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<b>consider </b><font color="Maroon">CPS</font> being  <a href="anproj_2.html#V4">Fanoian</a> <a href="anproj_2.html#V5">Desarguesian</a> <a href="anproj_2.html#NM2">CollProjectivePlane</a>;<br/>
<b>reconsider </b><font color="Maroon">CPS'</font> = <font color="Maroon">CPS</font> as   <a href="anproj_2.html#NM1">CollProjectiveSpace</a> ;<br/>
<b>take </b><font color="Maroon">X</font> =  <a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS'</font>;<br/>

<a NAME="E3:39_1_1"/>
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="pre_topc.html#NM2">Point</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> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D11">ANPROJ_2:def 11</a></i>;<br/>
<a NAME="E4:39_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font>, <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">X</font><br/> for <font color="Olive">L</font>, <font color="Olive">Q</font>, <font color="Olive">R</font>, <font color="Olive">S</font>, <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">X</font>  st not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">L</font> &amp; not <font color="Olive">r</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">L</font> &amp; not <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Olive">s</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">R</font> &amp; not <font color="Olive">r</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">R</font> &amp; not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">S</font> &amp; not <font color="Olive">s</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">S</font> &amp; <font color="Olive">a</font>,<font color="Olive">p</font>,<font color="Olive">s</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">L</font> &amp; <font color="Olive">a</font>,<font color="Olive">q</font>,<font color="Olive">r</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">b</font>,<font color="Olive">q</font>,<font color="Olive">s</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">R</font> &amp; <font color="Olive">b</font>,<font color="Olive">p</font>,<font color="Olive">r</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">S</font> &amp; <font color="Olive">c</font>,<font color="Olive">p</font>,<font color="Olive">q</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">A</font> &amp; <font color="Olive">c</font>,<font color="Olive">r</font>,<font color="Olive">s</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">B</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="incproj.html#R3" target="_self">on</a> <font color="Olive">C</font> holds <br/> not <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">C</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E4:39_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:39_1_1"/>
for <font color="Olive">o</font>, <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">p3</font>, <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">q3</font>, <font color="Olive">r1</font>, <font color="Olive">r2</font>, <font color="Olive">r3</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS'</font>  st <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q1</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q1</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q2</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q2</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q3</font> &amp; <font color="Olive">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q3</font> &amp; not <font color="Olive">o</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">o</font>,<font color="Olive">p1</font>,<font color="Olive">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">o</font>,<font color="Olive">p2</font>,<font color="Olive">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p2</font>,<font color="Olive">p3</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q2</font>,<font color="Olive">q3</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">p3</font>,<font color="Olive">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q1</font>,<font color="Olive">q3</font>,<font color="Olive">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">o</font>,<font color="Olive">p1</font>,<font color="Olive">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">o</font>,<font color="Olive">p2</font>,<font color="Olive">q2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">o</font>,<font color="Olive">p3</font>,<font color="Olive">q3</font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Olive">r1</font>,<font color="Olive">r2</font>,<font color="Olive">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D12">ANPROJ_2:def 12</a></i>;<br/>
<a NAME="E6:39_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
for <font color="Olive">o</font>, <font color="Olive">b1</font>, <font color="Olive">a1</font>, <font color="Olive">b2</font>, <font color="Olive">a2</font>, <font color="Olive">b3</font>, <font color="Olive">a3</font>, <font color="Olive">r</font>, <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">X</font><br/> for <font color="Olive">C1</font>, <font color="Olive">C2</font>, <font color="Olive">C3</font>, <font color="Olive">A1</font>, <font color="Olive">A2</font>, <font color="Olive">A3</font>, <font color="Olive">B1</font>, <font color="Olive">B2</font>, <font color="Olive">B3</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">X</font>  st <font color="Olive">o</font>,<font color="Olive">b1</font>,<font color="Olive">a1</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">C1</font> &amp; <font color="Olive">o</font>,<font color="Olive">a2</font>,<font color="Olive">b2</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">C2</font> &amp; <font color="Olive">o</font>,<font color="Olive">a3</font>,<font color="Olive">b3</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">C3</font> &amp; <font color="Olive">a3</font>,<font color="Olive">a2</font>,<font color="Olive">t</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">A1</font> &amp; <font color="Olive">a3</font>,<font color="Olive">r</font>,<font color="Olive">a1</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">A2</font> &amp; <font color="Olive">a2</font>,<font color="Olive">s</font>,<font color="Olive">a1</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">A3</font> &amp; <font color="Olive">t</font>,<font color="Olive">b2</font>,<font color="Olive">b3</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">B1</font> &amp; <font color="Olive">b1</font>,<font color="Olive">r</font>,<font color="Olive">b3</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">B2</font> &amp; <font color="Olive">b1</font>,<font color="Olive">s</font>,<font color="Olive">b2</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">B3</font> &amp; <font color="Olive">C1</font>,<font color="Olive">C2</font>,<font color="Olive">C3</font> <a href="incproj.html#R1" target="_self">are_mutually_different</a>  &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a1</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a2</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a3</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b1</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b2</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b3</font> &amp; <font color="Olive">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b1</font> &amp; <font color="Olive">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b2</font> &amp; <font color="Olive">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b3</font> holds <br/> ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">X</font> st <font color="Olive">r</font>,<font color="Olive">s</font>,<font color="Olive">t</font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">O</font>
 
<b>by </b><i><a class="ref" href="incproj.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:39_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS'</font>  ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS'</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">p</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/>
<a NAME="E8:39_1_1"/><b>then </b>
for <font color="Olive">M</font>, <font color="Olive">N</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">X</font>  ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:39_1_1"/>
( <font color="Maroon">X</font> is <a href="incproj.html#V10" target="_self">Desarguesian</a> &amp; <font color="Maroon">X</font> is <a href="incproj.html#V9" target="_self">Fanoian</a> &amp; <font color="Maroon">X</font> is <a href="incproj.html#V6" target="_self">2-dimensional</a> )
 <b>by </b><i>, , , , </i>;<br/>


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