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

<b>let </b><font color="Maroon">GF</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">GF</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>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="vectsp_4.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/>









<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v1</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="vectsp_4.html#K3" target="_self">+</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="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:65_1"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v1</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:65_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font>
 <b>and </b><br/><a NAME="E5:65_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>
<b>thus </b><a NAME="E6:65_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="vectsp_4.html#D2" target="_self">Def2</a>, , <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>


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

<b>assume </b><a NAME="E2:65"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:65"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v1</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:65"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>


</div>
