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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">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>assume </b><a NAME="E1:20"/><i><font color="Green">E40</font></i>: 
(  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> or  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">u</font> or  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> )
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">u</font>,<font color="Olive">v</font> holds <br/><font color="Olive">w</font>,<font color="Olive">u</font>,<font color="Olive">v</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:20_1"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<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><b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/>
<i><font color="Green">E43</font></i>:  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<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="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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>
<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">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="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> <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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E5:20_1"/>
0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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#T23">RLVECT_1:23</a></i>;<br/>
<a NAME="E6:20_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</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">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <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></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="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E7:20_1"/>
<font color="Maroon">w</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">u</font>
 ;<br/><a NAME="E2:20_2"/><b>then </b>
<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/><b>hence </b><a NAME="E3:20_2"/>
<font color="Maroon">w</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_3"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
 ;<br/><a NAME="E2:20_3"/><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>hence </b><a NAME="E3:20_3"/>
<font color="Maroon">w</font>,<font color="Maroon">u</font>,<font color="Maroon">v</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="E5:20"/>
<font color="Maroon">w</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i>, , </i>;<br/>


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