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<b>let </b><font color="Maroon">S</font> be   <a href="incsp_1.html#NM4">IncSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font>
 ;<br/>

<b>consider </b><font color="Maroon">B</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>, <font color="Maroon">C</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:44"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:44"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">A</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>let </b><font color="Maroon">Q</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">S</font>;<br/>

<b>thus </b><a NAME="E7:44"/>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font> implies <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:44_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E2:44_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E3:44_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="ref" href="incsp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="incsp_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:44_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:44_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">A</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, </i>;<br/>
<a NAME="E6:44_1"/>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">A</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="txt" href="incsp_1.html#E3:44"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:44_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="txt" href="incsp_1.html#E4:44"><i><font color="Green">E36</font></i></a>, <a class="ref" href="incsp_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E8:44"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/>
<a NAME="E9:44"/><b>then </b>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:44"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> implies ( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font> ) )
 <b>by </b><i><a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="ref" href="incsp_1.html#D5" target="_self">Def5</a></i>;<br/>


</div>
