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<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">W1</font> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="rmod_2.html#M1">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="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span> ;<br/>
<a NAME="E1:6"/><i><font color="Green">E30</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span>  )
 
<b>by </b><i/>;<br/>
<a NAME="E2:6"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_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="E1:6_1"/>
<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="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">u</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:6_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">u</font>
 <b>and </b><br/><a NAME="E4:6_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 ;<br/>
<b>thus </b><a NAME="E5:6_1"/>
<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="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><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="E1:6_2"/>
<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="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">u</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E36</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">u</font>
 <b>and </b><br/><a NAME="E4:6_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 ;<br/>
<b>thus </b><a NAME="E5:6_2"/>
<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="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:6"/>
<font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="rmod_2.html#T37">RMOD_2:37</a></i>;<br/>


</div>
