<?xml version="1.0"?>
<div class="add">

<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> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E31</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>3</sub></font>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>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="E4:45"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</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="E5:45"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</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#E2:45"><i><font color="Green">E32</font></i></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="E7:45"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:45"/><i><font color="Green">E36</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>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>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="E10:45"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</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="E11:45"/><i><font color="Green">E38</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>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>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><br/><a NAME="E13:45"/><i><font color="Green">E39</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>7</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E14:45"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>8</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E40</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>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:45_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>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>3</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E5:45"><i><font color="Green">E34</font></i></a>, <a class="txt" href="incsp_1.html#E11:45"><i><font color="Green">E38</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:45_1"/><b>then </b>
not <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>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#E1:45"><i><font color="Green">E31</font></i></a>, <a class="txt" href="incsp_1.html#E10:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:45_1"/><b>then </b>
( not <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> &amp; <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#R4" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="incsp_1.html#E8:45"><i><font color="Green">E36</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:45_1"/><b>then </b><i><font color="Green">E41</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="txt" href="incsp_1.html#E7:45"><i><font color="Green">E35</font></i></a>, <a class="ref" href="incsp_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E6:45_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>8</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>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="incsp_1.html#E8:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="incsp_1.html#E11:45"><i><font color="Green">E38</font></i></a>, <a class="txt" href="incsp_1.html#E1:45_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:45_1"/><b>then </b>
<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>8</sub></font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E8:45_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E13:45"><i><font color="Green">E39</font></i></a>, <a class="txt" href="incsp_1.html#E5:45_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>


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

<a NAME="E15:45"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R2" 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#R2" 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#R2" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E13:45"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E16:45"/><b>then </b>
( <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#R4" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>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>3</sub></font> &amp; <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>7</sub></font> &amp; <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>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>7</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="incsp_1.html#E5:45"><i><font color="Green">E34</font></i></a>, <a class="txt" href="incsp_1.html#E8:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="incsp_1.html#E11:45"><i><font color="Green">E38</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/>
<b>hence </b><a NAME="E17:45"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <b>by </b><i><a class="txt" href="incsp_1.html#E7:45"><i><font color="Green">E35</font></i></a>, <a class="txt" href="incsp_1.html#E10:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
