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<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><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">AFP</font> <a href="translac.html#NR1" target="_self">satisfies_Fano</a>  &amp; <font color="Maroon">AFP</font> <a href="aff_2.html#NR11" target="_self">satisfies_des</a>  )
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 ;<br/>

<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E26</font></i>: 
<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>
 ;<br/><b>set </b><font color="Maroon">g</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font>;<br/><a NAME="E2:26_1"/><i><font color="Green">E28</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>by </b><i><a class="ref" href="transgeo.html#T99">TRANSGEO:99</a></i>;<br/><a NAME="E3:26_1"/>
<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="transgeo.html#K1">*</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#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#T23">FUNCT_2:23</a></i>;<br/><b>hence </b><a NAME="E4:26_1"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font> st <br/>( <font color="Olive">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  &amp; <font color="Olive">g</font> <a href="transgeo.html#K1">*</a> <font color="Olive">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="translac.html#E2:26_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</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">AFP</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>;<br/><b>set </b><font color="Maroon">b</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/><a NAME="E3:26_2"/>
<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">a</font>
 <b>by </b><i><a class="txt" href="translac.html#E3:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="transgeo.html#D11">TRANSGEO:def 11</a></i>;<br/><b>then 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="E5:26_2"/><i><font color="Green">E31</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T22">AFF_1:22</a></i>;<br/><a NAME="E6:26_2"/><i><font color="Green">E32</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</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="E8:26_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><b>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="E10:26_2"/><i><font color="Green">E34</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E11:26_2"/><i><font color="Green">E35</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E1:26_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E13:26_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">h</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E14:26_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, </i>;<br/><a NAME="E15:26_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</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> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="translac.html#E1:26_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, <a class="txt" href="translac.html#E8:26_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="translac.html#E10:26_2"><i><font color="Green">E34</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">f1</font> being   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E17:26_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f1</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E18:26_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">f2</font> being   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E20:26_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f2</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E21:26_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</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="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">f3</font> being   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E23:26_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f3</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E24:26_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">f4</font> being   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E26:26_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f4</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E27:26_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, </i>;<br/><a NAME="E28:26_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a>  <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>by </b><i><a class="ref" href="translac.html#T2" target="_self">Th2</a>, <a class="ref" href="transgeo.html#T104">TRANSGEO:104</a></i>;<br/><a NAME="E29:26_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_2_1"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="translac.html#E17:26_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="transgeo.html#T4">TRANSGEO:4</a></i>;<br/>
<a NAME="E2:26_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="translac.html#E14:26_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/>
<a NAME="E3:26_2_1"/>
<font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a>  <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 
<b>by </b><i>, <a class="ref" href="transgeo.html#T104">TRANSGEO:104</a></i>;<br/>
<a NAME="E4:26_2_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 
<b>by </b><i><a class="txt" href="translac.html#E13:26_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="transgeo.html#T105">TRANSGEO:105</a></i>;<br/>
<a NAME="E5:26_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="txt" href="translac.html#E8:26_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="translac.html#E10:26_2"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="transgeo.html#T103">TRANSGEO:103</a></i>;<br/>
<a NAME="E6:26_2_1"/>
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="translac.html#E15:26_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="transgeo.html#T4">TRANSGEO:4</a></i>;<br/>
<a NAME="E7:26_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</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> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/>
<a NAME="E8:26_2_1"/>
<font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 
<b>by </b><i><a class="txt" href="translac.html#E18:26_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="translac.html#E20:26_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="transgeo.html#T105">TRANSGEO:105</a></i>;<br/>
<a NAME="E9:26_2_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="translac.html#E8:26_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="translac.html#E11:26_2"><i><font color="Green">E35</font></i></a>, , , <a class="ref" href="transgeo.html#T103">TRANSGEO:103</a></i>;<br/>
<a NAME="E10:26_2_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E11:26_2_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</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#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>;<br/>
<b>then </b><font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, , <a class="txt" href="translac.html#E20:26_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="translac.html#E2"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font></span>)</span> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/>
<b>then </b><font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font></span>)</span> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</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">f4</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font></span>)</span> <a href="transgeo.html#K1">*</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>
<b>by </b><i><a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>
<br/>.= 
<font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_2_1"/>
<font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f4</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f3</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E30:26_2"/><i><font color="Green">E84</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="translac.html#E6:26_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E31:26_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E32:26_2"/>
<font color="Maroon">p</font>,<font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="translac.html#E17:26_2"><i><font color="Green">E41</font></i></a>, , , <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E33:26_2"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f4</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="translac.html#E17:26_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="translac.html#E18:26_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="translac.html#E1:26"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E34:26_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="translac.html#E2:26"><i><font color="Green">E24</font></i></a>, , <a class="txt" href="translac.html#E18:26_2"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="translac.html#E2"><i><font color="Green">Lemma22</font></i></a></i>;<br/><a NAME="E35:26_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="translac.html#E5:26_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E36:26_2"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="translac.html#E14:26_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="translac.html#E15:26_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E37:26_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="translac.html#E13:26_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="translac.html#E14:26_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="translac.html#T2" target="_self">Th2</a>, <a class="txt" href="translac.html#E1:26"><i><font color="Green">E23</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2_2"/>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 ;<br/><a NAME="E2:26_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:26_2_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="translac.html#E5:26_2"><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/><a NAME="E4:26_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/><a NAME="E5:26_2_2"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E6:26_2_2"/><b>then </b>
( <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E7:26_2_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E8:26_2_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E9:26_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<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="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E10:26_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/></div><b>end;</b></div><a NAME="E39:26_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  )
 <b>by </b><i><a class="txt" href="translac.html#E13:26_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="translac.html#T2" target="_self">Th2</a>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="transgeo.html#T105">TRANSGEO:105</a></i>;<br/><a NAME="E40:26_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</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>
 <b>by </b><i><a class="ref" href="transgeo.html#D11">TRANSGEO:def 11</a></i>;<br/><a NAME="E41:26_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</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>
 <b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/><a NAME="E42:26_2"/><b>then </b>
<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="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>;<br/><a NAME="E43:26_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="transgeo.html#K1">*</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="transgeo.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="translac.html#E14:26_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="translac.html#E15:26_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="transgeo.html#T4">TRANSGEO:4</a></i>
;<br/><a NAME="E45:26_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f2</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  )
 <b>by </b><i><a class="ref" href="translac.html#T2" target="_self">Th2</a>, <a class="ref" href="transgeo.html#T105">TRANSGEO:105</a></i>;<br/><a NAME="E46:26_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="transgeo.html#K1">*</a> <font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="translac.html#E3:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="transgeo.html#T103">TRANSGEO:103</a></i>;<br/><b>hence </b><a NAME="E47:26_2"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font> st <br/>( <font color="Olive">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  &amp; <font color="Olive">g</font> <a href="transgeo.html#K1">*</a> <font color="Olive">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="translac.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:26"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font> st <br/>( <font color="Olive">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  &amp; <font color="Olive">g</font> <a href="transgeo.html#K1">*</a> <font color="Olive">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div>
