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

<b>let </b><font color="Maroon">V</font> be   <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 <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">o</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">w</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">w2</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">u1</font>,<font color="Maroon">v1</font>,<font color="Maroon">w1</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  )
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">w2</font> <a href="anproj_2.html#R4" target="_self">lie_on_an_angle</a> 
 <b>and </b><br/><a NAME="E3:43"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">w2</font> <a href="anproj_2.html#R5" target="_self">are_half_mutually_not_Prop</a> 
 <b>and </b><br/><a NAME="E4:43"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v2</font>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u2</font>,<font color="Maroon">v</font>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">w2</font>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">w</font>,<font color="Maroon">u2</font>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">w2</font>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">w</font>,<font color="Maroon">v2</font>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 ;<br/>

<a NAME="E5:43"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">o</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &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">w</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &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">u1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v1</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>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:43"/><i><font color="Green">E63</font></i>: 
( not <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">u2</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">w2</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:43"/><i><font color="Green">E69</font></i>: 
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font>,<font color="Maroon">v</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font>,<font color="Maroon">w</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font>,<font color="Maroon">v2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font>,<font color="Maroon">w2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">w2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v2</font>,<font color="Maroon">w2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:43"/><i><font color="Green">E70</font></i>: 
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font>,<font color="Maroon">u</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">u2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T17">ANPROJ_1:17</a></i>;<br/>
<a NAME="E9:43"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">w2</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:43"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">a3</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E13:43"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">b3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , , , </i>;<br/>
<b>consider </b><font color="Maroon">c2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:43"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">d2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> &amp; <font color="Maroon">c2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">d2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , , , </i>;<br/>
<b>consider </b><font color="Maroon">c3</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:43"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">d3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> &amp; <font color="Maroon">c3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">d3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , , , </i>;<br/>
<b>set </b><font color="Maroon">v'</font> = <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">w'</font> = <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">v2'</font> = <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">w2'</font> = <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>;<br/>
<a NAME="E18:43"/><i><font color="Green">E76</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E19:43"/><i><font color="Green">E77</font></i>: 
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E78</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> or  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> )
 ;<br/>

<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1"/>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 ;<br/><a NAME="E2:43_1_1"/><b>then </b>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T6">ANPROJ_1:6</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T6">ANPROJ_1:6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_2"/>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 ;<br/><a NAME="E2:43_1_2"/><b>then </b>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T6">ANPROJ_1:6</a></i>;<br/><b>hence </b><a NAME="E3:43_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T6">ANPROJ_1:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:43"><i><font color="Green">E59</font></i></a>, <a class="txt" href="anproj_2.html#E3:43"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:43"/><i><font color="Green">E80</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2"/>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v2</font>
 ;<br/>

<a NAME="E2:43_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">u</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>hence </b><a NAME="E3:43_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:43"/><i><font color="Green">E81</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E22:43"/><i><font color="Green">E82</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">w2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_3"/>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">w2</font>
 ;<br/>

<a NAME="E2:43_3"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">u</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>hence </b><a NAME="E3:43_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:43"/><i><font color="Green">E83</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E24:43"/>
not <font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_4"/>
<font color="Maroon">o</font>,<font color="Maroon">u2</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 ;<br/>

<a NAME="E2:43_4"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>
<a NAME="E3:43_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>
<a NAME="E4:43_4"/>
<font color="Maroon">o</font>,<font color="Maroon">v</font>,<font color="Maroon">o</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="ref" href="anproj_1.html#T16">ANPROJ_1:16</a></i>;<br/>
<b>hence </b><a NAME="E5:43_4"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="anproj_2.html#E6:43"><i><font color="Green">E63</font></i></a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="anproj_1.html#T19">ANPROJ_1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:43"/><b>then </b><i><font color="Green">E86</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w2</font>
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E26:43"/>
not <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">v2</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_5"/>
<font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">v2</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 ;<br/>

<a NAME="E2:43_5"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">v2</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>
<a NAME="E3:43_5"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>
<a NAME="E4:43_5"/>
<font color="Maroon">o</font>,<font color="Maroon">v2</font>,<font color="Maroon">o</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="ref" href="anproj_1.html#T16">ANPROJ_1:16</a></i>;<br/>
<b>hence </b><a NAME="E5:43_5"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="anproj_2.html#T2" target="_self">Th2</a>, <a class="ref" href="anproj_1.html#T19">ANPROJ_1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:43"/><b>then </b><i><font color="Green">E89</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v2</font>,<font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E28:43"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E29:43"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">u2</font>,<font color="Maroon">w</font>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T10">ANPROJ_1:10</a></i>;<br/>
<a NAME="E30:43"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E31:43"/><i><font color="Green">E101</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E2:43"><i><font color="Green">E59</font></i></a>, </i>;<br/>
<a NAME="E32:43"/><i><font color="Green">E102</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E3:43"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<a NAME="E33:43"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E34:43"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">u2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="txt" href="anproj_2.html#E4"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E35:43"/><i><font color="Green">E154</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E4:43"><i><font color="Green">E61</font></i></a>, </i>;<br/>
<a NAME="E36:43"/><i><font color="Green">E155</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u2</font>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E5:43"><i><font color="Green">E62</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">A3</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E38:43"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B3</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="anproj_2.html#T2" target="_self">Th2</a>, <a class="txt" href="anproj_2.html#E5"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="anproj_2.html#T3" target="_self">Th3</a>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon">A3'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B3'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E40:43"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A3'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B3'</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="anproj_2.html#T2" target="_self">Th2</a>, <a class="txt" href="anproj_2.html#E5"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="anproj_2.html#E7:43"><i><font color="Green">E69</font></i></a>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/>
<a NAME="E41:43"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">A3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E15:43"><i><font color="Green">E74</font></i></a>, </i>;<br/>
<a NAME="E42:43"/>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B3'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B3'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">A3'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E17:43"><i><font color="Green">E75</font></i></a>, </i>;<br/>
<a NAME="E43:43"/><b>then </b><i><font color="Green">E159</font></i>: 
( <font color="Maroon">B3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3'</font> &amp; <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">B3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A3'</font> )
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E18:43"><i><font color="Green">E76</font></i></a>, <a class="ref" href="anproj_1.html#T13">ANPROJ_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">w1'</font> = <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>;<br/>
<a NAME="E44:43"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_6"/>
not <font color="Maroon">B3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then </b><font color="Maroon">w1</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="anproj_2.html#E15:43"><i><font color="Green">E74</font></i></a>, <a class="txt" href="anproj_2.html#E19:43"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E3:43_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:43"/>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E18:43"><i><font color="Green">E76</font></i></a>, <a class="txt" href="anproj_2.html#E19:43"><i><font color="Green">E77</font></i></a>, </i>;<br/>
<a NAME="E46:43"/><b>then </b><i><font color="Green">E161</font></i>: 
 <a href="anproj_1.html#R1">are_Prop</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">w1</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">A2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E48:43"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E5"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="anproj_2.html#E8:43"><i><font color="Green">E70</font></i></a>, <a class="txt" href="anproj_2.html#E11:43"><i><font color="Green">E72</font></i></a>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon">A2'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E50:43"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B2'</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E5"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="anproj_2.html#E9:43"><i><font color="Green">E71</font></i></a>, <a class="txt" href="anproj_2.html#E13:43"><i><font color="Green">E73</font></i></a>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/>
<a NAME="E51:43"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E20:43"><i><font color="Green">E80</font></i></a>, </i>;<br/>
<a NAME="E52:43"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B2'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">A2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E21:43"><i><font color="Green">E81</font></i></a>, </i>;<br/>
<a NAME="E53:43"/><b>then </b><i><font color="Green">E167</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2'</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">B2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2'</font> )
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E22:43"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_1.html#T13">ANPROJ_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">v1'</font> = <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>;<br/>
<a NAME="E54:43"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_7"/>
not <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then </b><font color="Maroon">v1</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="anproj_2.html#E20:43"><i><font color="Green">E80</font></i></a>, <a class="txt" href="anproj_2.html#E23:43"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E3:43_7"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:43"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E22:43"><i><font color="Green">E82</font></i></a>, <a class="txt" href="anproj_2.html#E23:43"><i><font color="Green">E83</font></i></a>, </i>;<br/>
<a NAME="E56:43"/><b>then </b><i><font color="Green">E169</font></i>: 
 <a href="anproj_1.html#R1">are_Prop</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">v1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E57:43"/><i><font color="Green">E170</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="anproj_2.html#E6:43"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<a NAME="E58:43"/><i><font color="Green">E171</font></i>: 
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a>, </i>;<br/>
<a NAME="E59:43"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>,<font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">A1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E61:43"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="anproj_2.html#E5"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="anproj_2.html#E25:43"><i><font color="Green">E86</font></i></a>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon">A1'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B1'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E63:43"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A1'</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B1'</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="anproj_2.html#E5"><i><font color="Green">Lemma67</font></i></a>, , , <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/>
<a NAME="E64:43"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E27:43"><i><font color="Green">E89</font></i></a>, </i>;<br/>
<a NAME="E65:43"/>
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B1'</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B1'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E28:43"><i><font color="Green">E90</font></i></a>, </i>;<br/>
<a NAME="E66:43"/><b>then </b>
( <font color="Maroon">A1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B1'</font> &amp; <font color="Maroon">A1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">B1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font> )
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E29:43"><i><font color="Green">E91</font></i></a>, <a class="ref" href="anproj_1.html#T13">ANPROJ_1:13</a></i>;<br/>
<a NAME="E67:43"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E1:43"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E68:43"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">c3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E1:43"><i><font color="Green">E58</font></i></a>, <a class="txt" href="anproj_2.html#E29:43"><i><font color="Green">E91</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">u1'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>;<br/>
<a NAME="E69:43"/>
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_8"/>
not <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then </b><font color="Maroon">u1</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="anproj_2.html#E27:43"><i><font color="Green">E89</font></i></a>, <a class="txt" href="anproj_2.html#E7"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E3:43_8"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E70:43"/><b>then </b>
<font color="Maroon">B1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E1:43"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E71:43"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="anproj_1.html#R1">are_Prop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<font color="Maroon">u1</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">C2</font> = <font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font>;<br/>
<b>set </b><font color="Maroon">C3</font> =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span>;<br/>
<a NAME="E72:43"/>
( <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c3</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a3</font></span>)</span> )
 
;<br/>
<a NAME="E73:43"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E74:43"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">o</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c3</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="ref" href="anproj_1.html#D3">ANPROJ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E75:43"/>
<font color="Maroon">u1</font>,<font color="Maroon">v1</font>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


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