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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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1"/>
<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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</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="E3:67_1"/><i><font color="Green">E45</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>4</sub></font>
 <b>and </b><br/><a NAME="E4:67_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>4</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>4</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="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/><b>consider </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><br/><a NAME="E6:67_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:67_1"/><i><font color="Green">E48</font></i>: 
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>4</sub></font>,<font color="Maroon">c<sub>5</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="txt" href="incsp_1.html#E3:67_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="incsp_1.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E8:67_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>
 <b>by </b><i><a class="txt" href="incsp_1.html#E4:67_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="incsp_1.html#E6:67_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E9:67_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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; not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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="txt" href="incsp_1.html#E7:67_1"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_2"/><i><font color="Green">E45</font></i>: 
not <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>
 ;<br/><b>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#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:67_2"/><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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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>and </b><br/><a NAME="E4:67_2"/><i><font color="Green">E47</font></i>: 
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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#T71" target="_self">Th71</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E5:67_2"/>
<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="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#R5" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E3:67_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/><b>hence </b><a NAME="E6:67_2"/><i><font color="Green">E48</font></i>: 
<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>3</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self">Th19</a></i>;<br/><b>assume </b><a NAME="E7:67_2"/>
<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>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</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="E9:67_2"/><i><font color="Green">E49</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>7</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>9</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E10:67_2"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</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>9</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E9:67_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>;<br/><a NAME="E11:67_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<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="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</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>9</sub></font>
 <b>by </b><i><a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/><a NAME="E12:67_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <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#R4" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E4:67_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="incsp_1.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E13:67_2"/><b>then </b>
( <font color="Maroon">c<sub>9</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>2</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E6:67_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="incsp_1.html#E11:67_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E14:67_2"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:67_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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; not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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="txt" href="incsp_1.html#E1:67"><i><font color="Green">E44</font></i></a></i>;<br/>


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