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

<b>let </b><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:123"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="transgeo.html#NR10" title="TRANSGEO:NR.10">is_Col</a> 
 <b>and </b><br/><a NAME="E2:123"/><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E3:123"/><i><font color="Green" title="E81">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 <b>and </b><br/><a NAME="E4:123"/><i><font color="Green" title="E82">A4</font></i>: 
( not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> )
 ;<br/>

<a NAME="E5:123"/><i><font color="Green" title="E83">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">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" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E3:123_1_1"/><i><font color="Green" title="E84">A6</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E2:123"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/><b>set </b><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font>;<br/><b>set </b><font color="Maroon" title="c9">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>;<br/><a NAME="E4:123_1_1"/><i><font color="Green" title="E85">A7</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="transgeo.html#E4:123"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="transgeo.html#E3:123_1_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E6:123_1_1"/><i><font color="Green" title="E86">A8</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">C</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E8:123_1_1"/><i><font color="Green" title="E87">A9</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">D</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E4:123_1_1"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><a NAME="E9:123_1_1"/><i><font color="Green" title="E88">A10</font></i>: 
( <font color="Maroon" title="c10">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c11">D</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E10:123_1_1"/><i><font color="Green" title="E89">A11</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/><a NAME="E11:123_1_1"/><i><font color="Green" title="E90">A12</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/><a NAME="E12:123_1_1"/><i><font color="Green" title="E91">A13</font></i>: 
not <font color="Maroon" title="c10">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">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" title="c10">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 ;<br/>

<a NAME="E2:123_1_1_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:123_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="transgeo.html#E4:123"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="transgeo.html#E3:123_1_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="transgeo.html#E10:123_1_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="transgeo.html#E11:123_1_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:123_1_1"/><i><font color="Green" title="E92">A14</font></i>: 
not <font color="Maroon" title="c11">D</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">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" title="c11">D</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 ;<br/>

<a NAME="E2:123_1_1_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:123_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="transgeo.html#E4:123"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="transgeo.html#E3:123_1_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="transgeo.html#E10:123_1_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="transgeo.html#E11:123_1_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c12">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E15:123_1_1"/><i><font color="Green" title="E93">A15</font></i>: 
( <font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E2:123"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="transgeo.html#E9:123_1_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="transgeo.html#E12:123_1_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E17:123_1_1"/><i><font color="Green" title="E94">A16</font></i>: 
( <font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E2:123"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="transgeo.html#E9:123_1_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="transgeo.html#E13:123_1_1"><i><font color="Green" title="E92">A14</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><a NAME="E18:123_1_1"/><i><font color="Green" title="E95">A17</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">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" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:123"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="transgeo.html#T113" target="_self" title="TRANSGEO:th.113">Th113</a></i>;<br/>
<b>hence </b><a NAME="E2:123_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E3:123"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="transgeo.html#E4:123"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="transgeo.html#E3:123_1_1"><i><font color="Green" title="E84">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:123_1_1"/><i><font color="Green" title="E96">A18</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">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" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:123"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="transgeo.html#T113" target="_self" title="TRANSGEO:th.113">Th113</a></i>;<br/>
<b>hence </b><a NAME="E2:123_1_1_4"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E3:123"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="transgeo.html#E4:123"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="transgeo.html#E3:123_1_1"><i><font color="Green" title="E84">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:123_1_1"/><i><font color="Green" title="E97">A19</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:123"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="transgeo.html#T115" target="_self" title="TRANSGEO:th.115">Th115</a></i>;<br/>
<a NAME="E2:123_1_1_5"/><b>then </b><i><font color="Green" title="E98">A20</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">C</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="transgeo.html#E18:123_1_1"><i><font color="Green" title="E95">A17</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:123_1_1_5"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E3:123"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="transgeo.html#E15:123_1_1"><i><font color="Green" title="E93">A15</font></i></a></i>;<br/>
<a NAME="E4:123_1_1_5"/><b>then </b>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E15:123_1_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="ref" href="transgeo.html#T110" target="_self" title="TRANSGEO:th.110">Th110</a></i>;<br/>
<b>hence </b><a NAME="E5:123_1_1_5"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E15:123_1_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="transgeo.html#E2:123_1_1_5"><i><font color="Green" title="E98">A20</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:123_1_1"/><i><font color="Green" title="E98">A21</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font></span>)</span> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:123"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="transgeo.html#T115" target="_self" title="TRANSGEO:th.115">Th115</a></i>;<br/>
<a NAME="E2:123_1_1_6"/><b>then </b><i><font color="Green" title="E99">A22</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">D</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="transgeo.html#E19:123_1_1"><i><font color="Green" title="E96">A18</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:123_1_1_6"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">d</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E3:123"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="transgeo.html#E17:123_1_1"><i><font color="Green" title="E94">A16</font></i></a></i>;<br/>
<a NAME="E4:123_1_1_6"/><b>then </b>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E17:123_1_1"><i><font color="Green" title="E94">A16</font></i></a>, <a class="ref" href="transgeo.html#T110" target="_self" title="TRANSGEO:th.110">Th110</a></i>;<br/>
<b>hence </b><a NAME="E5:123_1_1_6"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E17:123_1_1"><i><font color="Green" title="E94">A16</font></i></a>, <a class="txt" href="transgeo.html#E2:123_1_1_6"><i><font color="Green" title="E99">A22</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:123_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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" title="E99">A23</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<a NAME="E2:123_1_1_7"/>
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> )
 
<b>by </b><i><a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="transgeo.html#E20:123_1_1"><i><font color="Green" title="E97">A19</font></i></a>, <a class="txt" href="transgeo.html#E21:123_1_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="transgeo.html#T110" target="_self" title="TRANSGEO:th.110">Th110</a></i>;<br/>
<a NAME="E3:123_1_1_7"/><b>then </b>
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="transgeo.html#E9:123_1_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="transgeo.html#E1:123_1_1_7"><i><font color="Green" title="E99">A23</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E4:123_1_1_7"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E6:123_1_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="transgeo.html#E8:123_1_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E5:123_1_1_7"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E10:123_1_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E6:123_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="transgeo.html#E2:123"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="transgeo.html#E4:123"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="transgeo.html#E3:123_1_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="transgeo.html#E9:123_1_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="transgeo.html#E10:123_1_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="transgeo.html#E11:123_1_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:123_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:123_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E3:123"><i><font color="Green" title="E81">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:123"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>;<br/>

<a NAME="E1:123_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E5:123"><i><font color="Green" title="E83">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:123_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:123"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
