<?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">q</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">p</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">B</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">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="E1:16"/><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">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">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</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> )
 ;<br/>

<a NAME="E2:16"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:16"/><b>then </b>
<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="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E4:16"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">pp'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green">E51</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="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_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="E2:16_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="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">f</font> =  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">g1</font> =  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">f1</font> =  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font>;<br/>
<a NAME="E6:16"/><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">Q</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:16"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>  st <font color="Olive">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/>

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">Q1</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q1</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">x'</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E5:16_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">x'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">Q3</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E7:16_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q3</font> &amp; <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q3</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E9:16_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q3</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:16_2_1"/><i><font color="Green">E76</font></i>: 
( <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="projred1.html#D1">PROJRED1:def 1</a></i>;<br/><a NAME="E3:16_2_1"/><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp'</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="projred1.html#D1">PROJRED1:def 1</a></i>;<br/><a NAME="E4:16_2_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">pp'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="projred2.html#D1" target="_self">Def1</a>, </i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E6:16_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:16_2_2"/>
( <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">pp'</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">pp'</font>,<font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">pp'</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">x'</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">Q3</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">x'</font>,<font color="Maroon">a</font> <a href="incproj.html#R4">on</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">A</font>,<font color="Maroon">O1</font>,<font color="Maroon">O2</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>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">R</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E4:16_2_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">x'</font>,<font color="Maroon">b</font> <a href="incproj.html#R4">on</a> <font color="Maroon">R</font>
 <b>by </b><i>, , , <a class="ref" href="projred1.html#T14">PROJRED1:14</a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E6:16_2_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x''</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x''</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/><a NAME="E7:16_2_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E8:16_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
( <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> )
 <b>by </b><i>, , , , , <a class="ref" href="projred1.html#D1">PROJRED1:def 1</a></i>;<br/><a NAME="E9:16_2_2"/><i><font color="Green">E85</font></i>: 
( <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="projred1.html#D1">PROJRED1:def 1</a></i>;<br/><a NAME="E10:16_2_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a>, , , </i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E12:16_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:16_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">X</font> =  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font>;<br/>
<a NAME="E8:16"/><i><font color="Green">E87</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="projred2.html#D1" target="_self">Def1</a>, <a class="ref" href="projred1.html#T25">PROJRED1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:16_3"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 ;<br/><b>hence </b><a NAME="E3:16_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:16"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
