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

<b>let </b><font color="Maroon">V</font> be  <a href="anproj_2.html#V1" target="_self">up-3-dimensional</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">o</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>, <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>, <font color="Maroon">p2</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>, <font color="Maroon">p3</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>, <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>, <font color="Maroon">q2</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>, <font color="Maroon">q3</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>, <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>, <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>, <font color="Maroon">r3</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D13" target="_self">ANPROJ_2:def 13</a><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> )
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E59</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E4:88"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p3</font>,<font color="Maroon">q1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">q3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">o'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">p1'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">p2'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:88"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">o'</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p1'</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p2'</font> &amp; <font color="Maroon">o'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p1'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p2'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o'</font>,<font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E81</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">o''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">p1''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">p3'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:88"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">o''</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p1''</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p3'</font> &amp; <font color="Maroon">o''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p1''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p3'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o''</font>,<font color="Maroon">p1''</font>,<font color="Maroon">p3'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E81</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E9:88"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">o'</font>,<font color="Maroon">p1'</font>,<font color="Maroon">p3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_1"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o''</font>,<font color="Maroon">o'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p1''</font>,<font color="Maroon">p1'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E84</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">p1'</font>,<font color="Maroon">p3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E84</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">o'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q1'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q2'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:88"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">o'''</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q1'</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q2'</font> &amp; <font color="Maroon">o'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q1'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q2'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o'''</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q2'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E81</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E12:88"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">o'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q2'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>proof </b></a><div class="add">

<a NAME="E1:88_2"/>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o'''</font>,<font color="Maroon">o'</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q2'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">o''''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q1''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q3'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E14:88"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">o''''</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q1''</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q3'</font> &amp; <font color="Maroon">o''''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q1''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q3'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o''''</font>,<font color="Maroon">q1''</font>,<font color="Maroon">q3'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E81</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E15:88"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">o'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><b>proof </b></a><div class="add">

<a NAME="E1:88_3"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o''''</font>,<font color="Maroon">o'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q1''</font>,<font color="Maroon">q1'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_3"/>
<font color="Maroon">o'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:88"/>
not <font color="Maroon">o'</font>,<font color="Maroon">p1'</font>,<font color="Maroon">q1'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E26:88"><i><font color="Green">E80</font></i></a>, <a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E17:88"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">o'</font>,<font color="Maroon">p1'</font>,<font color="Maroon">p2'</font>,<font color="Maroon">p3'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">q3'</font> <a href="anproj_2.html#R4" target="_self">lie_on_an_angle</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green">E85</font></i></a>, <a class="txt" href="anproj_2.html#E37:88"><i><font color="Green">E87</font></i></a>, , </i>;<br/>
<a NAME="E18:88"/>
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o'</font>,<font color="Maroon">p2'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o'</font>,<font color="Maroon">p3'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o'</font>,<font color="Maroon">q2'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o'</font>,<font color="Maroon">q3'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p1'</font>,<font color="Maroon">p3'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q1'</font>,<font color="Maroon">q2'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q1'</font>,<font color="Maroon">q3'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p2'</font>,<font color="Maroon">p3'</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q2'</font>,<font color="Maroon">q3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E25:88"><i><font color="Green">E79</font></i></a>, <a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E19:88"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">o'</font>,<font color="Maroon">p1'</font>,<font color="Maroon">p2'</font>,<font color="Maroon">p3'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">q3'</font> <a href="anproj_2.html#R5" target="_self">are_half_mutually_not_Prop</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E20:88"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font>,<font color="Maroon">p3'</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">q1'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">q3'</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E88</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">p1'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q2'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">r3'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E22:88"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p1'''</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q2'''</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">r3'</font> &amp; <font color="Maroon">p1'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q2'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r3'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p1'''</font>,<font color="Maroon">q2'''</font>,<font color="Maroon">r3'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E23:88"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p1'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">r3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_4"><b>proof </b></a><div class="add">

<a NAME="E1:88_4"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p1'''</font>,<font color="Maroon">p1'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q2'''</font>,<font color="Maroon">q2'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_4"/>
<font color="Maroon">p1'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">r3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">q1'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">p2'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">r3''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E25:88"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q1'''</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p2'''</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">r3''</font> &amp; <font color="Maroon">q1'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p2'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r3''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q1'''</font>,<font color="Maroon">p2'''</font>,<font color="Maroon">r3''</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E26:88"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">q1'</font>,<font color="Maroon">p2'</font>,<font color="Maroon">r3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_5"><b>proof </b></a><div class="add">

<a NAME="E1:88_5"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q1'''</font>,<font color="Maroon">q1'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p2'''</font>,<font color="Maroon">p2'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">r3''</font>,<font color="Maroon">r3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_5"/>
<font color="Maroon">q1'</font>,<font color="Maroon">p2'</font>,<font color="Maroon">r3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p1''''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q3''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">r2'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E28:88"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p1''''</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q3''</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">r2'</font> &amp; <font color="Maroon">p1''''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q3''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r2'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p1''''</font>,<font color="Maroon">q3''</font>,<font color="Maroon">r2'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E29:88"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">p1'</font>,<font color="Maroon">q3'</font>,<font color="Maroon">r2'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_6"><b>proof </b></a><div class="add">

<a NAME="E1:88_6"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p1''''</font>,<font color="Maroon">p1'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q3''</font>,<font color="Maroon">q3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E88</font></i></a>, , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_6"/>
<font color="Maroon">p1'</font>,<font color="Maroon">q3'</font>,<font color="Maroon">r2'</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>
<b>consider </b><font color="Maroon">p3''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q1'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">r2''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E31:88"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p3''</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q1'''</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">r2''</font> &amp; <font color="Maroon">p3''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q1'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r2''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p3''</font>,<font color="Maroon">q1'''</font>,<font color="Maroon">r2''</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E32:88"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">p3'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">r2'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_7"><b>proof </b></a><div class="add">

<a NAME="E1:88_7"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p3''</font>,<font color="Maroon">p3'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q1'''</font>,<font color="Maroon">q1'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">r2''</font>,<font color="Maroon">r2'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_7"/>
<font color="Maroon">p3'</font>,<font color="Maroon">q1'</font>,<font color="Maroon">r2'</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>
<b>consider </b><font color="Maroon">p2''''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q3'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">r1'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E34:88"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p2''''</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q3'''</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">r1'</font> &amp; <font color="Maroon">p2''''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q3'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r1'</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p2''''</font>,<font color="Maroon">q3'''</font>,<font color="Maroon">r1'</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E35:88"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">p2'</font>,<font color="Maroon">q3'</font>,<font color="Maroon">r1'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_8"><b>proof </b></a><div class="add">

<a NAME="E1:88_8"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p2''''</font>,<font color="Maroon">p2'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q3'''</font>,<font color="Maroon">q3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E88</font></i></a>, , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_8"/>
<font color="Maroon">p2'</font>,<font color="Maroon">q3'</font>,<font color="Maroon">r1'</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>
<b>consider </b><font color="Maroon">p3'''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">q2''''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">r1''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E37:88"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">p3'''</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">q2''''</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">r1''</font> &amp; <font color="Maroon">p3'''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q2''''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r1''</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">p3'''</font>,<font color="Maroon">q2''''</font>,<font color="Maroon">r1''</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E38:88"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p3'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">r1'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_9"><b>proof </b></a><div class="add">

<a NAME="E1:88_9"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p3'''</font>,<font color="Maroon">p3'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">q2''''</font>,<font color="Maroon">q2'</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">r1''</font>,<font color="Maroon">r1'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E86</font></i></a>, , , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_9"/>
<font color="Maroon">p3'</font>,<font color="Maroon">q2'</font>,<font color="Maroon">r1'</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="E39:88"/>
<font color="Maroon">r1'</font>,<font color="Maroon">r2'</font>,<font color="Maroon">r3'</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, , , </i>;<br/>
<a NAME="E40:88"/><b>then </b>
<font color="Maroon">r1'</font>,<font color="Maroon">r2'</font>,<font color="Maroon">r3'</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E83</font></i></a>, , , <a class="txt" href="anproj_2.html#E7"><i><font color="Green">Lemma92</font></i></a>, <a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="anproj_2.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E41:88"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, , , <a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E61</font></i></a></i>;<br/>


</div>
