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

<b>let </b><font color="Maroon">POS</font> be   <a href="analmetr.html#NM1">OrtAfSp</a>;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green">E38</font></i>: 
 <a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font> is   <a href="diraf.html#NM2">AffinPlane</a>
 ;<br/>

<a NAME="E2:63"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>;<br/><b>thus </b><a NAME="E1:63_1"/>
( ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> &amp; ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> implies ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> ) ) &amp; ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; not <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i><a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E2:63_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; not <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> )
 ;<br/>

<b>assume </b><a NAME="E2:63_1_1"/><i><font color="Green">E51</font></i>: 
( not <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font>, <font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/>
<a NAME="E4:63_1_1"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/>
<a NAME="E5:63_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">r'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E7:63_1_1"/><i><font color="Green">E61</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="analmetr.html#T7" target="_self">Th7</a>, <a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/>
<a NAME="E9:63_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="analmetr.html#E5:63_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:63_1_1"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q'</font>,<font color="Maroon">p'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">s'</font>,<font color="Maroon">r'</font>,<font color="Maroon">x'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E11:63_1_1"/><b>then </b>
( <font color="Maroon">q'</font>,<font color="Maroon">p'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">s'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">s'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">x'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E12:63_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">q'</font> &amp; <font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font> &amp; <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">p'</font> &amp; <font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E13:63_1_1"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">s</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E14:63_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E15:63_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">s</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 ;<br/><b>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" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E3:63_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">y'</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E4:63_1_1_1"/>
not <font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_1_1"/>
<font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>
 ;<br/>

<a NAME="E2:63_1_1_1_1"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E9:63_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:63_1_1_1_1"/><b>then </b>
<font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:63_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:63_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E6:63_1_1_1"/><i><font color="Green">E70</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p'</font>,<font color="Maroon">y'</font>,<font color="Maroon">z'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E8:63_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">z'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E9:63_1_1_1"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">z'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E10:63_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E11:63_1_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, </i>;<br/><a NAME="E12:63_1_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E7:63_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="analmetr.html#E9:63_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:63_1_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T12" target="_self">Th12</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E14:63_1_1_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">z</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="analmetr.html#E16:63_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E15:63_1_1_1"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">p'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E16:63_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">p'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:63_1_1_1"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">p'</font>,<font color="Maroon">q'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E18:63_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">q'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E9:63_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="analmetr.html#E1:63_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/><a NAME="E19:63_1_1_1"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E1:63_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E20:63_1_1_1"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E9:63_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E21:63_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> )
 ;<br/><b>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" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E3:63_1_1_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">y'</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E4:63_1_1_2"/>
not <font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_2_1"/>
<font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>
 ;<br/>

<a NAME="E2:63_1_1_2_1"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:63_1_1_2_1"/><b>then </b>
<font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:63_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:63_1_1_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E6:63_1_1_2"/><i><font color="Green">E77</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p'</font>,<font color="Maroon">y'</font>,<font color="Maroon">z'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E8:63_1_1_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">p'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">z'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E9:63_1_1_2"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">z'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E10:63_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E11:63_1_1_2"/>
<font color="Maroon">x</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E16:63_1_1"><i><font color="Green">E67</font></i></a>, </i>;<br/><a NAME="E12:63_1_1_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E7:63_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:63_1_1_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T12" target="_self">Th12</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E14:63_1_1_2"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">z</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, , <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E15:63_1_1_2"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">p'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E16:63_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E16:63_1_1_2"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">p'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:63_1_1_2"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">p'</font>,<font color="Maroon">q'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E18:63_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">q'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/><a NAME="E19:63_1_1_2"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E20:63_1_1_2"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E21:63_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_3"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 ;<br/><b>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" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E3:63_1_1_3"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y'</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E4:63_1_1_3"/>
not <font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_3_1"/>
<font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>
 ;<br/>

<a NAME="E2:63_1_1_3_1"/><b>then </b>
<font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:63_1_1_3_1"/><b>then </b>
<font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:63_1_1_3_1"/><b>then </b>
<font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:63_1_1_3_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E6:63_1_1_3"/><i><font color="Green">E163</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q'</font>,<font color="Maroon">y'</font>,<font color="Maroon">z'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E8:63_1_1_3"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">z'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E16:63_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E9:63_1_1_3"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">z'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E10:63_1_1_3"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E11:63_1_1_3"/>
<font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:63_1_1_3"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E7:63_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:63_1_1_3"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T12" target="_self">Th12</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E14:63_1_1_3"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">z</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, , <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E15:63_1_1_3"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E16:63_1_1_3"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">q'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:63_1_1_3"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">q'</font>,<font color="Maroon">p'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E18:63_1_1_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">r'</font>,<font color="Maroon">p'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/><a NAME="E19:63_1_1_3"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">r'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E20:63_1_1_3"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E21:63_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_4"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> )
 ;<br/><b>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" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E3:63_1_1_4"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y'</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E4:63_1_1_4"/>
not <font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_4_1"/>
<font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>
 ;<br/>

<a NAME="E2:63_1_1_4_1"/><b>then </b>
<font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:63_1_1_4_1"/><b>then </b>
<font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:63_1_1_4_1"/><b>then </b>
<font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:63_1_1_4_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E6:63_1_1_4"/><i><font color="Green">E169</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q'</font>,<font color="Maroon">y'</font>,<font color="Maroon">z'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E8:63_1_1_4"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">q'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">z'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E16:63_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E9:63_1_1_4"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">z'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E15:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E10:63_1_1_4"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E11:63_1_1_4"/>
<font color="Maroon">x</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:63_1_1_4"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E7:63_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:63_1_1_4"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T12" target="_self">Th12</a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E14:63_1_1_4"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">z</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, , <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E15:63_1_1_4"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E16:63_1_1_4"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">q'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:63_1_1_4"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">q'</font>,<font color="Maroon">p'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E18:63_1_1_4"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x'</font>,<font color="Maroon">s'</font>,<font color="Maroon">p'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/><a NAME="E19:63_1_1_4"/><b>then </b>
<font color="Maroon">x'</font>,<font color="Maroon">s'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E20:63_1_1_4"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, <a class="txt" href="analmetr.html#E14:63_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E21:63_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:63_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="analmetr.html#T7" target="_self">Th7</a>, <a class="ref" href="analmetr.html#T13" target="_self">Th13</a>, <a class="txt" href="analmetr.html#E9:63_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="analmetr.html#E12:63_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
<font color="Maroon">POS</font> is   <a href="analmetr.html#NM2">OrtAfPl</a>
 <b>by </b><i>, , </i>;<br/>


</div>
