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<div class="add">

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

<b>given </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>, <font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>, <font color="Maroon">r</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>, <font color="Maroon">r1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font><b> such that </b><a NAME="E1:18"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">s</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>  holds <br/>(  not <font color="Maroon">p</font>,<font color="Maroon">p1</font>,<font color="Olive">s</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">r</font>,<font color="Maroon">r1</font>,<font color="Olive">s</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>set </b><font color="Maroon">M''</font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">p</font>,<font color="Maroon">p1</font>;<br/>
<b>set </b><font color="Maroon">N''</font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">r</font>,<font color="Maroon">r1</font>;<br/>
<a NAME="E2:18"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r1</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"/><i><font color="Green">E28</font></i>: 
(  not <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> or  not <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r1</font> )
 ;<br/>

<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 ;<br/><a NAME="E2:18_1_1"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">p1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r</font>,<font color="Maroon">r1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="collsp.html#T7">COLLSP:7</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 ;<br/><a NAME="E2:18_1_2"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">p1</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r</font>,<font color="Maroon">r1</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="collsp.html#T7">COLLSP:7</a></i>;<br/><b>hence </b><a NAME="E3:18_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M'</font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">p</font>,<font color="Maroon">p1</font>, <font color="Maroon">N'</font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">r</font>,<font color="Maroon">r1</font> as    <a href="incproj.html#M1" target="_self">LINE</a> of <font color="Maroon">CPS</font> <b>by </b><i><a class="ref" href="collsp.html#D7">COLLSP:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M'</font>, <font color="Maroon">N</font> = <font color="Maroon">N'</font> as   <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> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>take </b>
<font color="Maroon">N</font>
;<br/>

<b>thus </b><a NAME="E5:18"/>
for <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>  holds <br/>(  not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> or  not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</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"/>
 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">M</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font> ;<br/>
<a NAME="E5:18_2"/>
( <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">p</font>,<font color="Maroon">p1</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">r</font>,<font color="Maroon">r1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:18_2"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">p1</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r</font>,<font color="Maroon">r1</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="collsp.html#T17">COLLSP:17</a></i>;<br/>
<b>hence </b><a NAME="E7:18_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


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