<?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">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><a NAME="E1:122"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>  st <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/>

<a NAME="E2:122"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/><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="Olive">a</font>,<font color="Olive">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="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_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">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>



<b>assume </b><a NAME="E1:122_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">x</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>;<br/>
<a NAME="E2:122_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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">b</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, <a class="ref" href="funct_2.html#T85">FUNCT_2:85</a></i>;<br/>
<b>set </b><font color="Maroon">K</font> =  <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">a</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>;<br/>
<b>set </b><font color="Maroon">M</font> = <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">a</font>,<font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E3:122_1"/><i><font color="Green">E30</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <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">a</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> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E4:122_1"/><b>then </b><i><font color="Green">E31</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">a</font>,<font color="Maroon">b</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="ref" href="transgeo.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E5:122_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</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">a</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="E6:122_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</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">a</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> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</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">a</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> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <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">a</font>,<font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:122_1"/>
<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">a</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">a</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#T56" target="_self">Th56</a>, <a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:122"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>  st <font color="Olive">A</font> <a href="aff_1.html#R4">//</a> <font color="Olive">C</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/>



<b>assume </b><a NAME="E1:122_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:122_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E3:122_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="transgeo.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E4:122_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> implies <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:122_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font>
 ;<br/>

<b>assume </b><a NAME="E2:122_2_1"/>
not <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E4:122_2_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</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="ref" href="transgeo.html#T56" target="_self">Th56</a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E6:122_2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <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">a</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="transgeo.html#T40" target="_self">Th40</a></i>;<br/>
<b>consider </b><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="E8:122_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <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">b</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="transgeo.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E9:122_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <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="funct_2.html#T85">FUNCT_2:85</a></i>;<br/>
<b>hence </b><a NAME="E10:122_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <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#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:122_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <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#T56" target="_self">Th56</a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:122"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  holds <br/><font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/>

<b>assume </b><a NAME="E1:122_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/>

<b>set </b><font color="Maroon">K</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E3:122_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T56" target="_self">Th56</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E4:122_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E6:122_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T40" target="_self">Th40</a></i>;<br/>
<b>consider </b><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="E8:122_3"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="transgeo.html#T40" target="_self">Th40</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E9:122_3"/><i><font color="Green">E93</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</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="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#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E10:122_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">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 
<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#T59" target="_self">Th59</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a></i>;<br/>
<b>hence </b><a NAME="E11:122_3"/>
<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#T61" target="_self">Th61</a>, <a class="ref" href="transgeo.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:122"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">C</font> holds <br/><font color="Olive">A</font> <a href="aff_1.html#R4">//</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/>



<b>assume </b><a NAME="E1:122_4"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 ;<br/>

<b>set </b><font color="Maroon">K</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">M</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>;<br/>
<a NAME="E2:122_4"/>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</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#T50">AFF_1:50</a></i>;<br/>
<a NAME="E3:122_4"/><b>then </b><i><font color="Green">E107</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_4_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:122_4_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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#D6" target="_self">Def6</a></i>;<br/><b>assume </b><a NAME="E3:122_4_1"/>
not <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">C</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E5:122_4_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>;<br/><a NAME="E6:122_4_1"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</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="ref" href="transgeo.html#T60" target="_self">Th60</a>, </i>;<br/><b>hence </b><a NAME="E7:122_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, <a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:122_4"/>
<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="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:122"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_5"><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">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>







<b>assume </b><a NAME="E1:122_5"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_5_1"/><i><font color="Green">E112</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">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/><b>set </b><font color="Maroon">K</font> = <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">a</font>,<font color="Maroon">b</font></span>)</span>;<br/><b>set </b><font color="Maroon">M</font> = <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">c</font>,<font color="Maroon">d</font></span>)</span>;<br/><a NAME="E2:122_5_1"/><i><font color="Green">E113</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/><a NAME="E3:122_5_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E4:122_5_1"/><b>then </b><i><font color="Green">E159</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <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">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <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">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <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">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E5:122_5_1"/>
<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">a</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> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, <a class="ref" href="transgeo.html#T60" target="_self">Th60</a></i>;<br/><b>hence </b><a NAME="E6:122_5_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T61" target="_self">Th61</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:122_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:122"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font> holds <br/><font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_6"><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">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>







<b>assume </b><a NAME="E1:122_6"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">t</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_6_1"/><i><font color="Green">E161</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">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/><b>set </b><font color="Maroon">K</font> = <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">a</font>,<font color="Maroon">b</font></span>)</span>;<br/><b>set </b><font color="Maroon">M</font> = <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">c</font>,<font color="Maroon">d</font></span>)</span>;<br/><a NAME="E2:122_6_1"/><i><font color="Green">E162</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E3:122_6_1"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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">b</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</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">d</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T59" target="_self">Th59</a>, <a class="ref" href="funct_2.html#T85">FUNCT_2:85</a></i>;<br/><a NAME="E4:122_6_1"/>
( <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">a</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">a</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> &amp; <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">c</font>,<font color="Maroon">d</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">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span> )
 <b>by </b><i><a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, <a class="ref" href="transgeo.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E5:122_6_1"/><b>then </b>
<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">a</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> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="transgeo.html#T58" target="_self">Th58</a>, <a class="ref" href="transgeo.html#T61" target="_self">Th61</a>, <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/><a NAME="E6:122_6_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E7:122_6_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">d</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T60" target="_self">Th60</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:122_6"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:122"/>
<font color="Maroon">f</font> <a href="transgeo.html#NR10" target="_self">is_Col</a> 
 <b>by </b><i><a class="ref" href="transgeo.html#T57" target="_self">Th57</a>, </i>;<br/>


</div>
