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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="incsp_1.html#NM4">IncSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:38"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:38"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:38"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:38"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>assume </b><a NAME="E5:38"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:38"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:38"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E50"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:38"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E11:38"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E50"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:38"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E14:38"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E50"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E15:38"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E3:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="incsp_1.html#E5:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="incsp_1.html#E8:38"><i><font color="Green">E33</font></i></a>, <a class="txt" href="incsp_1.html#E11:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E16:38"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E17:38"/><b>then </b>
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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><a class="ref" href="incsp_1.html#T38" target="_self">Th38</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:38"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E20:38"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E19:38"><i><font color="Green">E38</font></i></a>, <a class="ref" href="enumset1.html#T45">ENUMSET1:45</a></i>;<br/>
<a NAME="E21:38"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:38"><i><font color="Green">E28</font></i></a>, <a class="txt" href="incsp_1.html#E8:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="incsp_1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E22:38"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E7:38"><i><font color="Green">E32</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E23:38"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E3:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="incsp_1.html#E5:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="incsp_1.html#E8:38"><i><font color="Green">E33</font></i></a>, <a class="txt" href="incsp_1.html#E11:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E24:38"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></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<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E19:38"><i><font color="Green">E38</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/>
<a NAME="E25:38"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E26:38"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E10:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E27:38"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E19:38"><i><font color="Green">E38</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E28:38"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:38"><i><font color="Green">E30</font></i></a>, <a class="txt" href="incsp_1.html#E14:38"><i><font color="Green">E37</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E29:38"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E13:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E30:38"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="incsp_1.html#E22:38"><i><font color="Green">E39</font></i></a>, <a class="txt" href="incsp_1.html#E26:38"><i><font color="Green">E40</font></i></a></i>;<br/>


</div>
