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

<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">v1</font></span>)</span> where <font color="Olive">v1</font> is   <a href="vectsp_2.html#NM5">Vector</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>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <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="E1:55"/>
( <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="rmod_2.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> <font color="Maroon">u</font></span>)</span> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>thus </b><a NAME="E2:55_1"/>
<font color="Maroon">v</font> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">x</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="E2:55_1_1"/><b>then </b>
<font color="Maroon">x</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="vectsp_2.html#NM5">Vector</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="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:55_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</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="E5:55_1_1"/><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>
 ;<br/>
<a NAME="E6:55_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="rmod_2.html#E6:55_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rmod_2.html#E8:55_1"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</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="vectsp_1.html#T66">VECTSP_1:66</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="rmod_2.html#E7:55_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>
;<br/>
<a NAME="E8:55_1_1"/><b>then </b>
<font color="Maroon">x</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"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <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> 
 
<b>by </b><i><a class="txt" href="rmod_2.html#E2:55"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E4:55_1"/><b>then </b>
<font color="Maroon">x</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"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <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">v2</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:55_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</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> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>and </b><br/><a NAME="E7:55_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>
<a NAME="E8:55_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="rmod_2.html#E6:55_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rmod_2.html#E8:55_1"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E9:55_1"/>
<font color="Maroon">x</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"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_2.html#E7:55_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E10:55_1"/><b>then </b>
<font color="Maroon">x</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="vectsp_2.html#NM5">Vector</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> 
 
<b>by </b><i><a class="txt" href="rmod_2.html#E2:55"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:55_1"/>
<font color="Maroon">x</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/>


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

<b>assume </b><a NAME="E2:55"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">v</font> <a href="rmod_2.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> <font color="Maroon">u</font></span>)</span> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:55"/>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</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#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/>
<a NAME="E4:55"/><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"><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="vectsp_2.html#NM5">Vector</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/>
<a NAME="E5:55"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:55"/><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"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <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">u1</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:55"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">v</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> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
 <b>and </b><br/><a NAME="E9:55"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>
<a NAME="E10:55"/>
( <font color="Maroon">v</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#K1">0.</a> <font color="Maroon">V</font></span>)</span> &amp; <font color="Maroon">v</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"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/>
<a NAME="E11:55"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T22">RLVECT_1:22</a></i>;<br/>
<a NAME="E12:55"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">u1</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T63">VECTSP_1:63</a></i>;<br/>
<b>hence </b><a NAME="E13:55"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>


</div>
