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

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green" title="E87">A1</font></i>: 
 <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font> is <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green" title="E88">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>  ex <font color="Olive" title="b3">r</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="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D10" target="_self" title="ANPROJ_2:def.10">ANPROJ_2:def 10</a><br/>

<b>given </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q1</font>, <font color="Maroon" title="c4">q2</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><b> such that </b><a NAME="E4:97"/><i><font color="Green" title="E89">A3</font></i>: 
not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E5:97"/><i><font color="Green" title="E90">A4</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">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>  ex <font color="Olive" title="b3">q3</font>, <font color="Olive" title="b4">r3</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="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font>,<font color="Olive" title="b3">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b4">r3</font>,<font color="Olive" title="b3">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>,  <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>,  <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>] <b>means </b>for <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</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>  ex <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x1</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="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b4">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; $2,$3,<font color="Olive" title="b3">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; $1,<font color="Olive" title="b4">x1</font>,<font color="Olive" title="b3">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  );<br/>
<a NAME="E6:97"/><i><font color="Green" title="E91">A5</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</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 <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</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><a NAME="E1:97_1"/><i><font color="Green" title="E92">A6</font></i>: 
<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c5">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</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/><a NAME="E1:97_1_1"/>
( <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font>,<font color="Maroon" title="c9">y2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c5">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c9">y2</font>,<font color="Maroon" title="c5">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_1"><i><font color="Green" title="E92">A6</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="E2:97_1_1"/>
 ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">x1</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="c8">y1</font>,<font color="Maroon" title="c9">y2</font>,<font color="Olive" title="b2">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Olive" title="b2">x1</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:97"/><i><font color="Green" title="E92">A7</font></i>: 
for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font>, <font color="Olive" title="b5">q</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 not <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>,<font color="Olive" title="b5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; ( for <font color="Olive" title="b6">x</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>  holds <br/>(  not <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>,<font color="Olive" title="b6">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b6">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) holds <br/> ex <font color="Olive" title="b6">q3</font>, <font color="Olive" title="b7">p3</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="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b7">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>,<font color="Olive" title="b6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b6">q3</font>,<font color="Olive" title="b7">p3</font>,<font color="Olive" title="b5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q1</font>, <font color="Maroon" title="c6">q2</font>, <font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font>, <font color="Maroon" title="c9">q</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="E1:97_2"/><i><font color="Green" title="E93">A8</font></i>: 
( not <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E2:97_2"/>
for <font color="Olive" title="b1">x</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>  holds <br/>(  not <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<a NAME="E3:97_2"/><i><font color="Green" title="E94">A9</font></i>: 
( <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c8">p2</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/>
<a NAME="E4:97_2"/><i><font color="Green" title="E95">A10</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q1</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:97_2"><i><font color="Green" title="E93">A8</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<a NAME="E5:97_2"/>
(  not <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_2_1"/>
( <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<a NAME="E2:97_2_1"/><b>then </b><i><font color="Green" title="E96">A11</font></i>: 
( <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c7">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c8">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<a NAME="E3:97_2_1"/>
<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</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="E4:97_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_2"><i><font color="Green" title="E93">A8</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_2"><i><font color="Green" title="E95">A10</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_2_1"><i><font color="Green" title="E96">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>
<b>hence </b><a NAME="E6:97_2"/>
 ex <font color="Olive" title="b1">q3</font>, <font color="Olive" title="b2">p3</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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Olive" title="b2">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Olive" title="b1">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">q3</font>,<font color="Olive" title="b2">p3</font>,<font color="Maroon" title="c9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_2"><i><font color="Green" title="E94">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:97"/><i><font color="Green" title="E93">A12</font></i>: 
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>] &amp; not <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b6">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>,<font color="Olive" title="b1">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>,<font color="Olive" title="b6">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</font>, <font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font>, <font color="Maroon" title="c10">x</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="E1:97_3"/><i><font color="Green" title="E94">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>]
 <b>and </b><br/><a NAME="E2:97_3"/><i><font color="Green" title="E95">A14</font></i>: 
not <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_3"/><i><font color="Green" title="E96">A15</font></i>: 
<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c10">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97_3"/><i><font color="Green" title="E97">A16</font></i>: 
not <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E5:97_3"/><i><font color="Green" title="E98">A17</font></i>: 
<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c10">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<a NAME="E6:97_3"/><i><font color="Green" title="E99">A18</font></i>: 
( <font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:97_3"><i><font color="Green" title="E95">A14</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<a NAME="E7:97_3"/><i><font color="Green" title="E100">A19</font></i>: 
( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3"><i><font color="Green" title="E97">A16</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">y1</font>, <font color="Maroon" title="c12">y2</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/><div><i><font color="Green" title="E101">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_1"/>
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font>
 ;<br/><a NAME="E2:97_3_1_1"/><b>then </b>
( <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">p1</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="E3:97_3_1_1"/>
 ex <font color="Olive" title="b1">z2</font>, <font color="Olive" title="b2">z1</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b2">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">z1</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2"/><i><font color="Green" title="E102">A21</font></i>: 
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font>
 ;<br/><b>consider </b><font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c14">b'</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><b> such that </b><br/><a NAME="E3:97_3_1_2"/><i><font color="Green" title="E103">A22</font></i>: 
( <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c14">b'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c13">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c13">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3"><i><font color="Green" title="E94">A13</font></i></a></i>;<br/><a NAME="E4:97_3_1_2"/>
 ex <font color="Olive" title="b1">a</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="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_1"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E104">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_1_1"/><i><font color="Green" title="E105">A24</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p1</font>
 ;<br/><b>take </b><font color="Maroon" title="c15">p1</font> = <font color="Maroon" title="c8">p1</font>;<br/><a NAME="E2:97_3_1_2_1_1"/>
<font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c15">p1</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="E3:97_3_1_2_1_1"/>
 ex <font color="Olive" title="b1">a</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="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_1_1"><i><font color="Green" title="E105">A24</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_1_2"/><i><font color="Green" title="E105">A25</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p2</font>
 ;<br/><b>take </b><font color="Maroon" title="c15">p2</font> = <font color="Maroon" title="c9">p2</font>;<br/><a NAME="E2:97_3_1_2_1_2"/>
<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c15">p2</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="E3:97_3_1_2_1_2"/>
 ex <font color="Olive" title="b1">a</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="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_1_2"><i><font color="Green" title="E105">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3_1_2_1"/>
 ex <font color="Olive" title="b1">a</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="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3"><i><font color="Green" title="E97">A16</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_1"><i><font color="Green" title="E104">A23</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">x1</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><b> such that </b><br/><a NAME="E6:97_3_1_2"/><i><font color="Green" title="E104">A26</font></i>: 
( <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c15">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">x1</font> )
 ;<br/><a NAME="E7:97_3_1_2"/>
 ex <font color="Olive" title="b1">a</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_2"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E105">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_2_1"/><i><font color="Green" title="E106">A28</font></i>: 
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y1</font>
 ;<br/><b>take </b><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c11">y1</font>;<br/><a NAME="E2:97_3_1_2_2_1"/>
<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c16">y1</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="E3:97_3_1_2_2_1"/>
 ex <font color="Olive" title="b1">a</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_2_1"><i><font color="Green" title="E106">A28</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_2_2"/><i><font color="Green" title="E106">A29</font></i>: 
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font>
 ;<br/><b>take </b><font color="Maroon" title="c16">y2</font> = <font color="Maroon" title="c12">y2</font>;<br/><a NAME="E2:97_3_1_2_2_2"/>
<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c16">y2</font>,<font color="Maroon" title="c16">y2</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="E3:97_3_1_2_2_2"/>
 ex <font color="Olive" title="b1">a</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_2_2"><i><font color="Green" title="E106">A29</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3_1_2_2"/>
 ex <font color="Olive" title="b1">a</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b1">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2"><i><font color="Green" title="E102">A21</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_2"><i><font color="Green" title="E105">A27</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">x3</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><b> such that </b><br/><a NAME="E9:97_3_1_2"/><i><font color="Green" title="E105">A30</font></i>: 
( <font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x3</font> &amp; <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c16">x3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/><b>consider </b><font color="Maroon" title="c17">d</font>, <font color="Maroon" title="c18">d'</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><b> such that </b><br/><a NAME="E11:97_3_1_2"/><i><font color="Green" title="E106">A31</font></i>: 
( <font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c18">d'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c17">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">d'</font>,<font color="Maroon" title="c17">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3"><i><font color="Green" title="E94">A13</font></i></a></i>;<br/><a NAME="E12:97_3_1_2"/><i><font color="Green" title="E107">A32</font></i>: 
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c10">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_3"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_3"><i><font color="Green" title="E99">A18</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><div><i><font color="Green" title="E108">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_3"/><i><font color="Green" title="E109">A34</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">d</font>
 ;<br/><a NAME="E2:97_3_1_2_3"/><i><font color="Green" title="E110">A35</font></i>: 
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c14">b'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E3:97_3_1_2_3"/><i><font color="Green" title="E111">A36</font></i>: 
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">d'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_3"><i><font color="Green" title="E109">A34</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E4:97_3_1_2_3"/>
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c5">q</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/><a NAME="E5:97_3_1_2_3"/><b>then </b><i><font color="Green" title="E112">A37</font></i>: 
<font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c18">d'</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_3"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_3_1_2_3"><i><font color="Green" title="E110">A35</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2_3"><i><font color="Green" title="E111">A36</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E6:97_3_1_2_3"/>
<font color="Maroon" title="c18">d'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c15">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">z1</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><b> such that </b><br/><a NAME="E8:97_3_1_2_3"/><i><font color="Green" title="E113">A38</font></i>: 
( <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c19">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c19">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E5:97_3_1_2_3"><i><font color="Green" title="E112">A37</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E9:97_3_1_2_3"/>
( <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c11">y1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c12">y2</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/><a NAME="E10:97_3_1_2_3"/><b>then </b>
( <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c11">y1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c12">y2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2"><i><font color="Green" title="E102">A21</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green" title="E105">A30</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E11:97_3_1_2_3"/><b>then </b><i><font color="Green" title="E114">A39</font></i>: 
<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c19">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green" title="E105">A30</font></i></a>, <a class="txt" href="anproj_2.html#E8:97_3_1_2_3"><i><font color="Green" title="E113">A38</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E12:97_3_1_2_3"/>
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c19">z1</font>,<font color="Maroon" title="c15">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:97_3_1_2_3"><i><font color="Green" title="E113">A38</font></i></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="E13:97_3_1_2_3"/>
 ex <font color="Olive" title="b1">z2</font>, <font color="Olive" title="b2">z1</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b2">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">z1</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3_1_2"><i><font color="Green" title="E104">A26</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2_3"><i><font color="Green" title="E114">A39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_4"/><i><font color="Green" title="E109">A40</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">d</font>
 ;<br/><a NAME="E2:97_3_1_2_4"/>
not <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_3_1_2_4_1"/>
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<a NAME="E2:97_3_1_2_4_1"/><b>then </b><i><font color="Green" title="E110">A41</font></i>: 
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<a NAME="E3:97_3_1_2_4_1"/><i><font color="Green" title="E111">A42</font></i>: 
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_1_1"/>
<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c6">q1</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="E2:97_3_1_2_4_1_1"/>
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3"><i><font color="Green" title="E99">A18</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</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="E4:97_3_1_2_4_1"/>
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c7">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_1_2"/>
<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c7">q2</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="E2:97_3_1_2_4_1_2"/>
<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c7">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3"><i><font color="Green" title="E99">A18</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</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>
<b>hence </b><a NAME="E5:97_3_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_3"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_4"><i><font color="Green" title="E109">A40</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_3_1_2_4_1"><i><font color="Green" title="E110">A41</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2_4_1"><i><font color="Green" title="E111">A42</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><b>then consider </b><font color="Maroon" title="c19">o</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><b> such that </b><br/><a NAME="E4:97_3_1_2_4"/><i><font color="Green" title="E110">A43</font></i>: 
( <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c18">d'</font>,<font color="Maroon" title="c19">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c19">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</font></i></a>, <a class="txt" href="anproj_2.html#E12:97_3_1_2"><i><font color="Green" title="E107">A32</font></i></a>, <a class="txt" href="anproj_2.html#E92"><i><font color="Green" title="E85">Lm45</font></i></a></i>;<br/><a NAME="E5:97_3_1_2_4"/>
<font color="Maroon" title="c18">d'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c15">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green" title="E106">A31</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">z1</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><b> such that </b><br/><a NAME="E7:97_3_1_2_4"/><i><font color="Green" title="E111">A44</font></i>: 
( <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c20">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c20">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3_1_2_4"><i><font color="Green" title="E110">A43</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E8:97_3_1_2_4"/>
( <font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c20">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3_1_2_4"><i><font color="Green" title="E110">A43</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_3_1_2_4"><i><font color="Green" title="E111">A44</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">z2</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><b> such that </b><br/><a NAME="E10:97_3_1_2_4"/><i><font color="Green" title="E112">A45</font></i>: 
( <font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c21">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c20">z1</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E11:97_3_1_2_4"/><i><font color="Green" title="E113">A46</font></i>: 
<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c20">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_2"/>
( <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c11">y1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c12">y2</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/>
<a NAME="E2:97_3_1_2_4_2"/><b>then </b>
( <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c11">y1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c12">y2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2"><i><font color="Green" title="E102">A21</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green" title="E105">A30</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1_2_4_2"/>
<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c20">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green" title="E105">A30</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_3_1_2_4"><i><font color="Green" title="E111">A44</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="E12:97_3_1_2_4"/><i><font color="Green" title="E114">A47</font></i>: 
<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c21">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_3"/>
( <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c9">p2</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/>
<a NAME="E2:97_3_1_2_4_3"/><b>then </b>
( <font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c8">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E5:97_3"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_3"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_3_1_2"><i><font color="Green" title="E104">A26</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1_2_4_3"/>
<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c21">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3_1_2"><i><font color="Green" title="E104">A26</font></i></a>, <a class="txt" href="anproj_2.html#E10:97_3_1_2_4"><i><font color="Green" title="E112">A45</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="E13:97_3_1_2_4"/>
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">z1</font>,<font color="Maroon" title="c21">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E10:97_3_1_2_4"><i><font color="Green" title="E112">A45</font></i></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="E14:97_3_1_2_4"/>
 ex <font color="Olive" title="b1">z2</font>, <font color="Olive" title="b2">z1</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b2">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">z1</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2_4"><i><font color="Green" title="E113">A46</font></i></a>, <a class="txt" href="anproj_2.html#E12:97_3_1_2_4"><i><font color="Green" title="E114">A47</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:97_3_1_2"/>
 ex <font color="Olive" title="b1">z2</font>, <font color="Olive" title="b2">z1</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b2">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">z1</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green" title="E108">A33</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:97_3_1"/>
 ex <font color="Olive" title="b1">z2</font>, <font color="Olive" title="b2">z1</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="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Olive" title="b2">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">z1</font>,<font color="Olive" title="b1">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1"><i><font color="Green" title="E101">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:97_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:97"/><i><font color="Green" title="E94">A48</font></i>: 
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>] &amp; not <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>,<font color="Olive" title="b1">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; ( for <font color="Olive" title="b6">x</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>  holds <br/>(  not <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b6">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>,<font color="Olive" title="b6">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</font>, <font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</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="E1:97_4"/><i><font color="Green" title="E95">A49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>]
 <b>and </b><br/><a NAME="E2:97_4"/><i><font color="Green" title="E96">A50</font></i>: 
not <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_4"/><i><font color="Green" title="E97">A51</font></i>: 
not <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97_4"/><i><font color="Green" title="E98">A52</font></i>: 
for <font color="Olive" title="b1">x</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>  holds <br/>(  not <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">q3</font>, <font color="Maroon" title="c11">p3</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><b> such that </b><br/><a NAME="E6:97_4"/><i><font color="Green" title="E99">A53</font></i>: 
( <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c11">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c10">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c10">q3</font>,<font color="Maroon" title="c11">p3</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E7:97"><i><font color="Green" title="E92">A7</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_4"><i><font color="Green" title="E96">A50</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_4"><i><font color="Green" title="E97">A51</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_4"><i><font color="Green" title="E98">A52</font></i></a></i>;<br/>
<a NAME="E7:97_4"/><i><font color="Green" title="E100">A54</font></i>: 
( <font color="Maroon" title="c10">q3</font>,<font color="Maroon" title="c11">p3</font>,<font color="Maroon" title="c10">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">q3</font>,<font color="Maroon" title="c11">p3</font>,<font color="Maroon" title="c11">p3</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/>
<a NAME="E8:97_4"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c10">q3</font>,<font color="Maroon" title="c11">p3</font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:97"><i><font color="Green" title="E93">A12</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_4"><i><font color="Green" title="E95">A49</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_4"><i><font color="Green" title="E96">A50</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_4"><i><font color="Green" title="E99">A53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:97_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:97"><i><font color="Green" title="E93">A12</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_4"><i><font color="Green" title="E97">A51</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_4"><i><font color="Green" title="E99">A53</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_4"><i><font color="Green" title="E100">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:97"/><i><font color="Green" title="E95">A55</font></i>: 
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>] &amp; not <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/>for <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</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="E1:97_5"/><i><font color="Green" title="E96">A56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>]
 <b>and </b><br/><a NAME="E2:97_5"/><i><font color="Green" title="E97">A57</font></i>: 
not <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</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/>



<a NAME="E3:97_5"/><i><font color="Green" title="E98">A58</font></i>: 
( not <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp;  ex <font color="Olive" title="b1">x</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="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>] )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:97"><i><font color="Green" title="E93">A12</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_5"><i><font color="Green" title="E96">A56</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_5"><i><font color="Green" title="E97">A57</font></i></a></i>;<br/>
<a NAME="E4:97_5"/>
( not <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; ( for <font color="Olive" title="b1">x</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>  holds <br/>(  not <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>] )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E9:97"><i><font color="Green" title="E94">A48</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_5"><i><font color="Green" title="E96">A56</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_5"><i><font color="Green" title="E97">A57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:97_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_5"><i><font color="Green" title="E98">A58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:97"/><i><font color="Green" title="E96">A59</font></i>: 
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">q3</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>] &amp; not <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Olive" title="b4">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">q</font>,<font color="Olive" title="b5">q3</font>,<font color="Olive" title="b4">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b5">q3</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">q3</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="E1:97_6"/><i><font color="Green" title="E97">A60</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>]
 <b>and </b><br/><a NAME="E2:97_6"/><i><font color="Green" title="E98">A61</font></i>: 
not <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_6"/><i><font color="Green" title="E99">A62</font></i>: 
( <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c8">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c8">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</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>consider </b><font color="Maroon" title="c12">z2</font>, <font color="Maroon" title="c13">z1</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><b> such that </b><br/><a NAME="E2:97_6_1"/><i><font color="Green" title="E100">A63</font></i>: 
( <font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c13">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c12">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">z1</font>,<font color="Maroon" title="c12">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_6"><i><font color="Green" title="E97">A60</font></i></a></i>;<br/><div><i><font color="Green" title="E101">A64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_6_1_1"/><i><font color="Green" title="E102">A65</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font>
 ;<br/><a NAME="E2:97_6_1_1"/><b>then </b>
( <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c13">z1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c8">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green" title="E99">A62</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green" title="E100">A63</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/><a NAME="E3:97_6_1_1"/><b>then </b>
<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c13">z1</font>,<font color="Maroon" title="c12">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6"><i><font color="Green" title="E98">A61</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green" title="E99">A62</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_6_1_1"><i><font color="Green" title="E102">A65</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E4:97_6_1_1"/>
 ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">x1</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="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Olive" title="b2">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Olive" title="b2">x1</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green" title="E100">A63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_6_1_2"/><i><font color="Green" title="E102">A66</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">z2</font>
 ;<br/><a NAME="E2:97_6_1_2"/>
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">z1</font>,<font color="Maroon" title="c12">z2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green" title="E99">A62</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green" title="E100">A63</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">x2</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><b> such that </b><br/><a NAME="E4:97_6_1_2"/><i><font color="Green" title="E103">A67</font></i>: 
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c13">z1</font>,<font color="Maroon" title="c14">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c14">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E5:97_6_1_2"/><i><font color="Green" title="E104">A68</font></i>: 
<font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2</font>
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6"><i><font color="Green" title="E98">A61</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><a NAME="E6:97_6_1_2"/>
( <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c7">q2</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/><a NAME="E7:97_6_1_2"/><b>then </b>
( <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c7">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green" title="E99">A62</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green" title="E100">A63</font></i></a>, <a class="txt" href="anproj_2.html#E5:97_6_1_2"><i><font color="Green" title="E104">A68</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E8:97_6_1_2"/><b>then </b>
<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c14">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_6_1_2"><i><font color="Green" title="E102">A66</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_6_1_2"><i><font color="Green" title="E103">A67</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E9:97_6_1_2"/>
 ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">x1</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="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Olive" title="b2">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Olive" title="b2">x1</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green" title="E100">A63</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_6_1_2"><i><font color="Green" title="E103">A67</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:97_6_1"/>
 ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">x1</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="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Olive" title="b2">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Olive" title="b2">x1</font>,<font color="Olive" title="b1">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6_1"><i><font color="Green" title="E101">A64</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:97_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:97"/><i><font color="Green" title="E97">A69</font></i>: 
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">p</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 ( for <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>] ) holds <br/> ex <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</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">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>] &amp; not <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b2">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">p</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><a NAME="E1:97_7"/><i><font color="Green" title="E98">A70</font></i>: 
for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">x1</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><b> such that </b><br/><a NAME="E3:97_7"/><i><font color="Green" title="E99">A71</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97"><i><font color="Green" title="E88">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x2</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><b> such that </b><br/><a NAME="E5:97_7"/><i><font color="Green" title="E100">A72</font></i>: 
not <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green" title="E99">A71</font></i></a>, <a class="ref" href="collsp.html#T19" title="COLLSP:th.19">COLLSP:19</a></i>;<br/>
<a NAME="E6:97_7"/><i><font color="Green" title="E101">A73</font></i>: 
not <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E5:97_7"><i><font color="Green" title="E100">A72</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<a NAME="E7:97_7"/><i><font color="Green" title="E102">A74</font></i>: 
not <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_7_1"/>
<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<a NAME="E2:97_7_1"/><b>then </b>
( <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c7">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green" title="E99">A71</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="E3:97_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green" title="E99">A71</font></i></a>, <a class="txt" href="anproj_2.html#E5:97_7"><i><font color="Green" title="E100">A72</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="E8:97_7"/><i><font color="Green" title="E103">A75</font></i>: 
( <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">x1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green" title="E99">A71</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/>
<a NAME="E9:97_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:97_7"><i><font color="Green" title="E98">A70</font></i></a></i>;<br/>
<a NAME="E10:97_7"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E11:97"><i><font color="Green" title="E96">A59</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_7"><i><font color="Green" title="E102">A74</font></i></a>, <a class="txt" href="anproj_2.html#E8:97_7"><i><font color="Green" title="E103">A75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:97_7"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">p</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>] &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_7"><i><font color="Green" title="E101">A73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:97"/><i><font color="Green" title="E98">A76</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">z</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">z</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/>





<a NAME="E1:97_8"/>
not <font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E4:97"><i><font color="Green" title="E89">A3</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<a NAME="E2:97_8"/><b>then </b>
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E5:97"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="anproj_2.html#E10:97"><i><font color="Green" title="E95">A55</font></i></a></i>;<br/>
<a NAME="E3:97_8"/><b>then </b>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>] &amp; not <font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>,<font color="Maroon" title="c5">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E12:97"><i><font color="Green" title="E97">A69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:97_8"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z</font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E10:97"><i><font color="Green" title="E95">A55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:97"/><i><font color="Green" title="E99">A77</font></i>: 
for <font color="Olive" title="b1">p4</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q4</font>, <font color="Olive" title="b5">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>  ex <font color="Olive" title="b6">r</font>, <font color="Olive" title="b7">r1</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="Olive" title="b1">p4</font>,<font color="Olive" title="b3">q</font>,<font color="Olive" title="b6">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">q4</font>,<font color="Olive" title="b7">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">r2</font>,<font color="Olive" title="b6">r</font>,<font color="Olive" title="b7">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p4</font>, <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">q4</font>, <font color="Maroon" title="c9">r2</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/>









<a NAME="E1:97_9"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r</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="c5">p4</font>,<font color="Maroon" title="c7">q</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c8">q4</font>,<font color="Olive" title="b1">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">r2</font>,<font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">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#E13:97"><i><font color="Green" title="E98">A76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:97_9"/>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">r1</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="c5">p4</font>,<font color="Maroon" title="c7">q</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c8">q4</font>,<font color="Olive" title="b2">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">r2</font>,<font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c5">CS</font> =  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font> as   <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a> <b>by </b><i><a class="txt" href="anproj_2.html#E1:97"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="anproj_2.html#E2:97"><i><font color="Green" title="E88">A2</font></i></a>, <a class="ref" href="anproj_2.html#D10" target="_self" title="ANPROJ_2:def.10">Def10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">CS</font>
;<br/>

<b>thus </b><a NAME="E16:97"/>
( <font color="Maroon" title="c5">CS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c5">CS</font> is <a href="anproj_2.html#V8" title="ANPROJ_2:attr.8">at_most-3-dimensional</a> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E14:97"><i><font color="Green" title="E99">A77</font></i></a>, <a class="ref" href="anproj_2.html#D15" target="_self" title="ANPROJ_2:def.15">Def15</a></i>;<br/>


</div>
