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

<b>let </b><font color="Maroon">CPS</font> be   <a href="anproj_2.html#NM1">CollProjectiveSpace</a>;<br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">p1</font>, <font color="Olive">r2</font>, <font color="Olive">q</font>, <font color="Olive">r1</font>, <font color="Olive">q1</font>, <font color="Olive">p</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">CPS</font>  st <font color="Olive">p1</font>,<font color="Olive">r2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">r1</font>,<font color="Olive">q1</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">r1</font>,<font color="Olive">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">r2</font>,<font color="Olive">q1</font>,<font color="Olive">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">q1</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">r2</font>,<font color="Olive">r1</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">p1</font>,<font color="Olive">r2</font>,<font color="Olive">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">p1</font>,<font color="Olive">r2</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">p1</font>,<font color="Olive">r1</font>,<font color="Olive">q1</font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Olive">r2</font>,<font color="Olive">r1</font>,<font color="Olive">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">q</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">r</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">s</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">Q</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">R</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">S</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">B</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>, <font color="Maroon">C</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:22"/><i><font color="Green">E28</font></i>: 
( not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">s</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">s</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">S</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R3" target="_self">on</a> <font color="Maroon">C</font> )
 ;<br/>

<b>assume </b><a NAME="E4:22"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font>, <font color="Maroon">s'</font> = <font color="Maroon">s</font>, <font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font> ;<br/>
<a NAME="E6:22"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s</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">S</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">S</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">S</font> &amp; <font color="Maroon">c</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">A</font> &amp; <font color="Maroon">q</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">B</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:22"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p'</font>,<font color="Maroon">r'</font>,<font color="Maroon">b'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">s'</font>,<font color="Maroon">q'</font>,<font color="Maroon">b'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p'</font>,<font color="Maroon">s'</font>,<font color="Maroon">a'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r'</font>,<font color="Maroon">q'</font>,<font color="Maroon">a'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p'</font>,<font color="Maroon">q'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r'</font>,<font color="Maroon">s'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">p'</font>,<font color="Maroon">r'</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 ex <font color="Olive">K</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> st <br/>( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , </i>;<br/><b>thus </b><a NAME="E4:22_1"/>
contradiction
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , <a class="ref" href="incproj.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">p'</font>,<font color="Maroon">r'</font>,<font color="Maroon">s'</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:22_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 ex <font color="Olive">K</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> st <br/>( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_2"/>
<font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , </i>;<br/><b>thus </b><a NAME="E4:22_2"/>
contradiction
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon">p'</font>,<font color="Maroon">s'</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:22_3"/><b>then </b><i><font color="Green">E100</font></i>: 
 ex <font color="Olive">K</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> st <br/>( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_3"/>
<font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , </i>;<br/><b>thus </b><a NAME="E4:22_3"/>
contradiction
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_4"/>
<font color="Maroon">r'</font>,<font color="Maroon">s'</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:22_4"/><b>then </b><i><font color="Green">E101</font></i>: 
 ex <font color="Olive">K</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> st <br/>( <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_4"/>
<font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , </i>;<br/><b>thus </b><a NAME="E4:22_4"/>
contradiction
 <b>by </b><i><a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:22"/>
contradiction
 <b>by </b><i><a class="ref" href="incproj.html#T2" target="_self">Th2</a>, , <a class="txt" href="incproj.html#E2:22"><i><font color="Green">E28</font></i></a>, <a class="txt" href="incproj.html#E3:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="incproj.html#T6" target="_self">Th6</a>, </i>;<br/>


</div>
