<?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><a NAME="E1:11"/><i><font color="Green">E48</font></i>: 
 <a href="conmetr.html#NR3" target="_self">MH1_holds_in</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">a1</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#D7" target="_self">CONMETR:def 7</a><br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a4</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b4</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



















<b>assume </b><a NAME="E2:11"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">M</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font> )
 ;<br/>

<a NAME="E3:11"/><b>then </b><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>by </b><i><a class="ref" href="analmetr.html#T62">ANALMETR:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b4'</font> = <font color="Maroon">b4</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>, <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">a4'</font> = <font color="Maroon">a4</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="E6:11"/><i><font color="Green">E71</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="E7:11"/>
not <font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">M</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="E3:11_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="E8:11"/><b>then </b>
 ex <font color="Olive">o'</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> st <br/>( <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:11"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">o</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> )
 ;<br/>
<b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o</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/>
<div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font>
 ;<br/><a NAME="E2:11_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a1'</font>,<font color="Maroon">a4'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_2_1"/>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E2:11_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<b>hence </b><a NAME="E3:11_2_1"/>
<font color="Maroon">a1'</font>,<font color="Maroon">a4'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:11_2"/><i><font color="Green">E76</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a4'</font>,<font color="Maroon">a2'</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E4:11_2"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a4'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a4'</font>
 ;<br/>

<a NAME="E2:11_2_2"/><b>then </b>
 ex <font color="Olive">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> st <br/>( <font color="Olive">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="Olive">A'</font> &amp; <font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Olive">A'</font> &amp; <font color="Maroon">a4'</font> <a href="hidden.html#R2">in</a> <font color="Olive">A'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:11_2_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:11_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b4</font>
 ;<br/><div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:11_3_1"/><i><font color="Green">E80</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">a1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">a1'</font>
 ;<br/>

<a NAME="E2:11_3_1_1"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">a2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:11_3_1"/><i><font color="Green">E81</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>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E4:11_3_1"/>
<font color="Maroon">a2'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:11_3_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">a2'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E2:11_3_1_2"/>
<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:11_3_1_2"/><b>then </b>
<font color="Maroon">a2'</font>,<font color="Maroon">a3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E4:11_3_1_2"/>
<font color="Maroon">a2'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">a3'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:11_3_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_3_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:11_3_2"/>
 ex <font color="Olive">d4'</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> st <br/>( <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d4'</font> &amp; <font color="Olive">d4'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>then consider </b><font color="Maroon">d4</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:11_3_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">d4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">d4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> )
 ;<br/><b>reconsider </b><font color="Maroon">d4'</font> = <font color="Maroon">d4</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>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="E7:11_3_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> &amp; <font color="Maroon">e1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e1'</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/><b>consider </b><font color="Maroon">e2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:11_3_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d4</font>,<font color="Maroon">e2</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d4</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon">e2'</font> = <font color="Maroon">e2</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="E12:11_3_2"/>
not <font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d4'</font>,<font color="Maroon">e2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d4'</font>,<font color="Maroon">e2'</font>
 ;<br/>

<a NAME="E2:11_3_2_1"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d4</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:11_3_2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a1</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>
<a NAME="E4:11_3_2_1"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E5:11_3_2_1"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E6:11_3_2_1"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">e1</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/>
<a NAME="E7:11_3_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a1</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>
<a NAME="E8:11_3_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">a1</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="11_3_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_1_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:11_3_2_1_1"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:11_3_2_1_1"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:11_3_2_1_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">b1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:11_3_2_1_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:11_3_2_1_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">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:11_3_2_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">b3'</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="E8:11_3_2_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="E9:11_3_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">a4'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_1_2"/>
<font color="Maroon">a4'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font>
 ;<br/>

<a NAME="E2:11_3_2_1_2"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:11_3_2_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:11_3_2_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:11_3_2_1_2"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:11_3_2_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:11_3_2_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E8:11_3_2_1_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>
<a NAME="E10:11_3_2_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/>
<a NAME="E11:11_3_2_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T85">ANALMETR:85</a></i>;<br/>
<a NAME="E12:11_3_2_1"/><b>then </b>
<font color="Maroon">a4</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E13:11_3_2_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a4</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="E14:11_3_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a4'</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/>
<b>hence </b><a NAME="E15:11_3_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">d1'</font> being   <a href="struct_0.html#NM1">Element</a> of <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="E14:11_3_2"/><i><font color="Green">E91</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">d1'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d4'</font>,<font color="Maroon">e2'</font>,<font color="Maroon">d1'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1'</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="E16:11_3_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d1</font>,<font color="Maroon">d4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:11_3_2_2"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d4</font>,<font color="Maroon">e2</font>,<font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:11_3_2_2"/><b>then </b>
<font color="Maroon">d4</font>,<font color="Maroon">e2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d4</font>,<font color="Maroon">d1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:11_3_2_2"/><b>then </b>
<font color="Maroon">d1</font>,<font color="Maroon">d4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d4</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E4:11_3_2_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d1</font>,<font color="Maroon">d4</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:11_3_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d1</font>,<font color="Maroon">d4</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</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="E19:11_3_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> &amp; <font color="Maroon">e1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e1'</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/><b>consider </b><font color="Maroon">e2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E22:11_3_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d1</font>,<font color="Maroon">e2</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon">e2'</font> = <font color="Maroon">e2</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="E24:11_3_2"/>
not <font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2'</font>,<font color="Maroon">d1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_3"/>
<font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2'</font>,<font color="Maroon">d1'</font>
 ;<br/>

<a NAME="E2:11_3_2_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2</font>,<font color="Maroon">d1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:11_3_2_3"/>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">e2</font>,<font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/>
<a NAME="E4:11_3_2_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>
<a NAME="E5:11_3_2_3"/>
<font color="Maroon">N'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E6:11_3_2_3"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">a4'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E7:11_3_2_3"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E8:11_3_2_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>
<a NAME="E9:11_3_2_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_3_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a4</font>
 ;<br/>

<a NAME="E2:11_3_2_3_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:11_3_2_3_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:11_3_2_3_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:11_3_2_3_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:11_3_2_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:11_3_2_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E8:11_3_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="E10:11_3_2_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">a1'</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="11_3_2_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_3_2"/>
<font color="Maroon">a1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3'</font>
 ;<br/>

<a NAME="E2:11_3_2_3_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:11_3_2_3_2"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:11_3_2_3_2"/><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">b1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:11_3_2_3_2"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:11_3_2_3_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">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:11_3_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">b3'</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="E8:11_3_2_3_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:11_3_2_3"/>
<font color="Maroon">a3</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/>
<a NAME="E12:11_3_2_3"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T85">ANALMETR:85</a></i>;<br/>
<a NAME="E13:11_3_2_3"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E14:11_3_2_3"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E15:11_3_2_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E16:11_3_2_3"/>
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">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="E26:11_3_2"/><i><font color="Green">E100</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">d2'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">e2'</font>,<font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><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="E28:11_3_2"/>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d2</font>,<font color="Maroon">d1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:11_3_2_4"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font>,<font color="Maroon">e2'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E2:11_3_2_4"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:11_3_2_4"/><b>then </b>
<font color="Maroon">d1</font>,<font color="Maroon">d2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d1</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E4:11_3_2_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">d2</font>,<font color="Maroon">d1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2</font>,<font color="Maroon">d1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:11_3_2_4"/>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">e2</font>,<font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/>
<b>hence </b><a NAME="E6:11_3_2_4"/>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d2</font>,<font color="Maroon">d1</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:11_3_2"/><b>then </b><i><font color="Green">E112</font></i>: 
( <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d2</font>,<font color="Maroon">d1</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</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="E31:11_3_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1'</font> &amp; <font color="Maroon">e1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e1'</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/><b>consider </b><font color="Maroon">e2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E34:11_3_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d2</font>,<font color="Maroon">e2</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon">e2'</font> = <font color="Maroon">e2</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="E36:11_3_2"/>
not <font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2'</font>,<font color="Maroon">d2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_5"/>
<font color="Maroon">o'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2'</font>,<font color="Maroon">d2'</font>
 ;<br/>

<a NAME="E2:11_3_2_5"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e2</font>,<font color="Maroon">d2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:11_3_2_5"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d2</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E4:11_3_2_5"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">e1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>
<a NAME="E5:11_3_2_5"/>
<font color="Maroon">o</font>,<font color="Maroon">e1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, , , <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/>
<a NAME="E6:11_3_2_5"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T85">ANALMETR:85</a></i>;<br/>
<a NAME="E7:11_3_2_5"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E8:11_3_2_5"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a4</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="E9:11_3_2_5"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">a4'</font>,<font color="Maroon">a3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E10:11_3_2_5"/>
<font color="Maroon">a2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a4'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2_5_1"/>
<font color="Maroon">a2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a4'</font>
 ;<br/>

<a NAME="E2:11_3_2_5_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:11_3_2_5_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:11_3_2_5_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:11_3_2_5_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:11_3_2_5_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:11_3_2_5_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b2'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E8:11_3_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>
<b>hence </b><a NAME="E11:11_3_2_5"/>
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">d3'</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="E38:11_3_2"/><i><font color="Green">E118</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">d3'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">e2'</font>,<font color="Maroon">d2'</font>,<font color="Maroon">d3'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><font color="Maroon">d3</font> = <font color="Maroon">d3'</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="E40:11_3_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d3</font>,<font color="Maroon">d2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:11_3_2_6"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d2'</font>,<font color="Maroon">d3'</font>,<font color="Maroon">e2'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E2:11_3_2_6"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d2</font>,<font color="Maroon">d3</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:11_3_2_6"/><b>then </b>
<font color="Maroon">d2</font>,<font color="Maroon">d3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d2</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E4:11_3_2_6"/><b>then </b>
<font color="Maroon">d3</font>,<font color="Maroon">d2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d2</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E5:11_3_2_6"/>
<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d3</font>,<font color="Maroon">d2</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><a NAME="E41:11_3_2"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">d3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d3</font>,<font color="Maroon">d2</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E42:11_3_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d3</font>,<font color="Maroon">d4</font>
 <b>by </b><i>, , , , , , </i>;<br/><a NAME="E43:11_3_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d1</font>,<font color="Maroon">d4</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/><a NAME="E44:11_3_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d2</font>,<font color="Maroon">d1</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/><a NAME="E45:11_3_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d3</font>,<font color="Maroon">d2</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T84">ANALMETR:84</a></i>;<br/><a NAME="E46:11_3_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">d3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d4</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E47:11_3_2"/>
<font color="Maroon">b3</font>,<font color="Maroon">b4</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">d3</font>,<font color="Maroon">d4</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E48:11_3_2"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="analmetr.html#T85">ANALMETR:85</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:11_3"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:11"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i/>;<br/>


</div>
