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









<b>assume </b><a NAME="E1:97"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<b>assume </b><a NAME="E2:97"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>
 ;<br/>

<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:97_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>hence </b><a NAME="E3:97_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Olive">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Olive">x</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2"/><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/><b>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="E3:97_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">z</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E2:97_2_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T38" target="_self">Th38</a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:97_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:97_2_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">b</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="E5:97_2_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Olive">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Olive">x</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E2:97_2_2"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E44"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="transgeo.html#T37" target="_self">Th37</a>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E3:97_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:97_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E5:97_2_2"/><b>then </b>
<font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E6:97_2_2"/><b>then </b>
<font color="Maroon">z</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font><b> such that </b><br/><a NAME="E8:97_2_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">t</font> )
 <b>by </b><i>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a>, <a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E9:97_2_2"/>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T38" target="_self">Th38</a>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E10:97_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">t</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">t</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E11:97_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">t</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">d</font>,<font color="Maroon">b</font>,<font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E12:97_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">t</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E13:97_2_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Olive">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Olive">x</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:97_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Olive">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T39" target="_self">Th39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:97"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Olive">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Olive">x</font> )
 <b>by </b><i/>;<br/>


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