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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="collsp.html#V4">proper</a> <a href="collsp.html#NM2">CollSp</a>;<br/>

<b>consider </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> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:14"/><i><font color="Green">E12</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#D6">COLLSP:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:14"/>
<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>by </b><i><a class="txt" href="incproj.html#E2:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="collsp.html#T7">COLLSP:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> as    <a href="incproj.html#M1" target="_self">LINE</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="collsp.html#D7">COLLSP:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="incproj.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E7:14"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:14_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="incproj.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E2:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="collsp.html#T17">COLLSP:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:14"/>
not for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div>
