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







<a NAME="E1:33"/><i><font color="Green">E27</font></i>: 
(  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> implies  ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E39</font></i>: 
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/>

<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>;<br/><a NAME="E4:33_1_1"/>
(  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13" target="_self">Th13</a>, <a class="ref" href="aff_1.html#T14" target="_self">Th14</a>, , </i>;<br/><b>hence </b><a NAME="E5:33_1_1"/>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:33_1_2"/>
(  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E15"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="aff_1.html#T14" target="_self">Th14</a>, , , </i>;<br/><b>hence </b><a NAME="E3:33_1_2"/>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:33_1_3"/><i><font color="Green">E71</font></i>: 
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E15"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/><a NAME="E3:33_1_3"/>
(  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T14" target="_self">Th14</a>, <a class="txt" href="aff_1.html#E19"><i><font color="Green">Lemma35</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E4:33_1_3"/>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_1"/>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E16"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="aff_1.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33"/>
(  ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) implies  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:33_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:33_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E15"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/>
<a NAME="E5:33_2"/>
(  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E16"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="aff_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>hence </b><a NAME="E6:33_2"/>
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" target="_self">Th13</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
(  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> iff  ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) )
 <b>by </b><i><a class="ref" href="aff_1.html#T12" target="_self">Th12</a></i>;<br/>


</div>
