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

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

<b>assume </b><a NAME="E1:18"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="aff_2.html#NR11" target="_self">satisfies_des</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a4</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b4</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font> )
 ;<br/><a NAME="E2:18_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">M</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">N</font> is <a href="aff_1.html#V1">being_line</a> )
 <b>by </b><i><a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E3:18_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> or <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a4</font> )
 ;<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/><a NAME="E2:18_1_1_1"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/>

<a NAME="E2:18_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">a3</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E2:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E5:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:18_1_1_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18_1_1_1_1"/>
<font color="Maroon">b1</font>,<font color="Maroon">b3</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a4</font>
 ;<br/><a NAME="E2:18_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b2</font>,<font color="Maroon">b4</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E3:18_1_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a4</font>,<font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E4:18_1_1_2"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E5:18_1_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E6:18_1_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E7:18_1_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E8:18_1_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E9:18_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b4</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="E10:18_1_1_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a4</font> )
 ;<br/><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  holds <br/>(  not  <a href="aff_1.html#R1">LIN</a> <font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font> or  not <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> or  not <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> or  not <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> )
 ;<br/><a NAME="E2:18_1_2_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:18_1_2_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:18_1_2_1_1"/>
<font color="Maroon">a4</font>,<font color="Maroon">b4</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E4:18_1_2_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/><a NAME="E5:18_1_2_1_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E6:18_1_2_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E7:18_1_2_1_1"/>
<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E8:18_1_2_1_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/><a NAME="E9:18_1_2_1_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E10:18_1_2_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E11:18_1_2_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E12:18_1_2_1_1"/><b>then </b>
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/><b>hence </b><a NAME="E13:18_1_2_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:18_1_2_1_2"/>
<font color="Maroon">a2</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>,<font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:18_1_2_1_2_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_1_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:18_1_2_1_2_1"/>
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_1_2_1_1"/>
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/>

<a NAME="E2:18_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:18_1_2_1_2_1_1"/>
<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:18_1_2_1_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:18_1_2_1_2_1"/>
<font color="Maroon">a2</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E7:18_1_2_1_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E8:18_1_2_1_2_1"/>
<font color="Maroon">a4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_1_2_1_2"/>
<font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font>
 ;<br/>

<a NAME="E2:18_1_2_1_2_1_2"/><b>then </b>
<font color="Maroon">a4</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:18_1_2_1_2_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a4</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:18_1_2_1_2_1_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_2_1_2_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:18_1_2_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E10:18_1_2_1_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/><a NAME="E11:18_1_2_1_2_1"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E12:18_1_2_1_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_2_2"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:18_1_2_1_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:18_1_2_1_2_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><b>hence </b><a NAME="E4:18_1_2_1_2_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1_2_1_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> )
 ;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:18_1_2_2"/><i><font color="Green">E81</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="translac.html#T2">TRANSLAC:2</a></i>;<br/><a NAME="E4:18_1_2_2"/>
 ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:18_1_2_2_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E3:18_1_2_2_1"/>
( <font color="Maroon">Y</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:18_1_2_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">Y</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> )
 ;<br/><a NAME="E7:18_1_2_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E8:18_1_2_2"/>
 ex <font color="Olive">d1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Olive">d1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">e1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:18_1_2_2_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">e2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:18_1_2_2_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">e2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E5:18_1_2_2_2"/>
not <font color="Maroon">d</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_2_2_1"/>
<font color="Maroon">d</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">e2</font>
 ;<br/>

<a NAME="E2:18_1_2_2_2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">d</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:18_1_2_2_2_1"/>
<font color="Maroon">d</font>,<font color="Maroon">e1</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:18_1_2_2_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/>
<a NAME="E5:18_1_2_2_2_1"/>
<font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E7:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E8:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E9:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E10:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E11:18_1_2_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</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="E12:18_1_2_2_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">d1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:18_1_2_2_2"/><i><font color="Green">E89</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d</font>,<font color="Maroon">e1</font>,<font color="Maroon">d1</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">e2</font>,<font color="Maroon">d1</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/>
<b>take </b>
<font color="Maroon">d1</font>
;<br/>

<a NAME="E8:18_1_2_2_2"/>
<font color="Maroon">b1</font>,<font color="Maroon">e2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">d1</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="E9:18_1_2_2_2"/>
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">d1</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">d1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:18_1_2_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">d1</font> )
 ;<br/><a NAME="E11:18_1_2_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E12:18_1_2_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_2_3"/><i><font color="Green">E94</font></i>: 
(  not <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> or  not <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> or  not <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> )
 ;<br/>

<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2_3_1"/>
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:18_1_2_2_3_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_2_3_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">d</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E4:18_1_2_2_3_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E5:18_1_2_2_3_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E6:18_1_2_2_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>,<font color="Maroon">a1</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="E7:18_1_2_2_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2_3_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:18_1_2_2_3_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_2_3_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:18_1_2_2_3_2"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E5:18_1_2_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</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="E6:18_1_2_2_3_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_1_2_2_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_1_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E14:18_1_2_2"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E15:18_1_2_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E16:18_1_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">d1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:18_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E18:18_1_2_2"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E19:18_1_2_2"/><b>then </b>
( <font color="Maroon">a2</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">d1</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</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/><a NAME="E20:18_1_2_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">d</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="aff_2.html#D11">AFF_2:def 11</a></i>;<br/><a NAME="E21:18_1_2_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">d1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_2_2_4"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E2:18_1_2_2_4"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:18_1_2_2_4"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:18_1_2_2_4"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_2_2_4"/>
<font color="Maroon">a1</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">d1</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="E22:18_1_2_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">d</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , , , <a class="ref" href="aff_2.html#D11">AFF_2:def 11</a></i>;<br/><a NAME="E23:18_1_2_2"/>
<font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E24:18_1_2_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="aff_2.html#D11">AFF_2:def 11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<font color="Maroon">X</font> <a href="conmetr1.html#NR1" target="_self">satisfies_minor_SCH</a> 
 <b>by </b><i/>;<br/>


</div>
