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<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="E1:26"/><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="E2:26"/><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#R3" target="_self">are_perspective</a> 
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E60</font></i>: 
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">o</font>,<font color="Maroon">u2</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">u2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">v2</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">w2</font> )
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green">E61</font></i>: 
( not <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; not <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; not <font color="Maroon">o</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>and </b><br/><a NAME="E5:26"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font>,<font color="Maroon">w1</font> <a href="anproj_2.html#R2" target="_self">lie_on_a_triangle</a>  &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">w2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font>,<font color="Maroon">w1</font> <a href="anproj_2.html#R2" target="_self">lie_on_a_triangle</a>  )
 ;<br/>

<a NAME="E6:26"/><i><font color="Green">E63</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="E7:26"/><i><font color="Green">E69</font></i>: 
( <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">v</font>,<font color="Maroon">v2</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">w</font>,<font color="Maroon">w2</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:26"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">u</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">w</font>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u2</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">w2</font>,<font color="Maroon">v1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">v2</font>,<font color="Maroon">w2</font>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:26"/><i><font color="Green">E71</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">o</font>,<font color="Maroon">v</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T16">ANPROJ_1:16</a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">u</font>,<font color="Maroon">u2</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">v</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">w</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">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="E12:26"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</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">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <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="E14:26"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">b2</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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="E16:26"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">b3</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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>set </b><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">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>;<br/>
<a NAME="E17:26"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w1</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">v1</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u1</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">w1</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">u2</font>,<font color="Maroon">w2</font>,<font color="Maroon">v1</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">v2</font>,<font color="Maroon">w2</font>,<font color="Maroon">u1</font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:26"/><i><font color="Green">E77</font></i>: 
( 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">v</font>,<font color="Maroon">w</font> )
 
<b>by </b><i>, <a class="ref" href="anproj_1.html#T17">ANPROJ_1:17</a></i>;<br/>
<b>then 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="E20:26"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c3</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">d3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <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="E22:26"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c2</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">d2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">c1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E24:26"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E25:26"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w1</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">a1</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">w</font></span>)</span>,<font color="Maroon">v1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">w</font></span>)</span>,<font color="Maroon">u1</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1"/>
(  <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">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:26_1"/>
( <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w1</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">a1</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">w</font></span>)</span>,<font color="Maroon">v1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">w</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_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:26"/><i><font color="Green">E82</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">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">a1</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">w</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">w</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E27:26"/>
( <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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">w</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E28:26"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">o</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w1</font> <a href="anproj_2.html#R1" target="_self">are_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">a1</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">w</font></span>)</span>,<font color="Maroon">v1</font> <a href="anproj_2.html#R1" target="_self">are_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">v</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">w</font></span>)</span>,<font color="Maroon">u1</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">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="E30:26"/><i><font color="Green">E84</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> <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">a1</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">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">v</font></span>)</span></span>)</span></span>)</span>
 <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="E32:26"/><i><font color="Green">E85</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> <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">a1</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">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">w</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , </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="E34:26"/><i><font color="Green">E86</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">v</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E35:26"/><i><font color="Green">E87</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"><span class="p3">(<span class="default"><font color="Maroon">A3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B3</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">A3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</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">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">v</font></span>)</span> &amp; <font color="Maroon">v1</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">A2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B2</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">A2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</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">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">w</font></span>)</span> &amp; <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">v</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">a3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E36:26"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <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">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <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">c2</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">d2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <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">c3</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">d3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E37:26"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">A1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">A2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">A3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">B3</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="anproj_1.html#T13">ANPROJ_1:13</a></i>;<br/>
<a NAME="E38:26"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">B3</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">A3</font> )
 
;<br/>
<a NAME="E39:26"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">A3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E91</font></i>: 
(  not <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  not <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  not <font color="Maroon">A3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:26_2_1"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:26_2_1"/>
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>
<div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2_2"/>
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:26_2_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:26_2_2"/>
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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2_3"/>
<font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:26_2_3"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:26_2_3"/>
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>
<b>hence </b><a NAME="E5:26_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">u1'</font> = <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">v1'</font> = <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">w1'</font> = <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>;<br/>
<a NAME="E40:26"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A3</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E41:26"/><b>then </b><i><font color="Green">E102</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">u1</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">v1</font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w1</font> )
 
<b>by </b><i>, </i>;<br/>
<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"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></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"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</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"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</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"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <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">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</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"><span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></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"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</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">a1</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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</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#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</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#T16">RLVECT_1:16</a></i>
;<br/>
<a NAME="E43:26"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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="E44:26"/>
<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="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


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