<?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">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/>







<a NAME="E1:57"/><i><font color="Green">E32</font></i>: 
( (  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> implies <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> ) &amp; ( <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> implies  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> ) &amp; <font color="Maroon">v</font> <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> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:57"/>
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> implies <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:57"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> iff <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/>


</div>
