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<b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>





<a NAME="E1:57"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span><br/> for <font color="Olive">x'</font>, <font color="Olive">y'</font>, <font color="Olive">z'</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y'</font>
 
;<br/>
<b>thus </b><a NAME="E2:57"/>
<font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font>

<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


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

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>, <font color="Maroon">z</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font>

<br/>.= 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</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">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>

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

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span> ;<br/><b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span></span>)</span> = 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span><b> such that </b><br/><a NAME="E7:57"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K13" target="_self">.</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">b'</font>
;<br/>

<b>thus </b><a NAME="E9:57"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
