<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="clvect_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/>





<b>thus </b><a NAME="E1:90"/>
( <font color="Maroon">v</font> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font> implies <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon">v</font> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:90_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:90_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</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> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:90_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E6:90_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T8" target="_self">Th8</a>, <a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E8:90_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/>
<a NAME="E9:90_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">u</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:90_1"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E11:90_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">2,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="clvect_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E12:90_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#T10" target="_self">Th10</a>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:90"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:90"/><b>then </b>
( <font color="Maroon">v</font> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> &amp; <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 
<b>by </b><i>, <a class="ref" href="clvect_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:90"/>
<font color="Maroon">v</font> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="clvect_1.html#K4" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>


</div>
