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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">L1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="incsp_1.html#NM2">LINE</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>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">S</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">L1</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">L2</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font>
 ;<br/>

<b>given </b><font color="Maroon">Q</font> being   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E5:39"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">L1</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">L2</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>, <font color="Maroon">B</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:39"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:39"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L1</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">C</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>, <font color="Maroon">C1</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E10:39"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font>
 <b>and </b><br/><a NAME="E11:39"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C1</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L2</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">D</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>, <font color="Maroon">E</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E13:39"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">E</font>
 <b>and </b><br/><a NAME="E14:39"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">E</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/>
<a NAME="E15:39"/>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C1</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">E</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#T17" target="_self">Th17</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="E16:39"/><b>then </b>
( <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">C1</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">D</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">E</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">E</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">D</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">E</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#T17" target="_self">Th17</a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, , <a class="ref" href="incsp_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E18:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">E</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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#K7">{</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">D</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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#K7">{</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">E</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#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E19:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">Q</font> &amp; <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">E</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">Q</font> &amp; <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C1</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">Q</font> &amp; <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C1</font>,<font color="Maroon">E</font></span><a href="domain_1.html#K9">}</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="enumset1.html#T45">ENUMSET1:45</a></i>;<br/>
<a NAME="E20:39"/><b>then </b><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  &amp; <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">E</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  &amp; <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C1</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  &amp; <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C1</font>,<font color="Maroon">E</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
( <font color="Maroon">C</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">C1</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L1</font> )
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C1</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:39_1"/>
contradiction
 <b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="incsp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="incsp_1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:39"/><b>then </b><i><font color="Green">E67</font></i>: 
(  not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a>  or  not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C1</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#T18" target="_self">Th18</a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="txt" href="incsp_1.html#E2:39"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E23:39"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C1</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#T15" target="_self">Th15</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="E24:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">C1</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, , <a class="ref" href="incsp_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E25:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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">C</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> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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">C1</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#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E26:39"/><b>then </b><i><font color="Green">E68</font></i>: 
( <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</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> &amp; <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C1</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="enumset1.html#T43">ENUMSET1:43</a></i>;<br/>
<a NAME="E27:39"/>
not <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">E</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="txt" href="incsp_1.html#E3:39"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="incsp_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E28:39"/><b>then </b>
(  not <font color="Maroon">D</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> or  not <font color="Maroon">E</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E29:39"/>
contradiction
 <b>by </b><i><a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, , , <a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>


</div>
