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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR11" target="_self">satisfies_des</a> 
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 ;<br/>

<a NAME="E4:24"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="translac.html#E3:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="transgeo.html#D11">TRANSGEO:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:24_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:24_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:24_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="translac.html#E2:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E1:24_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="transgeo.html#D11">TRANSGEO:def 11</a></i>;<br/><a NAME="E5:24_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="translac.html#E3:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E2:24_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="transgeo.html#D11">TRANSGEO:def 11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E32</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="translac.html#E4:24_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="aff_1.html#T22">AFF_1:22</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:24_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E6:24_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="translac.html#E1:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="translac.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E7:24_1_1"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:24_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E3:24_1_1_1"/><i><font color="Green">E36</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:24_1_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="translac.html#E1:24_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="translac.html#E5:24_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="translac.html#E3:24_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:24_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="translac.html#E3:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E5:24_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="translac.html#E6:24_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="translac.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:24_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/>
<a NAME="E6:24_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="translac.html#E6:24_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/>
<a NAME="E7:24_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="translac.html#E3:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E4:24"><i><font color="Green">E26</font></i></a>, <a class="ref" href="transgeo.html#T85">TRANSGEO:85</a>, <a class="ref" href="transgeo.html#T100">TRANSGEO:100</a></i>;<br/>
<a NAME="E8:24_1_1_1"/><b>then </b>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="translac.html#E3:24_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="translac.html#E6:24_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="translac.html#E10"><i><font color="Green">Lemma8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="translac.html#E4:24_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="translac.html#E1:24_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="translac.html#E2:24_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:24_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="translac.html#E3:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E5:24_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a>, <a class="ref" href="transgeo.html#T105">TRANSGEO:105</a></i>;<br/><font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="translac.html#E2:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E3:24_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="translac.html#E5:24_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="translac.html#E6:24_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="translac.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="translac.html#E2:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E8:24_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="translac.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/><b>hence </b><a NAME="E10:24_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="transgeo.html#T13">TRANSGEO:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:24_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="translac.html#E2:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E3:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="translac.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="transgeo.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="transgeo.html#T11">TRANSGEO:11</a></i>;<br/>


</div>
