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<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">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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>] <b>means </b>( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a</font>,<font color="Maroon">a<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a<sub>2</sub></font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a<sub>1</sub></font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">q</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">p</font>,<font color="Maroon">a<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">q</font>,<font color="Maroon">a<sub>2</sub></font> ) ) );<br/>
<b>assume </b><a NAME="E1:20"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">AFP</font> <a href="aff_2.html#NR11" target="_self">satisfies_des</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y'</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font>
 
<b>by </b><i>, , <a class="ref" href="transgeo.html#T97">TRANSGEO:97</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x'</font>,<font color="Olive">y</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">x'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>





<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E29</font></i>: 
( ( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">q</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">q</font>,<font color="Maroon">y</font> ) ) ) &amp; ( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x'</font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">q</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">y</font> &amp; <font color="Olive">p</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Olive">q</font>,<font color="Maroon">y</font> ) ) ) )
 ;<br/>

<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E32</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a>, , </i>;<br/><a NAME="E3:20_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="translac.html#E3:20"><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="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E4:20_1_1"/>
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="txt" href="translac.html#E3:20"><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="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="txt" href="translac.html#E5:20"><i><font color="Green">E28</font></i></a>, <a class="txt" href="translac.html#E1:20_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="transgeo.html#T97">TRANSGEO:97</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font>
 ;<br/><a NAME="E2:20_1_2"/><i><font color="Green">E36</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> or  not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or  not <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_1"/><i><font color="Green">E37</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:20_1_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_2_1"/>
contradiction
 <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#E1:20_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:20_1_2"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x'</font> or  not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">y</font> or  not <font color="Maroon">a</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_2"/><i><font color="Green">E39</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:20_1_2_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_2_2"/>
contradiction
 <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#E1:20_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:20_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i>, <a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E5:20"><i><font color="Green">E28</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="txt" href="translac.html#E4:20"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:20"/>
 ex <font color="Olive">f</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/>for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<b>from </b><i><a class="ref" href="transgeo.html#S1">TRANSGEO:sch 1</a>(<a class="ref" href="translac.html#D1" target="_self">Def1</a>, <a class="txt" href="translac.html#E2"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="translac.html#E2:20"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E1:20"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">f</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="E8:20"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds <br/> ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> iff ( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">y</font> &amp; <font color="Maroon">a</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">y</font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">x</font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">q</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">y</font> &amp; <font color="Olive">p</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">q</font>,<font color="Olive">y</font> ) ) ) )
 ;<br/>
<a NAME="E9:20"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds  <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>

<b>set </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green">E44</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</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>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E3:20_2_1"/><i><font color="Green">E71</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E4:20_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</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#E5:20"><i><font color="Green">E28</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:20_2_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="translac.html#E5:20"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:20"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds  <font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>



<a NAME="E1:20_3"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="translac.html#E4:20"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:20_3"/>
<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>



<div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_4_1"/><i><font color="Green">E74</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font> )
 ;<br/><a NAME="E2:20_4_1"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:20_4_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_4_2"/><i><font color="Green">E76</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font> )
 ;<br/><a NAME="E2:20_4_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E3:20_4_2"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="translac.html#E4:20"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E4:20_4_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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#T18">AFF_1:18</a></i>;<br/><a NAME="E5:20_4_2"/><b>then </b>
<font color="Maroon">a</font>,<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">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><b>hence </b><a NAME="E6:20_4_2"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_4_3"/><i><font color="Green">E79</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font> )
 ;<br/><a NAME="E2:20_4_3"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E3:20_4_3"/>
 ex <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Olive">x'</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Olive">x'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E4:20_4_3"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E5:20_4_3"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_4_4"/><i><font color="Green">E84</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font> )
 ;<br/><a NAME="E2:20_4_4"/><b>then </b><i><font color="Green">E86</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E3:20_4_4"/>
 ex <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Olive">y'</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Olive">y'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><b>hence </b><a NAME="E4:20_4_4"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:20_4"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="translac.html#E5:20"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:20"/><b>then </b>
<font color="Maroon">f</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 
<b>by </b><i><a class="ref" href="transgeo.html#T85">TRANSGEO:85</a></i>;<br/>
<a NAME="E13:20"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 
<b>by </b><i><a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="transgeo.html#T100">TRANSGEO:100</a></i>;<br/>
<a NAME="E14:20"/>
<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">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<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="E2:20_5"/><i><font color="Green">E88</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T22">AFF_1:22</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E4:20_5"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E5:20_5"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</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></i>;<br/>
<a NAME="E6:20_5"/>
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<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#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:20_5"/>
<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">b</font>
 <b>by </b><i><a class="txt" href="translac.html#E3:20"><i><font color="Green">E25</font></i></a>, , <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:20"/>
 ex <font color="Olive">f</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">f</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a>  &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="translac.html#E5:20"><i><font color="Green">E28</font></i></a></i>;<br/>


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