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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rmod_2.html#M1" target="_self">Submodule</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">v2</font></span>)</span> where <font color="Olive">v2</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:63"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font> implies  ex <font color="Olive">v1</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">v1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
<font color="Maroon">u</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">v2</font></span>)</span> where <font color="Olive">v2</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Olive">v2</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="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:63_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u</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="E5:63_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>
<b>take </b><font color="Maroon">x</font> =  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v1</font>;<br/>

<b>thus </b><a NAME="E6:63_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E7:63_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E8:63_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
 ;<br/>


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

<b>given </b><font color="Maroon">v1</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E2:63"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v1</font> )
 ;<br/>

<a NAME="E3:63"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <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">v1</font></span>)</span> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:63"/><b>then </b>
<font color="Maroon">u</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">v2</font></span>)</span> where <font color="Olive">v2</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>


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