i><font color="Green">E78</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_1_10"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:30_2_1_10"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:30_2_1_10"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</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_2.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_2.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:30_2_1_10"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</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/>
<b>hence </b><a NAME="E5:30_2_1_10"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">T</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c'</font>,<font color="Maroon">x</font>;<br/>
<a NAME="E23:30_2_1"/><i><font color="Green">E79</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c'</font>,<font color="Maroon">x</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c'</font>,<font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="aff_2.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E24:30_2_1"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c'</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="aff_2.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_2.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_2.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_2.html#D7" target="_self">Def7</a>, </i>;<br/>
<a NAME="E25:30_2_1"/><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_2.html#D8" target="_self">Def8</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E26:30_2_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">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_2.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E27:30_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></div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon">AP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a>  iff <font color="Maroon">AP</font> <a href="aff_2.html#NR5" target="_self">satisfies_DES_1</a>  )
 <b>by </b><i/>;<br/>


</div>
