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

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

<a NAME="E1:61"/><i><font color="Green">E38</font></i>: 
 <a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font>,the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">POS</font> #)
 
;<br/>
<a NAME="E2:61"/><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="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E3:61"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">s</font> holds <br/><font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">q</font>,<font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </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>assume </b><a NAME="E1:61_1"/><i><font color="Green">E51</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">p</font>,<font color="Maroon">s</font> )
 ;<br/>

<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:61_1_1"/><b>then </b>
<font color="Maroon">q</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#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_1_1"/>
<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">s</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 ;<br/><a NAME="E2:61_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</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>, <a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</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="E4:61_1_2"/><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>
 <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#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E5:61_1_2"/>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E6:61_1_2"/><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">s'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E7:61_1_2"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E8:61_1_2"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:61_1_2"/>
<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">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="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:61_1"/>
<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">s</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#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:61"/><i><font color="Green">E64</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>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> 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="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">x</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">x</font>,<font color="Olive">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </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>;<br/>

<b>assume </b><a NAME="E1:61_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font><b> such that </b><br/><a NAME="E3:61_2"/><i><font color="Green">E66</font></i>: 
( <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">y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<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">c'</font> = <font color="Maroon">c</font>, <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" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/>
<a NAME="E5:61_2"/>
not <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_2_1"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">y'</font>
 ;<br/>

<a NAME="E2:61_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:61_2_1"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">y</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T7" target="_self">Th7</a>, <a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:61_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then 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="E7:61_2"/><i><font color="Green">E67</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">x'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">y'</font>,<font color="Maroon">x'</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">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="E9:61_2"/>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">c'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</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="E10:61_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">x</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E11:61_2"/>
<font color="Maroon">c</font>,<font color="Maroon">y</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="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E12:61_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">c</font>,<font color="Maroon">x</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="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E13:61_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">c</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:61_2"/>
 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="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Olive">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">x</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:61"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> holds <br/> ( ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; <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">c</font> &amp; ( <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">d</font> implies ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">d</font>,<font color="Olive">c</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> ) ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">s</font> implies <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">q</font>,<font color="Olive">s</font> ) )
 
<b>by </b><i>, <a class="txt" href="analmetr.html#E3:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:61"/>
<font color="Maroon">POS</font> is   <a href="analmetr.html#NM1">OrtAfSp</a>
 <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>
