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

<b>let </b><font color="Maroon">AFP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</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">AFP</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:123"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR10" target="_self">is_Col</a> 
 <b>and </b><br/><a NAME="E2:123"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E3:123"/><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">AFP</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E4:123"/><i><font color="Green">E28</font></i>: 
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <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> )
 ;<br/>

<a NAME="E5:123"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_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">AFP</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:123_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E3:123_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>;<br/><a NAME="E4:123_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, <a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E6:123_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E8:123_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E9:123_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">D</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E10:123_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E11:123_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E12:123_1_1"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:123_1_1_1"/>
<font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:123_1_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:123_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="transgeo.html#T62" target="_self">Th62</a>, <a class="txt" href="transgeo.html#E18:123_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:123_1_1"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">D</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:123_1_1_2"/>
<font color="Maroon">D</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:123_1_1_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:123_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="transgeo.html#T62" target="_self">Th62</a>, <a class="txt" href="transgeo.html#E18:123_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E15:123_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, <a class="ref" href="transgeo.html#T61" target="_self">Th61</a>, <a class="txt" href="transgeo.html#E19:123_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E17:123_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, <a class="ref" href="transgeo.html#T61" target="_self">Th61</a>, <a class="txt" href="transgeo.html#E20:123_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E18:123_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:123_1_1_3"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="transgeo.html#T44" target="_self">Th44</a>, <a class="ref" href="transgeo.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><a NAME="E2:123_1_1_3"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, , <a class="ref" href="transgeo.html#T57" target="_self">Th57</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:123_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:123_1_1_4"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="transgeo.html#T44" target="_self">Th44</a>, <a class="ref" href="transgeo.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><a NAME="E2:123_1_1_4"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, , <a class="ref" href="transgeo.html#T57" target="_self">Th57</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:123_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:123_1_1_5"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font></span>)</span> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T44" target="_self">Th44</a>, <a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E2:123_1_1_5"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#T63" target="_self">Th63</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:123_1_1_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, <a class="txt" href="transgeo.html#E2:123_1_1_5"><i><font color="Green">E93</font></i></a></i>;<br/>
<a NAME="E4:123_1_1_5"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:123_1_1_5"><i><font color="Green">E93</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:123_1_1_5"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E2:123_1_1_5"><i><font color="Green">E93</font></i></a>, <a class="ref" href="transgeo.html#T67" target="_self">Th67</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:123_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:123_1_1_6"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font></span>)</span> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T44" target="_self">Th44</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="transgeo.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E2:123_1_1_6"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="transgeo.html#T64" target="_self">Th64</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:123_1_1_6"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, <a class="txt" href="transgeo.html#E21:123_1_1"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E4:123_1_1_6"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E21:123_1_1"><i><font color="Green">E94</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:123_1_1_6"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E21:123_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="transgeo.html#T69" target="_self">Th69</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:123_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:123_1_1_7"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</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">x</font>
 ;<br/>

<a NAME="E2:123_1_1_7"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="transgeo.html#D9" target="_self">Def9</a>, <a class="ref" href="transgeo.html#T67" target="_self">Th67</a>, </i>;<br/>
<a NAME="E3:123_1_1_7"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="transgeo.html#T61" target="_self">Th61</a>, <a class="ref" href="transgeo.html#T69" target="_self">Th69</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E4:123_1_1_7"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E5:123_1_1_7"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T62" target="_self">Th62</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E6:123_1_1_7"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, , <a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="transgeo.html#T61" target="_self">Th61</a>, <a class="ref" href="transgeo.html#T62" target="_self">Th62</a>, <a class="txt" href="transgeo.html#E18:123_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:123_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:123_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:123"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_2"><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">AFP</font>;<br/>

<a NAME="E1:123_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T56" target="_self">Th56</a></i>;<br/>
<b>hence </b><a NAME="E2:123_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:123"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
