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

<b>let </b><font color="Maroon">G</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">E36</font></i>: 
<font color="Maroon">G</font> is   <a href="incproj.html#NM2">IncProjSp</a>
 ;<br/><a NAME="E2:9_1"/><b>then </b>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><br/> for <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  st <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</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">G</font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:9_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>  ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Olive">p</font>,<font color="Olive">q</font> <a href="incproj.html#R3">on</a> <font color="Olive">P</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">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/>



<b>consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E3:9_1_1"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font>
 <b>by </b><i>, <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">p</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">P</font>
 <b>by </b><i>, <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">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Olive">P</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">P</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:9_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D12">INCPROJ:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E3:9_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i>, <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">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font>
 <b>by </b><i>, <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">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><br/> for <font color="Olive">M</font>, <font color="Olive">N</font>, <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">p</font> <a href="incproj.html#R4">on</a> <font color="Olive">M</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">p</font> <a href="incproj.html#R4">on</a> <font color="Olive">N</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="incproj.html#R3">on</a> <font color="Olive">P</font> &amp; <font color="Olive">b</font>,<font color="Olive">d</font> <a href="incproj.html#R3">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">P</font> &amp; <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">Q</font> &amp; <font color="Olive">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">N</font> holds <br/> ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <font color="Olive">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">P</font>,<font color="Olive">Q</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">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:9_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="incproj.html#R4">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">p</font> <a href="incproj.html#R4">on</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E2:9_1_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">Q</font> )
 <b>and </b><br/><a NAME="E3:9_1_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E4:9_1_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E5:9_1_3"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:9_1_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E8:9_1_3"/>
<font color="Maroon">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>
 <b>by </b><i>, </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">E46</font></i>: 
<font color="Maroon">G</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">E47</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>  ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Olive">p</font>,<font color="Olive">q</font> <a href="incproj.html#R3">on</a> <font color="Olive">P</font>
 <b>and </b><br/><a NAME="E3:9_2"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">p</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">P</font>
 <b>and </b><br/><a NAME="E4:9_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Olive">P</font> )
 <b>and </b><br/><a NAME="E5:9_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><br/> for <font color="Olive">M</font>, <font color="Olive">N</font>, <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">p</font> <a href="incproj.html#R4">on</a> <font color="Olive">M</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">p</font> <a href="incproj.html#R4">on</a> <font color="Olive">N</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="incproj.html#R3">on</a> <font color="Olive">P</font> &amp; <font color="Olive">b</font>,<font color="Olive">d</font> <a href="incproj.html#R3">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">P</font> &amp; <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">Q</font> &amp; <font color="Olive">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">N</font> holds <br/> ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <font color="Olive">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">P</font>,<font color="Olive">Q</font>
 ;<br/><a NAME="E6:9_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><br/> for <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  st <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font>
 <b>by </b><i>, </i>;<br/><div><i><font color="Green">E52</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">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:9_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E3:9_2_1"/>
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">p</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:9_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><div><i><font color="Green">E55</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">P</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:9_2_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E3:9_2_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E4:9_2_2"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <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">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E11:9_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><br/> for <font color="Olive">M</font>, <font color="Olive">N</font>, <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> &amp; <font color="Olive">d</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> &amp; <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> &amp; <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">d</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; not <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">N</font> holds <br/> ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</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">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:9_2_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>and </b><br/><a NAME="E2:9_2_3"/><i><font color="Green">E59</font></i>: 
( not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E3:9_2_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="incproj.html#R4">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">p</font> <a href="incproj.html#R4">on</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E4:9_2_3"/>
( <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">Q</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:9_2_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">q1</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">q1</font>
;<br/>

<b>thus </b><a NAME="E7:9_2_3"/>
( <font color="Maroon">q1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:9_2"/>
<font color="Maroon">G</font> is   <a href="incproj.html#NM2">IncProjSp</a>
 <b>by </b><i>, , , , <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>
