<?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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">r</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="E1:6"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 )
 ;<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">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>



<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><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>, <font color="Maroon">c1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:6"/><i><font color="Green">E61</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"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E7:6"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; not <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:6"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">a3</font>, <font color="Olive">b3</font> being  <a href="real_1.html#NM1">Real</a> holds  <span class="p1">(<span class="default"><font color="Olive">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="Olive">b3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</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="Olive">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Olive">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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="Olive">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Olive">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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="Olive">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Olive">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a3</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b3</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<a NAME="E1:6_1"/>
<font color="Maroon">a3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</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">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><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> <font color="Maroon">u1</font></span>)</span> = 
<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">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</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"><span class="p4">(<span class="default"><font color="Maroon">b3</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">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, </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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b3</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">p</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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"><span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</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">c1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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"><span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</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">c1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</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"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</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">c1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</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"><span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">b3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E70</font></i>: 
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> or  not <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  or  not <font color="Maroon">u1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_1"/><i><font color="Green">E72</font></i>: 
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font>
 ;<br/><a NAME="E2:6_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">q</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/><a NAME="E3:6_2_1"/>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Maroon">q</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="E4:6_2_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_2"/>
not <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 ;<br/><a NAME="E2:6_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:6_2_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">q</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/><a NAME="E4:6_2_2"/>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T15">ANPROJ_1:15</a></i>;<br/><b>hence </b><a NAME="E5:6_2_2"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_3"/>
not <font color="Maroon">u1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 ;<br/><a NAME="E2:6_2_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<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><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:6_2_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">q</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/><a NAME="E4:6_2_3"/>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T15">ANPROJ_1:15</a></i>;<br/><b>hence </b><a NAME="E5:6_2_3"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:6_2"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3"/><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">u1</font> &amp; <font color="Maroon">u</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">c</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">a3</font> = 1;<br/><b>set </b><font color="Maroon">b3</font> = 0;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>;<br/><a NAME="E1:6_3_1"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> = 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">u</font> <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/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_3_1_1"/>
<span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E2:6_3_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E4:6_3_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/><a NAME="E5:6_3_1"/>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">1 <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">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/><b>hence </b><a NAME="E6:6_3_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:6_3"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_4"/><i><font color="Green">E83</font></i>: 
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">u</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">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_4_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>set </b><font color="Maroon">a3</font> = 0;<br/><b>set </b><font color="Maroon">b3</font> = 1;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>;<br/><a NAME="E2:6_4_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> = 
<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
<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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_4_1_1"/>
<span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:6_4_1"/>
<span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">0 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><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">0 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">0 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">0 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E5:6_4_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/><a NAME="E6:6_4_1"/>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">0 <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">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/><b>hence </b><a NAME="E7:6_4_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_4_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">a3</font> = 1;<br/><b>set </b><font color="Maroon">b3</font> =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>;<br/><a NAME="E2:6_4_2"/>
<font color="Maroon">c1</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/><a NAME="E3:6_4_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <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="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:6_4_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_4_2_1"/>
not <span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 ;<br/>

<b>then </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38">RLVECT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <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"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T39">RLVECT_1:39</a></i>
;<br/>
<a NAME="E3:6_4_2_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>;<br/>
<a NAME="E4:6_4_2_1"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T31">RLVECT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:6_4_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="anproj_1.html#D2">ANPROJ_1:def 2</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">c</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">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> = 
<font color="Maroon">c</font> <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> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c</font> <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> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
0

;<br/><b>then </b><span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">1 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E7:6_4_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/><a NAME="E8:6_4_2"/>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T11">ANPROJ_1:11</a></i>;<br/><b>hence </b><a NAME="E9:6_4_2"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:6_4"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:6"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, </i>;<br/>


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