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

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">p1</font>, <font color="Olive">q</font>, <font color="Olive">q1</font>, <font color="Olive">r2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>  ex <font color="Olive">r</font>, <font color="Olive">r1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font> st <br/>( <font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">q1</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">r2</font>,<font color="Olive">r</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</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">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>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font> ;<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="incproj.html#M1" target="_self">LINE</a> of <font color="Maroon">CPS</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>, <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="E5:19"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">M'</font> <a href="hidden.html#R1">=</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="collsp.html#D7">COLLSP:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>, <font color="Maroon">q1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font><b> such that </b><br/><a NAME="E7:19"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">N'</font> <a href="hidden.html#R1">=</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="collsp.html#D7">COLLSP:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">b'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>, <font color="Maroon">c'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font><b> such that </b><br/><a NAME="E9:19"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">b'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E10:19"/><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/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b'</font>, <font color="Maroon">c</font> = <font color="Maroon">c'</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="E12:19"/>
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 
<b>by </b><i>, , , <a class="ref" href="collsp.html#T17">COLLSP:17</a></i>;<br/>
<a NAME="E13:19"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E14:19"/>
 ex <font color="Olive">S</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">S</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">S</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">S</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E1:19"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E15:19"/>
 ex <font color="Olive">b</font>, <font color="Olive">c</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> ex <font color="Olive">S</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">S</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">S</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">S</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="incproj.html#T2" target="_self">Th2</a></i>;<br/>


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