<?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">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">u</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="E1:11"/><i><font color="Green">E40</font></i>: 
( <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>  )
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E41</font></i>: 
not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="E4:11"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  implies  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</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="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 
<b>by </b><i/>;<br/>
<b>then 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="E4:11_1"/><i><font color="Green">E45</font></i>: 
<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">u</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">v</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">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:11_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/>
<a NAME="E6:11_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><i><font color="Green">E58</font></i>:  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<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"><font color="Maroon">b</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">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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"><font color="Maroon">b</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">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E3:11_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:11_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or  <a href="real_1.html#K1">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:11_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E3:11_1_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:11_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b</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#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">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E3:11_1_1_3"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:11_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:11_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</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"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <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="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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="E8:11_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</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="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:11"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</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="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> implies <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>given </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><b> such that </b><a NAME="E1:11_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</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">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 ;<br/>

 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<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">v</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">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></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">a</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"><font color="Maroon">b</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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
;<br/>
<b>hence </b><a NAME="E3:11_2"/>
<font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:11"/>
( <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  iff  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</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="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
