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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> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</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="E2:72"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:72"/>
<font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:72"><i><font color="Green">E46</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:72"/><i><font color="Green">E47</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="E6:72"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E7:72"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T80" target="_self">Th80</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E8:72"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E7:72"><i><font color="Green">E49</font></i></a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>7</sub></font> being   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E9:72"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<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="E11:72"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</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="E12:72"/><i><font color="Green">E52</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>5</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>9</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="E14:72"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E15:72"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>6</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="E16:72"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</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>5</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</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>6</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E7:72"><i><font color="Green">E49</font></i></a>, <a class="txt" href="incsp_1.html#E12:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="incsp_1.html#E15:72"><i><font color="Green">E54</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E17:72"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</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>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</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>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E6:72"><i><font color="Green">E48</font></i></a>, <a class="txt" href="incsp_1.html#E9:72"><i><font color="Green">E50</font></i></a>, <a class="txt" href="incsp_1.html#E15:72"><i><font color="Green">E54</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, <a class="ref" href="incsp_1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E18:72"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</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>9</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R5" 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>2</sub></font>,<font color="Maroon">c<sub>8</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>9</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E19:72"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>10</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:72_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</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>10</sub></font>
 ;<br/><a NAME="E2:72_1"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:72_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="enumset1.html#T98">ENUMSET1:98</a></i>;<br/><a NAME="E3:72_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</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>8</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>10</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</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>9</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>10</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:72_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/><a NAME="E4:72_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</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>10</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</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>10</sub></font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E5:72_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E11:72"><i><font color="Green">E51</font></i></a>, <a class="txt" href="incsp_1.html#E14:72"><i><font color="Green">E53</font></i></a>, <a class="txt" href="incsp_1.html#E16:72"><i><font color="Green">E55</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E6:72_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E5:72"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:72"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</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/>
<b>hence </b><a NAME="E22:72"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E6:72"><i><font color="Green">E48</font></i></a>, <a class="txt" href="incsp_1.html#E9:72"><i><font color="Green">E50</font></i></a>, <a class="txt" href="incsp_1.html#E19:72"><i><font color="Green">E56</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
