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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="incsp_1.html#L1">IncProjStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="incproj.html#NM2">IncProjSp</a>
 ;<br/><a NAME="E2:9_1"/><b>then </b>
for <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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>by </b><i><a class="ref" href="projpl_1.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><a NAME="E4:9_1"/>
for <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>ex <font color="Olive">b<sub>3</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<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#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>4</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="E2:9_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:9_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E3:9_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:9_1"/>
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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:9_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="incproj.html#D11">INCPROJ:def 11</a></i>;<br/><b>thus </b><a NAME="E6:9_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</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>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <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="E2:9_1_2"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:9_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="incproj.html#D12">INCPROJ:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<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>thus </b><a NAME="E3:9_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:9_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:9_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>10</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>10</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E1:9_1_3"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E2:9_1_3"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E3:9_1_3"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>10</sub></font> &amp; <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> )
 ;<br/>

<a NAME="E4:9_1_3"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="projpl_1.html#E1:9_1_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E5:9_1_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</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>5</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="projpl_1.html#E2:9_1_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E7:9_1_3"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</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="projpl_1.html#E1:9_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="projpl_1.html#E3:9_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projpl_1.html#E4:9_1_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E8:9_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E7:9_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:9_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>and </b><br/><a NAME="E2:9_2"/><i><font color="Green">E10</font></i>: 
for <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>ex <font color="Olive">b<sub>3</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:9_2"/><i><font color="Green">E11</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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:9_2"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</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>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E5:9_2"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>10</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>10</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font>
 ;<br/><a NAME="E6:9_2"/><i><font color="Green">E14</font></i>: 
for <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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:9_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>now </b></a><div class="add"><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#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>4</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="E2:9_2_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_2"><i><font color="Green">E10</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E3:9_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>3</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="E9:9_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:9_2"><i><font color="Green">E11</font></i></a></i>;<br/><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </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> 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="E2:9_2_2"/><i><font color="Green">E18</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="incproj.html#R1">are_mutually_different</a>  &amp; <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="incproj.html#R4">on</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E4:9_2"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E3:9_2_2"/><i><font color="Green">E19</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> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_2_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E4:9_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:9_2_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><b>hence </b><a NAME="E5:9_2_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:9_2_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:9_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font>, <font color="Olive">b<sub>10</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>10</sub></font> &amp; not <font color="Olive">b<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>9</sub></font> &amp; not <font color="Olive">b<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>8</sub></font> holds <br/>ex <font color="Olive">b<sub>11</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>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_3"><b>proof </b></a><div class="add">

<b>let </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>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E1:9_2_3"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>and </b><br/><a NAME="E2:9_2_3"/><i><font color="Green">E19</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<a NAME="E3:9_2_3"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="projpl_1.html#E1:9_2_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E4:9_2_3"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="projpl_1.html#E1:9_2_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</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="E6:9_2_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E5:9_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projpl_1.html#E2:9_2_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projpl_1.html#E3:9_2_3"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>thus </b><a NAME="E7:9_2_3"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E6:9_2_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:9_2"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="incproj.html#NM2">IncProjSp</a>
 <b>by </b><i><a class="txt" href="projpl_1.html#E6:9_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projpl_1.html#E7:9_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projpl_1.html#E9:9_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projpl_1.html#E10:9_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a>, <a class="ref" href="incproj.html#D10">INCPROJ:def 10</a>, <a class="ref" href="incproj.html#D11">INCPROJ:def 11</a>, <a class="ref" href="incproj.html#D12">INCPROJ:def 12</a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/></div>
<b>end;</b></div>


</div>
