<?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>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#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/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>













<b>assume </b><a NAME="E1:13"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E14</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>
 ;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E2:13_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E1:13_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="collsp.html#T16">COLLSP:16</a>, <a class="ref" href="collsp.html#T17">COLLSP:17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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="txt" href="incproj.html#E1:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="collsp.html#D7">COLLSP:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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/><a NAME="E5:13_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="incproj.html#E2:13_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="incproj.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1"/>
ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:13_1_2"/>
ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E3:13_1_2"/>
ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="incproj.html#E1:13_1_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1"/>
ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E2:13_1"><i><font color="Green">E13</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>8</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><b> such that </b><a NAME="E3:13"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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="incproj.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:13"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="incproj.html#E3:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="incproj.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#T25">COLLSP:25</a></i>;<br/>


</div>
