<?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">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E39</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:5"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <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><b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E4:5_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E6:5_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#D1">ANALMETR:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:5"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <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><b> such that </b><br/><a NAME="E3:5_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E4:5_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E6:5_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="analmetr.html#D1">ANALMETR:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:5"/>
( <font color="Maroon">x</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">y</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/>;<br/>


</div>
