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

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

<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E48</font></i>: 
 <a href="conmetr.html#NR1" target="_self">OPAP_holds_in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E49</font></i>: 
 <a href="conaffm.html#NR1" target="_self">DES_holds_in</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="conmetr.html#D2" target="_self">CONMETR:def 2</a><br/><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">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">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><b>that </b><br/><a NAME="E3:15"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">M</font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="analmetr.html#NR6" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o</font>, <font color="Maroon">a1'</font> = <font color="Maroon">a1</font>, <font color="Maroon">a2'</font> = <font color="Maroon">a2</font>, <font color="Maroon">a3'</font> = <font color="Maroon">a3</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b1</font>, <font color="Maroon">b2'</font> = <font color="Maroon">b2</font>, <font color="Maroon">b3'</font> = <font color="Maroon">b3</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font>, <font color="Maroon">N'</font> = <font color="Maroon">N</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green">E72</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="analmetr.html#T58">ANALMETR:58</a></i>;<br/>
<a NAME="E8:15"/><b>then </b><i><font color="Green">E73</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">M</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">N</font>
 ;<br/><div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E2:15_1_1"/><i><font color="Green">E77</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E3:15_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b3'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:15_1_1"/>
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/>

<a NAME="E2:15_1_1_2_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</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="E3:15_1_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</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/>
<b>hence </b><a NAME="E4:15_1_1_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>
<a NAME="E2:15_1_1_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1_2"/>
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:15_1_1"/><b>then </b>
<font color="Maroon">b2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/><a NAME="E6:15_1_1"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b2'</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#T11">AFF_1:11</a></i>;<br/><b>hence </b><a NAME="E7:15_1_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/><a NAME="E2:15_1_2"/><i><font color="Green">E82</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E3:15_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_2_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</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><b>end;</b></div><a NAME="E4:15_1_2"/>
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3'</font>,<font color="Maroon">b1'</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E5:15_1_2"/><b>then </b>
<font color="Maroon">b2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/><b>hence </b><a NAME="E6:15_1_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</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="analmetr.html#T81">ANALMETR:81</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/><a NAME="E2:15_1_3"/><i><font color="Green">E86</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_3_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>
 ;<br/>

<a NAME="E2:15_1_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_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><a NAME="E3:15_1_3"/><i><font color="Green">E87</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">b1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_3_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_3_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b3'</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><b>end;</b></div><a NAME="E4:15_1_3"/>
<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3'</font>,<font color="Maroon">b1'</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><b>hence </b><a NAME="E5:15_1_3"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</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#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> )
 ;<br/><a NAME="E2:15_1_4"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">b1</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="15_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> or <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> )
 ;<br/>

<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:15_1_4_1_1"/><i><font color="Green">E93</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">a2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">a2'</font>
 ;<br/>

<a NAME="E2:15_1_4_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_1_1_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><a NAME="E3:15_1_4_1_1"/><i><font color="Green">E94</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_1_1_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">a2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_4_1_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<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/>


</div><b>end;</b></div><a NAME="E4:15_1_4_1_1"/>
<font color="Maroon">b2'</font>,<font color="Maroon">a2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_1_1_3"/>
<font color="Maroon">b2</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_4_1_1_3"/>
<font color="Maroon">b2'</font>,<font color="Maroon">a2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">a3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_4_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:15_1_4_1_2"/><i><font color="Green">E97</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_1_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font>
 ;<br/>

<a NAME="E2:15_1_4_1_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E3:15_1_4_1_2_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</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="E4:15_1_4_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</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/>
<b>hence </b><a NAME="E5:15_1_4_1_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><a NAME="E3:15_1_4_1_2"/><i><font color="Green">E99</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_1_2_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_4_1_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<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/>


</div><b>end;</b></div><a NAME="E4:15_1_4_1_2"/>
<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">a2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_1_2_3"/>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E2:15_1_4_1_2_3"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_1_2_3"/>
<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">a2'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_4_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_1_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:15_1_4_1_3"/><i><font color="Green">E101</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_1_3_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font>
 ;<br/>

<a NAME="E2:15_1_4_1_3_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E3:15_1_4_1_3_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</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="E4:15_1_4_1_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</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/>
<b>hence </b><a NAME="E5:15_1_4_1_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><a NAME="E3:15_1_4_1_3"/><i><font color="Green">E113</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_1_3_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_4_1_3_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</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/>


</div><b>end;</b></div><a NAME="E4:15_1_4_1_3"/>
<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_1_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_1_3_3"/>
<font color="Maroon">b1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">a3</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_4_1_3_3"/>
<font color="Maroon">b1'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">a3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_4_1_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_1_4_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:15_1_4_2"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E5:15_1_4_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">x'</font>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>then consider </b><font color="Maroon">A'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E7:15_1_4_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">A'</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A'</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/><a NAME="E9:15_1_4_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">A</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_2_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="analmetr.html#K4">Line</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_2_1_1"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">e1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">e1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:15_1_4_2_1_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">e1'</font> = <font color="Maroon">e1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<a NAME="E3:15_1_4_2_1_1_1"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E4:15_1_4_2_1_1_1"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">e1'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E5:15_1_4_2_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">x'</font>,<font color="Maroon">e1'</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="E6:15_1_4_2_1_1_1"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_4_2_1_1"/>
for <font color="Olive">e1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">e1</font> holds <br/><font color="Olive">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:15_1_4_2_1_1_2"/><i><font color="Green">E123</font></i>: 
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">e1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">e1'</font> = <font color="Maroon">e1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<a NAME="E3:15_1_4_2_1_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">x'</font>,<font color="Maroon">e1'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_4_2_1_1_2"/>
<font color="Maroon">e1</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_1_4_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="analmetr.html#K4">Line</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#D12">ANALMETR:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_4_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">x</font> <a href="analmetr.html#R6">_|_</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , , <a class="ref" href="analmetr.html#T77">ANALMETR:77</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_2_1"/>
<font color="Maroon">A</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#D15">ANALMETR:def 15</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">e1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E11:15_1_4_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">e1'</font> &amp; <font color="Maroon">b3'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E12:15_1_4_2"/>
not <font color="Maroon">b3'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">b3'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 ;<br/>

<b>consider </b><font color="Maroon">d1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">d2'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:15_1_4_2_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">d1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d2'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E4:15_1_4_2_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">d1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">d2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E5:15_1_4_2_2"/>
<font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E6:15_1_4_2_2"/><b>then </b>
<font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D4">AFF_1:def 4</a></i>;<br/>
<a NAME="E7:15_1_4_2_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">b3'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1'</font>, <font color="Maroon">d2</font> = <font color="Maroon">d2'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<a NAME="E9:15_1_4_2_2"/>
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E10:15_1_4_2_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E11:15_1_4_2_2"/>
<font color="Maroon">d1</font>,<font color="Maroon">d2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, , , <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/>
<a NAME="E12:15_1_4_2_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>
<a NAME="E13:15_1_4_2_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T85">ANALMETR:85</a></i>;<br/>
<a NAME="E14:15_1_4_2_2"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E15:15_1_4_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E16:15_1_4_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E17:15_1_4_2_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>then consider </b><font color="Maroon">c3'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E14:15_1_4_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b3'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">c3'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><b>reconsider </b><font color="Maroon">c3</font> = <font color="Maroon">c3'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E16:15_1_4_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">b3'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:15_1_4_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>consider </b><font color="Maroon">e1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E19:15_1_4_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> &amp; <font color="Maroon">a2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E20:15_1_4_2"/>
not <font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_3"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 ;<br/>

<a NAME="E2:15_1_4_2_3"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E3:15_1_4_2_3"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:15_1_4_2_3"/><b>then </b>
<font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E5:15_1_4_2_3"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:15_1_4_2_3"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">o'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:15_1_4_2_3"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c3'</font>,<font color="Maroon">o'</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="E8:15_1_4_2_3"/>
<font color="Maroon">c3'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_3_1"/>
<font color="Maroon">c3'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 ;<br/>

<a NAME="E2:15_1_4_2_3_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">o'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:15_1_4_2_3_1"/>
<font color="Maroon">b3'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/>
<a NAME="E4:15_1_4_2_3_1"/><b>then </b>
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</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="E5:15_1_4_2_3_1"/><b>then </b>
<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E6:15_1_4_2_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b3'</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/>
<b>hence </b><a NAME="E7:15_1_4_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>
<a NAME="E9:15_1_4_2_3"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">a3'</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="E10:15_1_4_2_3"/>
<font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E11:15_1_4_2_3"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E12:15_1_4_2_3"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E13:15_1_4_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c2'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E22:15_1_4_2"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">c2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">c2'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">c2'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E24:15_1_4_2"/>
<font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">c2'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E25:15_1_4_2"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">c2'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E26:15_1_4_2"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">c2</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E27:15_1_4_2"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><b>consider </b><font color="Maroon">e1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E29:15_1_4_2"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> &amp; <font color="Maroon">a1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E30:15_1_4_2"/>
not <font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 ;<br/>

<a NAME="E2:15_1_4_2_4"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E3:15_1_4_2_4"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:15_1_4_2_4"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E5:15_1_4_2_4"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:15_1_4_2_4"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">o'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:15_1_4_2_4"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c3'</font>,<font color="Maroon">o'</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="E8:15_1_4_2_4"/>
<font color="Maroon">c3'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_4_1"/>
<font color="Maroon">c3'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 ;<br/>

<a NAME="E2:15_1_4_2_4_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">o'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:15_1_4_2_4_1"/>
<font color="Maroon">b3'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/>
<a NAME="E4:15_1_4_2_4_1"/><b>then </b>
<font color="Maroon">a3'</font>,<font color="Maroon">b2'</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="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E5:15_1_4_2_4_1"/><b>then </b>
<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E6:15_1_4_2_4_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b3'</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/>
<b>hence </b><a NAME="E7:15_1_4_2_4_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>
<a NAME="E9:15_1_4_2_4"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">a3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E10:15_1_4_2_4"/>
<font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E11:15_1_4_2_4"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E12:15_1_4_2_4"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E13:15_1_4_2_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E32:15_1_4_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">c1'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c1'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E34:15_1_4_2"/>
<font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">c1'</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E35:15_1_4_2"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">c1'</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E36:15_1_4_2"/><b>then </b><i><font color="Green">E148</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font> )
 <b>by </b><i><a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E37:15_1_4_2"/><b>then </b><i><font color="Green">E149</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E38:15_1_4_2"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_5"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> )
 ;<br/>

<div><i><font color="Green">E155</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_2_5_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font>
 ;<br/><a NAME="E2:15_1_4_2_5_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E3:15_1_4_2_5_1"/>
<font color="Maroon">b3'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/><a NAME="E4:15_1_4_2_5_1"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">o</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E5:15_1_4_2_5_1"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b2</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="analmetr.html#T82">ANALMETR:82</a></i>;<br/><a NAME="E6:15_1_4_2_5_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E7:15_1_4_2_5_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">a3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E8:15_1_4_2_5_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">a3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><b>hence </b><a NAME="E9:15_1_4_2_5_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="15_1_4_2_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_2_5_2"/>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> )
 ;<br/><a NAME="E2:15_1_4_2_5_2"/><b>then </b><i><font color="Green">E157</font></i>: 
( <font color="Maroon">a3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">o</font> or <font color="Maroon">a3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E3:15_1_4_2_5_2"/>
( <font color="Maroon">a2'</font>,<font color="Maroon">o'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">a3'</font> &amp; <font color="Maroon">a1'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/><a NAME="E4:15_1_4_2_5_2"/><b>then </b>
( <font color="Maroon">a2</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">o</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E5:15_1_4_2_5_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">c3</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="analmetr.html#T82">ANALMETR:82</a></i>;<br/><a NAME="E6:15_1_4_2_5_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">c3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E7:15_1_4_2_5_2"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E8:15_1_4_2_5_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E9:15_1_4_2_5_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E10:15_1_4_2_5_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/><a NAME="E11:15_1_4_2_5_2"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 <b>by </b><i>, , , , <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/><a NAME="E12:15_1_4_2_5_2"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/><b>hence </b><a NAME="E13:15_1_4_2_5_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_4_2_5"/>
contradiction
 <b>by </b><i><a class="ref" href="conmetr.html#D8" target="_self">Def8</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E39:15_1_4_2"/><i><font color="Green">E161</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b3</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_6"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b3</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font> or  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font> )
 ;<br/>

<a NAME="E2:15_1_4_2_6"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b3'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a3'</font> or  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a3'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_2_6"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E40:15_1_4_2"/><i><font color="Green">E162</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_7"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font> or  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font> )
 ;<br/>

<a NAME="E2:15_1_4_2_7"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">c3'</font> or  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c3'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_2_7"/><b>then </b><i><font color="Green">E163</font></i>: 
( <font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> or <font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:15_1_4_2_7"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E5:15_1_4_2_7"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E6:15_1_4_2_7"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_4_2_7"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E41:15_1_4_2"/><i><font color="Green">E165</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_8"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_2_8"/>
( <font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</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="E2:15_1_4_2_8"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">a1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">b1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">a2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_2_8"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E42:15_1_4_2"/><i><font color="Green">E166</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_9"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_2_9"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E2:15_1_4_2_9"/><b>then </b>
( <font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c2'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c2'</font> )
 
<b>by </b><i>, , , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E3:15_1_4_2_9"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>,<font color="Maroon">c1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>,<font color="Maroon">c2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font>,<font color="Maroon">c2'</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="E4:15_1_4_2_9"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:15_1_4_2"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">c1</font>
 <b>by </b><i>, , <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#D8" target="_self">Def8</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, , <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/><a NAME="E44:15_1_4_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">c3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E45:15_1_4_2"/>
<font color="Maroon">b2</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">c2</font>
 <b>by </b><i>, , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#D8" target="_self">Def8</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, , <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/><a NAME="E46:15_1_4_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">c3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E47:15_1_4_2"/><i><font color="Green">E170</font></i>: 
not <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_10"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:15_1_4_2_10"/>
<font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_2_10"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_2_10"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_2_10"/>
contradiction
 <b>by </b><i>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E48:15_1_4_2"/>
not <font color="Maroon">c3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_11"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">c3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:15_1_4_2_11"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_2_11"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_2_11"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_2_11"/>
contradiction
 <b>by </b><i>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E49:15_1_4_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">c1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , , , , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="txt" href="conmetr.html#E2:15"><i><font color="Green">E49</font></i></a>, <a class="txt" href="conmetr.html#E3:15"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E50:15_1_4_2"/><i><font color="Green">E174</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_12"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font>
 ;<br/>

<a NAME="E2:15_1_4_2_12"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_2_12"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:15_1_4_2_12"/>
<font color="Maroon">o'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E5:15_1_4_2_12"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E6:15_1_4_2_12"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_4_2_12"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E51:15_1_4_2"/><i><font color="Green">E176</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_13"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_13"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:15_1_4_2_13"/><b>then </b><i><font color="Green">E177</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c1'</font>,<font color="Maroon">c2'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_2_13"/>
<font color="Maroon">c1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_13_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2'</font>
 ;<br/>

<a NAME="E2:15_1_4_2_13_1"/>
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_13_1_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font>
 ;<br/>

<a NAME="E2:15_1_4_2_13_1_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_2_13_1_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_2_13_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_2_13_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15_1_4_2_13_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:15_1_4_2_13_1"/><b>then </b>
<font color="Maroon">c1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:15_1_4_2_13_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c1</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:15_1_4_2_13_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c1'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:15_1_4_2_13_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E8:15_1_4_2_13_1"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E9:15_1_4_2_13_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E10:15_1_4_2_13_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E11:15_1_4_2_13_1"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E12:15_1_4_2_13_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_4_2_13"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E52:15_1_4_2"/>
<font color="Maroon">c1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">a2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_14"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_2_14"/>
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_2_14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_2_14_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font>
 ;<br/>

<a NAME="E2:15_1_4_2_14_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_2_14_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_2_14_1"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_2_14_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_4_2_14"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">c2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_2_14"/>
<font color="Maroon">c1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E53:15_1_4_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</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="conmetr.html#D7" target="_self">Def7</a>, , <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="conmetr.html#T2" target="_self">Th2</a>, <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_3"/><i><font color="Green">E182</font></i>: 
not <font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:15_1_4_3"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E5:15_1_4_3"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">x'</font>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>then consider </b><font color="Maroon">A'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E7:15_1_4_3"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">A'</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A'</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/><a NAME="E9:15_1_4_3"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E10:15_1_4_3"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">A</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_3_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="analmetr.html#K4">Line</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_3_1_1"/><i><font color="Green">E188</font></i>: 
for <font color="Olive">e1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">e1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:15_1_4_3_1_1_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">e1'</font> = <font color="Maroon">e1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<a NAME="E3:15_1_4_3_1_1_1"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E4:15_1_4_3_1_1_1"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">e1'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E5:15_1_4_3_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">x'</font>,<font color="Maroon">e1'</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="E6:15_1_4_3_1_1_1"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_4_3_1_1"/>
for <font color="Olive">e1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">e1</font> holds <br/><font color="Olive">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:15_1_4_3_1_1_2"/><i><font color="Green">E190</font></i>: 
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">e1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">e1'</font> = <font color="Maroon">e1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<a NAME="E3:15_1_4_3_1_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">x'</font>,<font color="Maroon">e1'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_4_3_1_1_2"/>
<font color="Maroon">e1</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_1_4_3_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="analmetr.html#K4">Line</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#D12">ANALMETR:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_4_3_1"/>
<font color="Maroon">o</font>,<font color="Maroon">x</font> <a href="analmetr.html#R6">_|_</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , , <a class="ref" href="analmetr.html#T77">ANALMETR:77</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_3_1"/>
<font color="Maroon">A</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#D15">ANALMETR:def 15</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">e1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E12:15_1_4_3"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">e1'</font> &amp; <font color="Maroon">b2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E13:15_1_4_3"/>
not <font color="Maroon">b2'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">b2'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 ;<br/>

<b>consider </b><font color="Maroon">d1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">d2'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:15_1_4_3_2"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">d1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d2'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E4:15_1_4_3_2"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">d1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">d2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E5:15_1_4_3_2"/>
<font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E6:15_1_4_3_2"/><b>then </b>
<font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D4">AFF_1:def 4</a></i>;<br/>
<a NAME="E7:15_1_4_3_2"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">b2'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1'</font>, <font color="Maroon">d2</font> = <font color="Maroon">d2'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<a NAME="E9:15_1_4_3_2"/>
<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E10:15_1_4_3_2"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E11:15_1_4_3_2"/>
<font color="Maroon">d1</font>,<font color="Maroon">d2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, , , <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/>
<b>hence </b><a NAME="E12:15_1_4_3_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c3'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E15:15_1_4_3"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">c3'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><b>reconsider </b><font color="Maroon">c3</font> = <font color="Maroon">c3'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E17:15_1_4_3"/>
<font color="Maroon">b2'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E18:15_1_4_3"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>consider </b><font color="Maroon">e1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E20:15_1_4_3"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> &amp; <font color="Maroon">a1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E21:15_1_4_3"/>
not <font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_3"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 ;<br/>

<a NAME="E2:15_1_4_3_3"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E3:15_1_4_3_3"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:15_1_4_3_3"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E5:15_1_4_3_3"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:15_1_4_3_3"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">o'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:15_1_4_3_3"/><b>then </b><i><font color="Green">E202</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c3'</font>,<font color="Maroon">o'</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="E8:15_1_4_3_3"/>
<font color="Maroon">c3'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_3_1"/>
<font color="Maroon">c3'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 ;<br/>

<a NAME="E2:15_1_4_3_3_1"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:15_1_4_3_3_1"/>
<font color="Maroon">b2'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/>
<a NAME="E4:15_1_4_3_3_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">o</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E5:15_1_4_3_3_1"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b3</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="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E6:15_1_4_3_3_1"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E7:15_1_4_3_3_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b3</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E8:15_1_4_3_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b3'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E9:15_1_4_3_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>
<a NAME="E9:15_1_4_3_3"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">a3'</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="E10:15_1_4_3_3"/>
<font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E11:15_1_4_3_3"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E12:15_1_4_3_3"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E13:15_1_4_3_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E23:15_1_4_3"/><i><font color="Green">E205</font></i>: 
( <font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">c1'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c1'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E25:15_1_4_3"/>
<font color="Maroon">a1'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">c1'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E26:15_1_4_3"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">c1'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E27:15_1_4_3"/><b>then </b><i><font color="Green">E207</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><b>consider </b><font color="Maroon">e1'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E29:15_1_4_3"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> &amp; <font color="Maroon">a2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E30:15_1_4_3"/>
not <font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_4"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 ;<br/>

<a NAME="E2:15_1_4_3_4"/>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E3:15_1_4_3_4"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:15_1_4_3_4"/><b>then </b>
<font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E5:15_1_4_3_4"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:15_1_4_3_4"/><b>then </b>
<font color="Maroon">c3'</font>,<font color="Maroon">o'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c3'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:15_1_4_3_4"/><b>then </b><i><font color="Green">E210</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c3'</font>,<font color="Maroon">o'</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="E8:15_1_4_3_4"/>
<font color="Maroon">c3'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_4_1"/>
<font color="Maroon">c3'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 ;<br/>

<a NAME="E2:15_1_4_3_4_1"/><b>then </b><i><font color="Green">E211</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:15_1_4_3_4_1"/>
<font color="Maroon">b2'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/>
<a NAME="E4:15_1_4_3_4_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">o</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E5:15_1_4_3_4_1"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b3</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="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E6:15_1_4_3_4_1"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E7:15_1_4_3_4_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b3</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E8:15_1_4_3_4_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b3'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E9:15_1_4_3_4_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>
<a NAME="E9:15_1_4_3_4"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">a3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E10:15_1_4_3_4"/>
<font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E11:15_1_4_3_4"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E12:15_1_4_3_4"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E13:15_1_4_3_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c2'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E32:15_1_4_3"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">c2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">c2'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">c2'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E34:15_1_4_3"/>
<font color="Maroon">a2'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">c2'</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E35:15_1_4_3"/><b>then </b><i><font color="Green">E214</font></i>: 
<font color="Maroon">c3'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">c2'</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E36:15_1_4_3"/><b>then </b><i><font color="Green">E215</font></i>: 
( <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E37:15_1_4_3"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_5"/><i><font color="Green">E217</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> )
 ;<br/>

<div><i><font color="Green">E218</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_3_5_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font>
 ;<br/><a NAME="E2:15_1_4_3_5_1"/>
<font color="Maroon">b2'</font>,<font color="Maroon">o'</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#T53">AFF_1:53</a></i>;<br/><a NAME="E3:15_1_4_3_5_1"/><b>then </b>
<font color="Maroon">a3'</font>,<font color="Maroon">b3'</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="E4:15_1_4_3_5_1"/><b>then </b>
<font color="Maroon">b3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">a3'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E5:15_1_4_3_5_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b3'</font>,<font color="Maroon">b2'</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/><b>hence </b><a NAME="E6:15_1_4_3_5_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="15_1_4_3_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4_3_5_2"/>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> )
 ;<br/><a NAME="E2:15_1_4_3_5_2"/><b>then </b><i><font color="Green">E220</font></i>: 
( <font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">o</font> or <font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E3:15_1_4_3_5_2"/>
( <font color="Maroon">a1</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_5_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_3_5_2_1"/>
( <font color="Maroon">a1'</font>,<font color="Maroon">o'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">a3'</font> &amp; <font color="Maroon">a2'</font>,<font color="Maroon">o'</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/>
<b>hence </b><a NAME="E2:15_1_4_3_5_2_1"/>
( <font color="Maroon">a1</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">o</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:15_1_4_3_5_2"/><b>then </b>
<font color="Maroon">c3</font>,<font color="Maroon">a3</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="analmetr.html#T82">ANALMETR:82</a></i>;<br/><a NAME="E5:15_1_4_3_5_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">c3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E6:15_1_4_3_5_2"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E7:15_1_4_3_5_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E8:15_1_4_3_5_2"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E9:15_1_4_3_5_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/><a NAME="E10:15_1_4_3_5_2"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 <b>by </b><i>, , , , <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/><a NAME="E11:15_1_4_3_5_2"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/><b>hence </b><a NAME="E12:15_1_4_3_5_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_4_3_5"/>
contradiction
 <b>by </b><i><a class="ref" href="conmetr.html#D8" target="_self">Def8</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E38:15_1_4_3"/><i><font color="Green">E222</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_6"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font>
 ;<br/>

<a NAME="E2:15_1_4_3_6"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_3_6"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:15_1_4_3_6"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E5:15_1_4_3_6"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E6:15_1_4_3_6"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_4_3_6"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E39:15_1_4_3"/><i><font color="Green">E224</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b3</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_7"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b3</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:15_1_4_3_7"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b3'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_3_7"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E40:15_1_4_3"/><i><font color="Green">E225</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_8"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_3_8"/>
( <font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</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="E2:15_1_4_3_8"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">a1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">b1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">a2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b3'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_3_8"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E41:15_1_4_3"/><i><font color="Green">E226</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_9"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_3_9"/>
( <font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c2'</font> &amp; <font color="Maroon">o'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c2'</font> )
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E2:15_1_4_3_9"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>,<font color="Maroon">c1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>,<font color="Maroon">c2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font>,<font color="Maroon">c2'</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="E3:15_1_4_3_9"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c3</font>,<font color="Maroon">c2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E42:15_1_4_3"/>
<font color="Maroon">a3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E43:15_1_4_3"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">c1</font>
 <b>by </b><i>, , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#D8" target="_self">Def8</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, , <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/><a NAME="E44:15_1_4_3"/><b>then </b><i><font color="Green">E227</font></i>: 
<font color="Maroon">c3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E45:15_1_4_3"/><i><font color="Green">E228</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_10"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:15_1_4_3_10"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_3_10"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E46:15_1_4_3"/><i><font color="Green">E229</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_11"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font>
 ;<br/>

<a NAME="E2:15_1_4_3_11"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_3_11"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">c3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:15_1_4_3_11"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E5:15_1_4_3_11"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E6:15_1_4_3_11"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_4_3_11"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E47:15_1_4_3"/>
<font color="Maroon">a3</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E48:15_1_4_3"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">c3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, <a class="txt" href="conmetr.html#E3:15"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/><a NAME="E49:15_1_4_3"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">c3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E50:15_1_4_3"/><i><font color="Green">E232</font></i>: 
not <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_12"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:15_1_4_3_12"/>
<font color="Maroon">o'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_3_12"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , , <a class="ref" href="conmetr.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_3_12"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_3_12"/>
contradiction
 <b>by </b><i>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E51:15_1_4_3"/>
not <font color="Maroon">c3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_13"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_13"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">c3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:15_1_4_3_13"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_3_13"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_3_13"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_3_13"/>
contradiction
 <b>by </b><i>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E52:15_1_4_3"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">c1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , , , , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, <a class="txt" href="conmetr.html#E2:15"><i><font color="Green">E49</font></i></a>, , , </i>;<br/><a NAME="E53:15_1_4_3"/><i><font color="Green">E236</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_14"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font>
 ;<br/>

<a NAME="E2:15_1_4_3_14"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_3_14"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:15_1_4_3_14"/>
<font color="Maroon">o'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E5:15_1_4_3_14"/><b>then </b>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E6:15_1_4_3_14"/><b>then </b>
<font color="Maroon">M</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_4_3_14"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E54:15_1_4_3"/><i><font color="Green">E238</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_15"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:15_1_4_3_15"/><b>then </b><i><font color="Green">E239</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c1'</font>,<font color="Maroon">c2'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:15_1_4_3_15"/>
<font color="Maroon">c1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_15_1"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2'</font>
 ;<br/>

<a NAME="E2:15_1_4_3_15_1"/>
<font color="Maroon">c3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_15_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_15_1_1"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">c3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:15_1_4_3_15_1_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">c3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_3_15_1_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_3_15_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_3_15_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15_1_4_3_15_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:15_1_4_3_15_1"/><b>then </b>
<font color="Maroon">c1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:15_1_4_3_15_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:15_1_4_3_15_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">a1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:15_1_4_3_15_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E8:15_1_4_3_15_1"/><b>then </b><i><font color="Green">E242</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E9:15_1_4_3_15_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E10:15_1_4_3_15_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , <a class="ref" href="conmetr.html#D7" target="_self">Def7</a>, , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E11:15_1_4_3_15_1"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E12:15_1_4_3_15_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:15_1_4_3_15"/><b>then </b>
<font color="Maroon">b2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E5:15_1_4_3_15"/><b>then </b>
<font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:15_1_4_3_15"/><b>then </b>
<font color="Maroon">A'</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="E7:15_1_4_3_15"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E8:15_1_4_3_15"/>
contradiction
 <b>by </b><i>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E55:15_1_4_3"/>
<font color="Maroon">c1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">a2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_16"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_4_3_16"/>
<font color="Maroon">c3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4_3_16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_4_3_16_1"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">c3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:15_1_4_3_16_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>
<a NAME="E3:15_1_4_3_16_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E4:15_1_4_3_16_1"/><b>then </b>
<font color="Maroon">A</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_4_3_16_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_4_3_16"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">c2</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_4_3_16"/>
<font color="Maroon">c1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c2</font>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E56:15_1_4_3"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</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="conmetr.html#D7" target="_self">Def7</a>, , <a class="txt" href="conmetr.html#E1:15"><i><font color="Green">E48</font></i></a>, <a class="ref" href="conmetr.html#T2" target="_self">Th2</a>, , , <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_4"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:15_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:15"/>
<font color="Maroon">a1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , </i>;<br/>


</div>
