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

<b>let </b><font color="Maroon">S</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">AS</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="analoaf.html#L1">AffinStruct</a> ;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">AS</font> <a href="hidden.html#R1">=</a>  <a href="diraf.html#K2" target="_self">Lambda</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:49"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">AS</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">S</font>,<span class="p1">(<span class="default"><a href="diraf.html#K1" target="_self">lambda</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">S</font></span>)</span> #)
 
;<br/>
<b>hence </b><a NAME="E3:49"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font>
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>

<b>thus </b><a NAME="E4:49"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font>, <font color="Olive">t</font>, <font color="Olive">u</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> holds <br/> ( <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">y</font>,<font color="Olive">x</font> &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">z</font>,<font color="Olive">z</font> &amp; ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">z</font>,<font color="Olive">t</font> &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">u</font>,<font color="Olive">w</font> implies <font color="Olive">z</font>,<font color="Olive">t</font> <a href="analoaf.html#R2">//</a> <font color="Olive">u</font>,<font color="Olive">w</font> ) &amp; ( <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">z</font> implies <font color="Olive">y</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">y</font>,<font color="Olive">z</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </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">AS</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font>, <font color="Maroon">t'</font> = <font color="Maroon">t</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E2:49_1"/>
( <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">z'</font>,<font color="Maroon">z'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:49_1"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/>

<a NAME="E4:49_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">z'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">u'</font>,<font color="Maroon">w'</font> implies <font color="Maroon">z'</font>,<font color="Maroon">t'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">u'</font>,<font color="Maroon">w'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:49_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> implies <font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> )
 <b>by </b><i>, </i>;<br/>

<a NAME="E6:49_1"/>
( <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> implies <font color="Maroon">y'</font>,<font color="Maroon">x'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:49_1"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> implies <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:49"/>
not for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E2:49_2"/><i><font color="Green">E27</font></i>: 
not <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font>, <font color="Maroon">y</font> = <font color="Maroon">y'</font>, <font color="Maroon">z</font> = <font color="Maroon">z'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i/>;<br/>
<a NAME="E4:49_2"/>
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:49_2"/>
not for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">z</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:49"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">x</font>,<font color="Olive">z</font> <a href="analoaf.html#R2">//</a> <font color="Olive">y</font>,<font color="Olive">t</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </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">AS</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:49_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">y'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t'</font> )
 <b>by </b><i><a class="txt" href="diraf.html#E1:49"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i/>;<br/>
<a NAME="E5:49_3"/>
( <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:49_3"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Olive">t</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">t</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:49"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">z</font>,<font color="Olive">t</font> &amp; <font color="Olive">x</font>,<font color="Olive">z</font> <a href="analoaf.html#R2">//</a> <font color="Olive">y</font>,<font color="Olive">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </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">AS</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:49_4"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">z'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">y'</font>,<font color="Maroon">t'</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i/>;<br/>
<a NAME="E5:49_4"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">t</font> )
 
<b>by </b><i>, , <a class="txt" href="diraf.html#E3:49_4"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:49_4"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Olive">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Olive">t</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:49"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">z</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">t</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/> ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">y</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">u</font> &amp; <font color="Olive">y</font>,<font color="Olive">z</font> <a href="analoaf.html#R2">//</a> <font color="Olive">t</font>,<font color="Olive">u</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5"><b>proof </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">AS</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>

<b>assume </b><a NAME="E1:49_5"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">z</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font>, <font color="Maroon">t'</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:49_5"/>
( <font color="Maroon">z'</font>,<font color="Maroon">x'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">x'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z'</font> )
 
<b>by </b><i><a class="txt" href="diraf.html#E5:49_5"><i><font color="Green">E32</font></i></a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:49_5"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y'</font>,<font color="Maroon">x'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">x'</font>,<font color="Maroon">u'</font> &amp; <font color="Maroon">y'</font>,<font color="Maroon">z'</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">t'</font>,<font color="Maroon">u'</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i/>;<br/>
<a NAME="E7:49_5"/>
( <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">t</font>,<font color="Maroon">u</font> )
 
<b>by </b><i><a class="txt" href="diraf.html#E5:49_5"><i><font color="Green">E32</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E8:49_5"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Olive">u</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">t</font>,<font color="Olive">u</font> )
 ;<br/>


</div><b>end;</b></div>


</div>
