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

<b>let </b><font color="Maroon">IPP</font> be  <a href="incproj.html#V6">2-dimensional</a> <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">pp'</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">O</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">O1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">O2</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">O3</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>































<b>assume </b><a NAME="E1:18"/><i><font color="Green">E47</font></i>: 
( not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; not <font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">O</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">p</font> <a href="incproj.html#R3">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">pp'</font> <a href="incproj.html#R3">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">d</font>,<font color="Maroon">p</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">pp'</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">pp'</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O3</font> )
 ;<br/>

<a NAME="E2:18"/><b>then </b><i><font color="Green">E49</font></i>: 
( not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; not <font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">O</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">pp'</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>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E9:18"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">pp'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E10:18"/>
<font color="Maroon">pp'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E11:18"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">O2</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E12:18"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">O2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
not <font color="Maroon">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">O2</font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:18_1"/><b>then </b>
<font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1"/>
contradiction
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:18"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
not <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:18_2"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O1</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2"/>
contradiction
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:18"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">pp'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3"/>
not <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">pp'</font>
 ;<br/>

<a NAME="E2:18_3"/><b>then </b>
<font color="Maroon">O3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:18_3"/>
contradiction
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:18"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_4"/>
<font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:18_4"/><b>then </b>
<font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:18_4"/>
contradiction
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:18"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_5"/>
not <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:18_5"/><b>then </b>
<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O3</font>
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:18_5"/>
contradiction
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:18"/>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <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="projred2.html#D1" target="_self">Def1</a>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div>
