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

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







<b>assume </b><a NAME="E1:77"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">f</font> <a href="transgeo.html#NR7" target="_self">is_Tr</a>  &amp; <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> &amp;  <a href="diraf.html#R3">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">b</font> )
 ;<br/>

<a NAME="E2:77"/><b>then </b>
<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">OAS</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E3:77"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, </i>;<br/>
<a NAME="E4:77"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR6" target="_self">is_Dil</a> 
 
<b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:77_1"/><i><font color="Green">E30</font></i>: 
(  <a href="diraf.html#R1">Mid</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">b</font> implies <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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, <a class="ref" href="transgeo.html#T26" target="_self">Th26</a></i>;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="diraf.html#R1">Mid</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">b</font>
 ;<br/><a NAME="E2:77_1_1"/>
(  <a href="diraf.html#R3">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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> &amp;  <a href="diraf.html#R3">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">a</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, , , <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E3:77_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/><a NAME="E2:77_1_1_1"/><b>then </b>
<font color="Maroon">b</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">b</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="transgeo.html#T29" target="_self">Th29</a>, <a class="ref" href="transgeo.html#T30" target="_self">Th30</a>, <a class="ref" href="transgeo.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E3:77_1_1_1"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_2"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:77_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">b</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">b</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, <a class="ref" href="transgeo.html#T30" target="_self">Th30</a>, <a class="ref" href="transgeo.html#D3" target="_self">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_2_1"/>
<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">b</font>
 ;<br/><a NAME="E2:77_1_1_2_1"/><b>then </b>
( <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">b</font> &amp; <font color="Maroon">a</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">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, <a class="ref" href="transgeo.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E3:77_1_1_2_1"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:77_1_1_2"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T35" target="_self">Th35</a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_3"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/><a NAME="E2:77_1_1_3"/><b>then </b>
(  <a href="diraf.html#R1">Mid</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">b</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</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="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E3:77_1_1_3"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">b</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">b</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</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="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E4:77_1_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</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">b</font> &amp; <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">a</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E5:77_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</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">b</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E4:77_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E6:77_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><b>hence </b><a NAME="E7:77_1_1_3"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T35" target="_self">Th35</a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:77_1_1"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="transgeo.html#D4" target="_self">Def4</a>, <a class="ref" href="diraf.html#T35">DIRAF:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="diraf.html#R1">Mid</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">a</font>
 ;<br/><a NAME="E2:77_1_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> implies <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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, <a class="ref" href="transgeo.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E3:77_1_2"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, , <a class="ref" href="transgeo.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T29" target="_self">Th29</a>, , , <a class="ref" href="diraf.html#T35">DIRAF:35</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:77"/>
<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">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T7">DIRAF:7</a></i>;<br/>


</div>
