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<b>let </b><font color="Maroon">AFS</font> be   <a href="diraf.html#NM1">AffinSpace</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">AFS</font>;<br/>



<b>assume </b><a NAME="E1:98"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 ;<br/>

<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_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">AFS</font> or for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</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">AFS</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/><a NAME="E2:98_1"/><i><font color="Green">E28</font></i>: 
<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="transgeo.html#D4" target="_self">Def4</a></i>;<br/><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_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">AFS</font>
 ;<br/><a NAME="E2:98_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="E3:98_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="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#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_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">AFS</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="E2:98_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="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>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font><b> such that </b><br/><a NAME="E4:98_1_2"/><i><font color="Green">E35</font></i>: 
(  <a href="aff_1.html#R1">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="aff_1.html#R1">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="transgeo.html#T38" target="_self">Th38</a>, <a class="txt" href="transgeo.html#E43"><i><font color="Green">Lemma62</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="E5:98_1_2"/>
 <a href="aff_1.html#R1">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>, , </i>;<br/><a NAME="E6:98_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="analoaf.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="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E7:98_1_2"/>
(  <a href="aff_1.html#R1">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="aff_1.html#R1">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>, , , </i>;<br/><a NAME="E8:98_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="analoaf.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="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E9:98_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="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#T13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E10:98_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:98_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="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="transgeo.html#T37" target="_self">Th37</a>, <a class="ref" href="transgeo.html#T39" target="_self">Th39</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_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">AFS</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>
 ;<br/><b>assume </b><a NAME="E2:98_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">AFS</font>
 ;<br/><b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font><b> such that </b><a NAME="E3:98_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">AFS</font><b> such that </b><br/><a NAME="E5:98_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">AFS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T38" target="_self">Th38</a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><a NAME="E6:98_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="E7:98_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="transgeo.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E8:98_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="aff_1.html#R1">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="ref" href="transgeo.html#T39" target="_self">Th39</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E9:98_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="aff_1.html#R1">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="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E10:98_2"/>
 <a href="aff_1.html#R1">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="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E11:98_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">y</font>,<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#D1">AFF_1:def 1</a></i>;<br/><a NAME="E12:98_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="ref" href="transgeo.html#T39" target="_self">Th39</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">AFS</font><b> such that </b><br/><a NAME="E14:98_2"/><i><font color="Green">E91</font></i>: 
not  <a href="aff_1.html#R1">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="aff_1.html#T22">AFF_1:22</a></i>;<br/><a NAME="E15:98_2"/>
<font color="Maroon">x</font>,<font color="Maroon">z</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">z</font>
 <b>by </b><i>, <a class="ref" href="transgeo.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E16:98_2"/><b>then </b>
 <a href="aff_1.html#R1">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="ref" href="transgeo.html#T39" target="_self">Th39</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E17:98_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="aff_1.html#R1">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="aff_1.html#T15">AFF_1:15</a></i>;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_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="E2:98_2_1"/><b>then </b>
<font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="analoaf.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>, <a class="ref" href="transgeo.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:98_2_1"/><b>then </b>
 <a href="aff_1.html#R1">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="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:98_2_1"/><b>then </b>
(  <a href="aff_1.html#R1">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="aff_1.html#R1">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="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>hence </b><a NAME="E5:98_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/></div><b>end;</b></div><a NAME="E19:98_2"/>
 <a href="aff_1.html#R1">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="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E20:98_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="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T40" target="_self">Th40</a>, <a class="ref" href="aff_1.html#T19">AFF_1:19</a></i>;<br/><a NAME="E21:98_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="analoaf.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#T37" target="_self">Th37</a></i>;<br/><a NAME="E22:98_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="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="transgeo.html#D6" target="_self">Def6</a>, <a class="ref" href="transgeo.html#T42" target="_self">Th42</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E23:98_2"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E24:98_2"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E25:98_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:98"/>
( ( <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">AFS</font> or for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</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">AFS</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> )
 <b>by </b><i/>;<br/>


</div>
