<?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="E1:14"/><i><font color="Green">E48</font></i>: 
 <a href="conmetr.html#NR7" target="_self">OSCH_holds_in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E49</font></i>: 
 <a href="conmetr.html#NR5" target="_self">TDES_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#D6" target="_self">CONMETR:def 6</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><b>that </b><br/><a NAME="E3:14"/><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="E4:14"/><i><font color="Green">E71</font></i>: 
( <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/>

<b>reconsider </b><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>, <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">b4'</font> = <font color="Maroon">b4</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="E7:14"/><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="E8:14"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> implies ( <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/>

<b>assume </b><a NAME="E2:14_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> or <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> or <font color="Maroon">a4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b4</font> )
 ;<br/>

<div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:14_1_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E3:14_1_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E4:14_1_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">b2'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E5:14_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">a1'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E6:14_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>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b4</font>
 ;<br/><a NAME="E2:14_1_2"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a4'</font>,<font color="Maroon">b4'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E4:14_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a4'</font>,<font color="Maroon">b4'</font>,<font color="Maroon">a1'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E5:14_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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:14_1_3"/>
<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E3:14_1_3"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E4:14_1_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E5:14_1_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">a2'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E6:14_1_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>hence </b><a NAME="E6:14_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:14_2"/><b>then </b>
<font color="Maroon">a3'</font>,<font color="Maroon">a4'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">b4'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/><b>hence </b><a NAME="E3:14_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="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:14"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> implies ( <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">a4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b4</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 ;<br/>

<b>assume </b><a NAME="E2:14_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> or <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> or <font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font> )
 ;<br/>

<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_3_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:14_3_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E3:14_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E4:14_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a2'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E5:14_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_3_2"/>
<font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b4</font>
 ;<br/><a NAME="E2:14_3_2"/><b>then </b>
<font color="Maroon">a4</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E3:14_3_2"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a4</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E4:14_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a4'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E5:14_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a4'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E6:14_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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_3_3"/>
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:14_3_3"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E3:14_3_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E4:14_3_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</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="E5:14_3_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>hence </b><a NAME="E6:14_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 ;<br/><div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a4</font>
 ;<br/><a NAME="E2:14_4_1"/><i><font color="Green">E90</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">b2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">b2'</font>
 ;<br/>

<a NAME="E2:14_4_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</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="E3:14_4_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="E3:14_4_1"/><i><font color="Green">E91</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b4'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_1_2"/>
<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="E2:14_4_1_2"/><b>then </b>
<font color="Maroon">a2'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">b4'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b4'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:14_4_1"/>
<font color="Maroon">b1'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">b4'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_1_3"/>
<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="E2:14_4_1_3"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</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#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E3:14_4_1_3"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E4:14_4_1_3"/>
<font color="Maroon">b1'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">b4'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:14_4_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><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/><a NAME="E2:14_4_2"/><i><font color="Green">E94</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font>
 ;<br/>

<a NAME="E2:14_4_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</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="E3:14_4_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="E3:14_4_2"/><i><font color="Green">E95</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">b3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_2_2"/>
<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="E2:14_4_2_2"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">b3'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:14_4_2"/>
<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_2_3"/>
<font color="Maroon">a2</font>,<font color="Maroon">a1</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#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E2:14_4_2_3"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_2_3"/>
<font color="Maroon">b2'</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:14_4_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="14_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a4</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> )
 ;<br/><div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">M</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</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="E3:14_4_3_1"/><i><font color="Green">E99</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </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/><b>consider </b><font color="Maroon">e1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:14_4_3_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">e1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</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="E8:14_4_3_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">e2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon">e1'</font> = <font color="Maroon">e1</font>, <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="E10:14_4_3_1"/>
not <font color="Maroon">b1'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">e2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_1"/>
<font color="Maroon">b1'</font>,<font color="Maroon">e1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">e2'</font>
 ;<br/>

<a NAME="E2:14_4_3_1_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:14_4_3_1_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">e2</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:14_4_3_1_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E5:14_4_3_1_1"/><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">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E6:14_4_3_1_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</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="E7:14_4_3_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E8:14_4_3_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><b>then consider </b><font color="Maroon">y'</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="E12:14_4_3_1"/><i><font color="Green">E112</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">e1'</font>,<font color="Maroon">y'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">e2'</font>,<font color="Maroon">y'</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">y</font> = <font color="Maroon">y'</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="E14:14_4_3_1"/><i><font color="Green">E113</font></i>: 
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_2"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">e1</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:14_4_3_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:14_4_3_1_2"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<b>hence </b><a NAME="E4:14_4_3_1_2"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:14_4_3_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_3"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">e2</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:14_4_3_1_3"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">e2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_3_1_3"/>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:14_4_3_1"/><i><font color="Green">E115</font></i>: 
(  <a href="conmetr.html#NR5" target="_self">TDES_holds_in</a> <font color="Maroon">X</font> implies  <a href="conmetr.html#NR8" target="_self">des_holds_in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_4"/>
 <a href="conmetr.html#NR5" target="_self">TDES_holds_in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:14_4_3_1_4"/><b>then </b>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:14_4_3_1_4"/><b>then </b>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR11" target="_self">satisfies_des</a> 
 
<b>by </b><i><a class="ref" href="aff_2.html#T28">AFF_2:28</a></i>;<br/>
<b>hence </b><a NAME="E4:14_4_3_1_4"/>
 <a href="conmetr.html#NR8" target="_self">des_holds_in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E17:14_4_3_1"/><i><font color="Green">E116</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_5"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:14_4_3_1_5"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:14_4_3_1_5"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">x'</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="E4:14_4_3_1_5"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E5:14_4_3_1_5"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">x'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E6:14_4_3_1_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><a NAME="E18:14_4_3_1"/><i><font color="Green">E119</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_6"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a4</font>
 ;<br/>

<a NAME="E2:14_4_3_1_6"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a4'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_3_1_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="E19:14_4_3_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_7"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E2:14_4_3_1_7"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4'</font>,<font color="Maroon">b4'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_3_1_7"/>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:14_4_3_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_8"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E2:14_4_3_1_8"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</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="E3:14_4_3_1_8"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:14_4_3_1_8"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<b>hence </b><a NAME="E5:14_4_3_1_8"/>
<font color="Maroon">a1</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:14_4_3_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , , , , </i>;<br/><a NAME="E22:14_4_3_1"/><i><font color="Green">E126</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_9"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:14_4_3_1_9"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:14_4_3_1_9"/>
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_9_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:14_4_3_1_9_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:14_4_3_1_9_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:14_4_3_1_9_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E5:14_4_3_1_9_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="E4:14_4_3_1_9"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</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="E5:14_4_3_1_9"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E6:14_4_3_1_9"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">x'</font>,<font color="Maroon">a1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E7:14_4_3_1_9"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:14_4_3_1"/><i><font color="Green">E129</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_10"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:14_4_3_1_10"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</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/>
<a NAME="E3:14_4_3_1_10"/>
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_10_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:14_4_3_1_10_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:14_4_3_1_10_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:14_4_3_1_10_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E5:14_4_3_1_10_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="E4:14_4_3_1_10"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:14_4_3_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_11"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_11"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E2:14_4_3_1_11"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E3:14_4_3_1_11"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E4:14_4_3_1_11"/>
<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:14_4_3_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">a2</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="14_4_3_1_12"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_12"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E2:14_4_3_1_12"/><b>then </b>
<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E3:14_4_3_1_12"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E4:14_4_3_1_12"/>
<font color="Maroon">a2</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#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:14_4_3_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_13"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_13"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:14_4_3_1_13"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:14_4_3_1_13"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</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="E4:14_4_3_1_13"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</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="E5:14_4_3_1_13"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E6:14_4_3_1_13"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E7:14_4_3_1_13"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_13_1"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:14_4_3_1_13_1"/><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">x</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:14_4_3_1_13_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E4:14_4_3_1_13_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E5:14_4_3_1_13_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">x'</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="E6:14_4_3_1_13_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:14_4_3_1_13_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">x'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E8:14_4_3_1_13_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="conmetr.html#E1:14"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:14_4_3_1_13"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b1'</font>,<font color="Maroon">b2'</font>,<font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E9:14_4_3_1_13"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">b2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E10:14_4_3_1_13"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E11:14_4_3_1_13"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</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="E12:14_4_3_1_13"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E13:14_4_3_1_13"/>
<font color="Maroon">a2</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:14_4_3_1"/>
<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E28:14_4_3_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</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#D4" target="_self">Def4</a>, <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>, </i>;<br/><a NAME="E29:14_4_3_1"/><i><font color="Green">E138</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_14"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:14_4_3_1_14"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</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="E3:14_4_3_1_14"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:14_4_3_1_14_1"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:14_4_3_1_14_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">a1</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="E4:14_4_3_1_14_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E5:14_4_3_1_14_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E6:14_4_3_1_14_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">x'</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="E7:14_4_3_1_14_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E8:14_4_3_1_14_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E9:14_4_3_1_14_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="conmetr.html#E3:14"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:14_4_3_1_14"/>
<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="E5:14_4_3_1_14"/><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">b1'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E6:14_4_3_1_14"/><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">b1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E7:14_4_3_1_14"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">y</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="E8:14_4_3_1_14"/><b>then </b>
<font color="Maroon">x</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="txt" href="conmetr.html#E1:14"><i><font color="Green">E48</font></i></a>, <a class="txt" href="conmetr.html#E2:14"><i><font color="Green">E49</font></i></a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E9:14_4_3_1_14"/><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">x</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E10:14_4_3_1_14"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E11:14_4_3_1_14"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E12:14_4_3_1_14"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">x'</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="E13:14_4_3_1_14"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E14:14_4_3_1_14"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">x'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E15:14_4_3_1_14"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="conmetr.html#E3:14"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:14_4_3_1"/><i><font color="Green">E143</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_15"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a4</font>
 ;<br/>

<a NAME="E2:14_4_3_1_15"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">a4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:14_4_3_1_15"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_1_15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:14_4_3_1_15_1"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:14_4_3_1_15_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">a1</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="E4:14_4_3_1_15_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E5:14_4_3_1_15_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E6:14_4_3_1_15_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">x'</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="E7:14_4_3_1_15_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E8:14_4_3_1_15_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E9:14_4_3_1_15_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="E4:14_4_3_1_15"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E5:14_4_3_1_15"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b1'</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="E6:14_4_3_1_15"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b1</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="E7:14_4_3_1_15"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E8:14_4_3_1_15"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:14"><i><font color="Green">E49</font></i></a>, <a class="txt" href="conmetr.html#E3:14"><i><font color="Green">E50</font></i></a>, <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>
<a NAME="E9:14_4_3_1_15"/><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">x</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>
<a NAME="E10:14_4_3_1_15"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a4</font>,<font color="Maroon">a2</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E11:14_4_3_1_15"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a4'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E12:14_4_3_1_15"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">x'</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="E13:14_4_3_1_15"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font>,<font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E14:14_4_3_1_15"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a2'</font>,<font color="Maroon">x'</font>,<font color="Maroon">a1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E15:14_4_3_1_15"/>
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="E31:14_4_3_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</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="14_4_3_1_16"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_16"/>
<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="E2:14_4_3_1_16"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b1'</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="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E3:14_4_3_1_16"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b1</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/>
<b>hence </b><a NAME="E4:14_4_3_1_16"/>
<font color="Maroon">x</font>,<font color="Maroon">y</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/>


</div><b>end;</b></div><a NAME="E32:14_4_3_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_1_17"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_1_17"/>
<font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E2:14_4_3_1_17"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4'</font>,<font color="Maroon">b4'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E3:14_4_3_1_17"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E4:14_4_3_1_17"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a4</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , <a class="ref" href="analmetr.html#T82">ANALMETR:82</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E33:14_4_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="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, , <a class="txt" href="conmetr.html#E1:14"><i><font color="Green">E48</font></i></a>, <a class="txt" href="conmetr.html#E2:14"><i><font color="Green">E49</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2"/>
not <font color="Maroon">M</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E2:14_4_3_2"/><b>then </b>
not <font color="Maroon">M'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/><b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E4:14_4_3_2"/><i><font color="Green">E149</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> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o'</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">K</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:14_4_3_2"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">N</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">K</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K</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="E9:14_4_3_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">K</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/><a NAME="E10:14_4_3_2"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">K'</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font>
 ;<br/><div><i><font color="Green">E157</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be   <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>;<br/><b>consider </b><font color="Maroon">D'</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="E2:14_4_3_2_1_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> &amp; <font color="Maroon">N'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D'</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</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="E4:14_4_3_2_1_1"/>
<font color="Maroon">N</font> <a href="analmetr.html#R9">//</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/><a NAME="E5:14_4_3_2_1_1"/><b>then </b>
<font color="Maroon">K</font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:14_4_3_2_1_1"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T88">ANALMETR:88</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</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>take </b><font color="Maroon">y'</font> = <font color="Maroon">y'</font>;<br/><b>thus </b><a NAME="E9:14_4_3_2_1_1"/>
( <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/></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="E4:14_4_3_2_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">d1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">a1'</font>,<font color="Maroon">d1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font> )
 ;<br/><b>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="E6:14_4_3_2_1"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">d3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">a3'</font>,<font color="Maroon">d3'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">e3'</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="E8:14_4_3_2_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">e3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">b3'</font>,<font color="Maroon">e3'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font> )
 <b>by </b><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="E10:14_4_3_2_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">e1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">b1'</font>,<font color="Maroon">e1'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1'</font>, <font color="Maroon">d3</font> = <font color="Maroon">d3'</font>, <font color="Maroon">e1</font> = <font color="Maroon">e1'</font>, <font color="Maroon">e3</font> = <font color="Maroon">e3'</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="E12:14_4_3_2_1"/><i><font color="Green">E168</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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_2_1_2"/><i><font color="Green">E169</font></i>: 
(  not <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font> or  not <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e3</font> or  not <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d1</font> or  not <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d3</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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">y'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:14_4_3_2_1_2_1"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font>
 <b>and </b><br/><a NAME="E2:14_4_3_2_1_2_1"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> )
 <b>and </b><br/><a NAME="E3:14_4_3_2_1_2_1"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 ;<br/><b>assume </b><a NAME="E4:14_4_3_2_1_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E5:14_4_3_2_1_2_1"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">x'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N'</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_1.html#T48">AFF_1:48</a></i>;<br/><b>hence </b><a NAME="E6:14_4_3_2_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_4_3_2_1_2"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/>


</div><b>end;</b></div><a NAME="E13:14_4_3_2_1"/><i><font color="Green">E173</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a4</font>,<font color="Maroon">b4</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font> )
 <b>by </b><i>, , , </i>;<br/><a NAME="E14:14_4_3_2_1"/><i><font color="Green">E174</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">d1</font>,<font color="Maroon">e1</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">d3</font>,<font color="Maroon">e3</font> )
 <b>by </b><i>, , , , , , </i>;<br/><a NAME="E15:14_4_3_2_1"/><i><font color="Green">E175</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">d1</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a2</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a4</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">d3</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">a2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_3"><b>proof </b></a><div class="add">

<div><i><font color="Green">E176</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2_1_3_1"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">d1</font>
 ;<br/><a NAME="E2:14_4_3_2_1_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">d1'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E3:14_4_3_2_1_3_1"/><b>then </b>
<font color="Maroon">d1</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/><b>hence </b><a NAME="E4:14_4_3_2_1_3_1"/>
contradiction
 <b>by </b><i>, , , , , , , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E177</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2_1_3_2"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a2</font>
 ;<br/><a NAME="E2:14_4_3_2_1_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</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="E3:14_4_3_2_1_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>
<div><i><font color="Green">E178</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2_1_3_3"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a4</font>
 ;<br/><a NAME="E2:14_4_3_2_1_3_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a4'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><b>hence </b><a NAME="E3:14_4_3_2_1_3_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>
<div><i><font color="Green">E179</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_3_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2_1_3_4"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">d3</font>
 ;<br/><a NAME="E2:14_4_3_2_1_3_4"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">d3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E3:14_4_3_2_1_3_4"/><b>then </b>
<font color="Maroon">d3</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/><b>hence </b><a NAME="E4:14_4_3_2_1_3_4"/>
contradiction
 <b>by </b><i>, , , , , , , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_3_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_4_3_2_1_3_5"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">a2</font>
 ;<br/><a NAME="E2:14_4_3_2_1_3_5"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">b3'</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="E3:14_4_3_2_1_3_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>hence </b><a NAME="E6:14_4_3_2_1_3"/>
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">d1</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a2</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a4</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">d3</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">a2</font> )
 <b>by </b><i><a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <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></i>;<br/>


</div><b>end;</b></div><a NAME="E16:14_4_3_2_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">d1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e1</font>,<font color="Maroon">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_2_1_4"/>
<font color="Maroon">a1'</font>,<font color="Maroon">d1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">e1'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E2:14_4_3_2_1_4"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">d1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">e1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_3_2_1_4"/>
<font color="Maroon">d1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e1</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:14_4_3_2_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">d1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_2_1_5"/>
<font color="Maroon">a1</font>,<font color="Maroon">d1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a2</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:14_4_3_2_1_5"/>
<font color="Maroon">d1</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="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:14_4_3_2_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</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="E19:14_4_3_2_1"/><b>then </b>
<font color="Maroon">d1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e1</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, , , , , <a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="conmetr.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E20:14_4_3_2_1"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">d1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E21:14_4_3_2_1"/>
<font color="Maroon">d1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_2_1_6"/>
<font color="Maroon">a1</font>,<font color="Maroon">d1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a4</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:14_4_3_2_1_6"/>
<font color="Maroon">d1</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a4</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:14_4_3_2_1"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">d1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e1</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , , <a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E23:14_4_3_2_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">d3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e3</font>,<font color="Maroon">b3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_2_1_7"/>
<font color="Maroon">a3'</font>,<font color="Maroon">d3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3'</font>,<font color="Maroon">e3'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E2:14_4_3_2_1_7"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">d3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">e3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E3:14_4_3_2_1_7"/>
<font color="Maroon">d3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e3</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><a NAME="E24:14_4_3_2_1"/>
<font color="Maroon">d3</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_2_1_8"/>
<font color="Maroon">a3</font>,<font color="Maroon">d3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a2</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:14_4_3_2_1_8"/>
<font color="Maroon">d3</font>,<font color="Maroon">a3</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="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:14_4_3_2_1"/><b>then </b><i><font color="Green">E185</font></i>: 
<font color="Maroon">d3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e3</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, , , , , <a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="conmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E26:14_4_3_2_1"/><i><font color="Green">E186</font></i>: 
( not <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">d3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">e3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/><a NAME="E27:14_4_3_2_1"/>
( not <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_9"><b>proof </b></a><div class="add">

<div><i><font color="Green">E187</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:14_4_3_2_1_9_1"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">x</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">x</font> )
 ;<br/><b>assume </b><a NAME="E2:14_4_3_2_1_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E3:14_4_3_2_1_9_1"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">x</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="conmetr.html#E3:14"><i><font color="Green">E50</font></i></a>, <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/><b>hence </b><a NAME="E4:14_4_3_2_1_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E3:14"><i><font color="Green">E50</font></i></a>, <a class="ref" href="analmetr.html#T51">ANALMETR:51</a></i>;<br/></div><b>end;</b></div>
<b>assume </b><a NAME="E2:14_4_3_2_1_9"/>
( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> or <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> or <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> or <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/>

<b>hence </b><a NAME="E3:14_4_3_2_1_9"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="conmetr.html#E2:14"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:14_4_3_2_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">d3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e3</font>,<font color="Maroon">b4</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="txt" href="conmetr.html#E1:14"><i><font color="Green">E48</font></i></a>, </i>;<br/><a NAME="E29:14_4_3_2_1"/><i><font color="Green">E190</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d3</font>,<font color="Maroon">e3</font>,<font color="Maroon">a3</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d3</font>,<font color="Maroon">e3</font>,<font color="Maroon">a4</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_10"><b>proof </b></a><div class="add">

<a NAME="E1:14_4_3_2_1_10"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">d3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4_3_2_1_10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_4_3_2_1_10_1"/>
not <font color="Maroon">d3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e3</font>
 ;<br/>

<a NAME="E2:14_4_3_2_1_10_1"/><b>then </b>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">e3</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font>
 
<b>by </b><i><a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:14_4_3_2_1_10_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">e3'</font>,<font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:14_4_3_2_1_10_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font>,<font color="Maroon">e3'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:14_4_3_2_1_10_1"/><b>then </b>
<font color="Maroon">e3'</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/>
<b>hence </b><a NAME="E6:14_4_3_2_1_10_1"/>
contradiction
 <b>by </b><i>, , , , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E2:14_4_3_2_1_10"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d3</font>,<font color="Maroon">e3</font>,<font color="Maroon">a3</font> or  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">d3</font>,<font color="Maroon">e3</font>,<font color="Maroon">a4</font> )
 ;<br/>

<a NAME="E3:14_4_3_2_1_10"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d3'</font>,<font color="Maroon">e3'</font>,<font color="Maroon">a3'</font> or  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d3'</font>,<font color="Maroon">e3'</font>,<font color="Maroon">a4'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:14_4_3_2_1_10"/><b>then </b>
( <font color="Maroon">a3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> or <font color="Maroon">a4'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:14_4_3_2_1_10"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:14_4_3_2_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">d3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">e3</font>
 <b>by </b><i><a class="ref" href="conmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E31:14_4_3_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">d3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a4</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E32:14_4_3_2_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , , <a class="ref" href="conmetr.html#D1" target="_self">Def1</a>, <a class="txt" href="conmetr.html#E2:14"><i><font color="Green">E49</font></i></a>, <a class="txt" href="conmetr.html#E3:14"><i><font color="Green">E50</font></i></a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:14_4_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>, , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:14_4_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="E5:14_4"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:14"/>
<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>
