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

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green" title="E68">A3</font></i>: 
<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<div><i><font color="Green" title="E69">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E70">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><b>take </b><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c5">r</font>;<br/><a NAME="E2:70_1"/><i><font color="Green" title="E71">A6</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:70_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><a NAME="E3:70_1"/>
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:70_1"/>
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:70_1"><i><font color="Green" title="E71">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2"/><i><font color="Green" title="E70">A7</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><div><i><font color="Green" title="E71">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2_1"/><i><font color="Green" title="E72">A9</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 ;<br/><b>take </b><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c5">r</font>;<br/><a NAME="E2:70_2_1"/><i><font color="Green" title="E73">A10</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_2_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:70"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="anproj_2.html#E2:70"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="anproj_2.html#E1:70_2_1"><i><font color="Green" title="E72">A9</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:70_2_1_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:70_2_1"/>
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:70_2_1"/>
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:70_2_1"><i><font color="Green" title="E73">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2_2"/><i><font color="Green" title="E72">A11</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><b>take </b><font color="Maroon" title="c7">r1</font> = <font color="Maroon" title="c6">r1</font>;<br/><a NAME="E2:70_2_2"/><i><font color="Green" title="E73">A12</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c7">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_2_2_1"/>
( <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c7">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:70"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="anproj_2.html#E3:70"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E2:70_2_2_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c7">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:70_2_2"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:70_2_2"/>
<font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c7">r1</font>,<font color="Maroon" title="c7">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:70_2_2"/>
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:70_2_2"><i><font color="Green" title="E73">A12</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:70_2"/>
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:70_2"><i><font color="Green" title="E70">A7</font></i></a>, <a class="txt" href="anproj_2.html#E2:70_2"><i><font color="Green" title="E71">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:70"/>
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:70"><i><font color="Green" title="E69">A4</font></i></a></i>;<br/>


</div>
