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



















<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E21</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> )
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E28</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E29</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green">E55</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E5:16"/><i><font color="Green">E56</font></i>: 
( <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="E6:16"/><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> )
 <b>and </b><br/><a NAME="E7:16"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 ;<br/>

<b>reconsider </b><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>, <font color="Maroon">d'</font> = <font color="Maroon">d</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font> ;<br/>
<a NAME="E9:16"/>
( <font color="Maroon">b'</font>,<font color="Maroon">p'</font>,<font color="Maroon">a'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p'</font>,<font color="Maroon">d'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">q'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font><b> such that </b><br/><a NAME="E11:16"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">b'</font>,<font color="Maroon">d'</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a'</font>,<font color="Maroon">c'</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9">ANPROJ_2:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q'</font> as   <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/>
<a NAME="E13:16"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , , , , , </i>;<br/>
<a NAME="E14:16"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , , , , , </i>;<br/>
<a NAME="E15:16"/><i><font color="Green">E72</font></i>: 
 ex <font color="Olive">P1</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">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P1</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P1</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P1</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E1:16"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E16:16"/>
 ex <font color="Olive">P2</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">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P2</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P2</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P2</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E1:16"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E17:16"/><b>then </b>
( <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#T2" target="_self">Th2</a>, <a class="ref" href="incproj.html#D2" target="_self">Def2</a>, , </i>;<br/>
<b>hence </b><a NAME="E18:16"/>
 ex <font color="Olive">q</font> being  <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> st <br/>( <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 ;<br/>


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