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

<b>let </b><font color="Maroon">V</font> be  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>















<b>assume </b><a NAME="E1:85"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p1</font>,<font color="Maroon">r2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">r1</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r2</font>,<font color="Maroon">r1</font>,<font color="Maroon">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:85"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">u</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">v</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">z</font> &amp; <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">z</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">z</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:85"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">w</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">w</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">z1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">z1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">w1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:85"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">x</font> &amp; <font color="Maroon">u1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">w1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">x</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u1</font>,<font color="Maroon">w1</font>,<font color="Maroon">x</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:85"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">v1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">y1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">x1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">u2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:85"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">u2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">t</font> &amp; <font color="Maroon">u2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">y2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">t</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u2</font>,<font color="Maroon">y2</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">v2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">w2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">t1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E13:85"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">w2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">v2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">w2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">t1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v2</font>,<font color="Maroon">w2</font>,<font color="Maroon">t1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">t2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E15:85"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">x2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">z2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">t2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">x2</font>,<font color="Maroon">z2</font>,<font color="Maroon">t2</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E16:85"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">z</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">x</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">y</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<a NAME="E1:85_1"/><i><font color="Green">E71</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">z1</font>,<font color="Maroon">z</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w1</font>,<font color="Maroon">w</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:85_1"><i><font color="Green">E77</font></i></a>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E2:85_1"/><i><font color="Green">E72</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">y1</font>,<font color="Maroon">y</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">x1</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E3:85_1"/><i><font color="Green">E73</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">u</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">y2</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:85_1"><i><font color="Green">E77</font></i></a>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E4:85_1"/><i><font color="Green">E74</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v2</font>,<font color="Maroon">v</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w2</font>,<font color="Maroon">w</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:85_1"><i><font color="Green">E77</font></i></a>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E5:85_1"/><i><font color="Green">E75</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">t1</font>,<font color="Maroon">t</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">x2</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, , , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E6:85_1"/><i><font color="Green">E76</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">z2</font>,<font color="Maroon">z</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">t2</font>,<font color="Maroon">t</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:85_1"><i><font color="Green">E77</font></i></a>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E7:85_1"/><i><font color="Green">E77</font></i>: 
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:85_1"><i><font color="Green">E77</font></i></a>, , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>thus </b><a NAME="E8:85_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">z</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E7:85_1"><i><font color="Green">E77</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:85_1"/>
<font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>

<b>thus </b><a NAME="E10:85_1"/>
<font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">x</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>

<b>thus </b><a NAME="E11:85_1"/>
<font color="Maroon">v</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , , <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>

<b>thus </b><a NAME="E12:85_1"/>
<font color="Maroon">u</font>,<font color="Maroon">y</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>

<b>thus </b><a NAME="E13:85_1"/>
<font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , , <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>

<b>thus </b><a NAME="E14:85_1"/>
<font color="Maroon">x</font>,<font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , , <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:85"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  or <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2">are_LinDep</a>  or <font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  or <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, , , <a class="ref" href="anproj_1.html#T23">ANPROJ_1:23</a></i>;<br/>
<b>hence </b><a NAME="E18:85"/>
( <font color="Maroon">p1</font>,<font color="Maroon">r2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">p1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">p1</font>,<font color="Maroon">r1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">r2</font>,<font color="Maroon">r1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E7:85"><i><font color="Green">E61</font></i></a></i>;<br/>


</div>
