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



<b>assume </b><a NAME="E1:62"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR6" target="_self">is_Dil</a> 
 ;<br/>

<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_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">OAS</font> or for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>  holds <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> )
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><a NAME="E2:62_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="diraf.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><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1"/>
<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">OAS</font>
 ;<br/><a NAME="E2:62_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="diraf.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="diraf.html#T25">DIRAF:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>  holds <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font>
 ;<br/><b>assume </b><a NAME="E2:62_1_2"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="diraf.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>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E4:62_1_2"/><i><font color="Green">E35</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E13"><i><font color="Green">Lemma38</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">t</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>;<br/><a NAME="E5:62_1_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<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">z</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E4:62_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="transgeo.html#T23" target="_self">Th23</a>, </i>;<br/><a NAME="E6:62_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T23" target="_self">Th23</a>, <a class="ref" href="diraf.html#T40">DIRAF:40</a></i>;<br/><a NAME="E7:62_1_2"/>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<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">z</font> &amp; <font color="Maroon">y</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">y</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E4:62_1_2"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="transgeo.html#T23" target="_self">Th23</a>, </i>;<br/><a NAME="E8:62_1_2"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="diraf.html#T40">DIRAF:40</a></i>;<br/><a NAME="E9:62_1_2"/><b>then </b>
( <font color="Maroon">z</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">z</font> &amp; <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="diraf.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="diraf.html#T27">DIRAF:27</a></i>;<br/><b>hence </b><a NAME="E10:62_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#D2" target="_self">Def2</a>, <a class="ref" href="transgeo.html#T24" target="_self">Th24</a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:62_1"/>
<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="diraf.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="transgeo.html#E14"><i><font color="Green">Lemma39</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E37</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">OAS</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="diraf.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>
 ;<br/><b>assume </b><a NAME="E2:62_2"/><i><font color="Green">E49</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">OAS</font>
 ;<br/><b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><a NAME="E3:62_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E5:62_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">OAS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E13"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><a NAME="E6:62_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E7:62_2"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="diraf.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/><a NAME="E8:62_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<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="transgeo.html#E14"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E9:62_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E10:62_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<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="transgeo.html#T23" target="_self">Th23</a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E11:62_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="diraf.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="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E12:62_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E14"><i><font color="Green">Lemma39</font></i></a>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E14:62_2"/><i><font color="Green">E91</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="diraf.html#T43">DIRAF:43</a></i>;<br/><a NAME="E15:62_2"/>
<font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="diraf.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">z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E16:62_2"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E14"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E17:62_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E2:62_2_1"/><b>then </b>
<font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:62_2_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">z</font>,<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="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E4:62_2_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">y</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><b>hence </b><a NAME="E5:62_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#D2" target="_self">Def2</a>, <a class="ref" href="transgeo.html#T24" target="_self">Th24</a>, <a class="ref" href="transgeo.html#D3" target="_self">Def3</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/></div><b>end;</b></div><a NAME="E19:62_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E20:62_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T28" target="_self">Th28</a>, <a class="ref" href="diraf.html#T40">DIRAF:40</a></i>;<br/><a NAME="E21:62_2"/>
<font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E1:62_2"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E22:62_2"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">x</font>,<font color="Maroon">z</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="diraf.html#T28">DIRAF:28</a></i>;<br/><a NAME="E23:62_2"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#D2" target="_self">Def2</a>, <a class="ref" href="transgeo.html#T26" target="_self">Th26</a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/><a NAME="E24:62_2"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/><b>hence </b><a NAME="E25:62_2"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#D3" target="_self">Def3</a>, <a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62"/>
( ( <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">OAS</font> or for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>  holds <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> ) iff for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</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="diraf.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> )
 <b>by </b><i/>;<br/>


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