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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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">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="Maroon">A</font> )
 ;<br/>

<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</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="E3:37_1"/><i><font color="Green">E50</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="ref" href="aff_1.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E4:37_1"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_1.html#T15" target="_self">Th15</a>, <a class="ref" href="aff_1.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_1.html#E16"><i><font color="Green">Lemma32</font></i></a></i>;<br/><a NAME="E5:37_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:37_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T16" target="_self">Th16</a>, <a class="ref" href="aff_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">A</font>
 ;<br/><b>then 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="E3:37_2"/><i><font color="Green">E52</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> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E4:37_2"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15" target="_self">Th15</a>, </i>;<br/><b>hence </b><a NAME="E5:37_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_1.html#T15" target="_self">Th15</a>, <a class="txt" href="aff_1.html#E16"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E19"><i><font color="Green">Lemma35</font></i></a></i>;<br/>


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