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

<b>let </b><font color="Maroon">OAS</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">OAS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR6" target="_self">is_Dil</a> 
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> holds  <font color="Maroon">p</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E4:84"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>  st not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Olive">y</font>,<font color="Olive">z</font> holds <br/><font color="Olive">y</font>,<font color="Olive">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/>



<b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E30</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:84_1"/>
( <font color="Maroon">p</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="transgeo.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>hence </b><a NAME="E3:84_1"/>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="pasch.html#T20">PASCH:20</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/>



<a NAME="E5:84"/>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font> implies <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E31</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E2:84_2_1"/><b>then </b>
<font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="transgeo.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:84_2_1"/>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E3:84_2_2"/><i><font color="Green">E36</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">y</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="diraf.html#T43">DIRAF:43</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E5:84_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E6:84_2_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">z</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="diraf.html#T31">DIRAF:31</a></i>;<br/><a NAME="E7:84_2_2"/><i><font color="Green">E50</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2_2_1"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:84_2_2_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">q</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">p</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:84_2_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:84_2_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E43"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="transgeo.html#E44"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="pasch.html#T21">PASCH:21</a></i>;<br/><a NAME="E9:84_2_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:84_2_2_2"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">y</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E44"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E3:84_2_2_2"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:84_2_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:84_2_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T85">FUNCT_2:85</a></i>;<br/><a NAME="E11:84_2_2"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2_2_3"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:84_2_2_3"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E3:84_2_2_3"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">p</font>,<font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E4:84_2_2_3"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E5:84_2_2_3"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E6:84_2_2_3"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="txt" href="transgeo.html#E43"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E7:84_2_2_3"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">y</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="transgeo.html#T37" target="_self">Th37</a>, <a class="ref" href="transgeo.html#T39" target="_self">Th39</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E8:84_2_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="transgeo.html#T37" target="_self">Th37</a>, <a class="ref" href="transgeo.html#T39" target="_self">Th39</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:84_2_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E13:84_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="transgeo.html#E43"><i><font color="Green">Lemma62</font></i></a>, </i>;<br/><a NAME="E14:84_2_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="transgeo.html#E44"><i><font color="Green">Lemma63</font></i></a>, </i>;<br/><a NAME="E15:84_2_2"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="txt" href="transgeo.html#E1:84_2"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E16:84_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="pasch.html#T21">PASCH:21</a></i>;<br/><a NAME="E17:84_2_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E18:84_2_2"/>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="transgeo.html#T37" target="_self">Th37</a>, <a class="ref" href="transgeo.html#T39" target="_self">Th39</a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><b>hence </b><a NAME="E19:84_2_2"/>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T38" target="_self">Th38</a>, , <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:84_2"/>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="transgeo.html#D4" target="_self">Def4</a>, <a class="ref" href="transgeo.html#D5" target="_self">Def5</a>, <a class="ref" href="diraf.html#T7">DIRAF:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:84"/>
<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T35" target="_self">Th35</a></i>;<br/>


</div>
